CHAPTER III K_{1} AND K_{2} OF A RING

Let R be an associative ring with unit. In this chapter, we introduce the classical definitions of the groups K_{1}(R) and K_{2}(R). These definitions use only linear algebra and elementary group theory, as applied to the groups GL(R) and E(R). We also define relative groups for K_{1} and K_{2}, as well as the negative K-groups K_{−n}(R) and the Milnor K-groups K_{n}^{M}(R).

In the next chapter we will give another definition: K_{n}(R) = π_{n}K(R) for all n ≥ 0, where K(R) is a certain topological space built using the category P(R) of finitely generated projective R-modules. We will then have to prove that these topologically defined groups agree with the definition of K_{0}(R) in chapter II, as well as with the classical constructions of K_{1}(R) and K_{2}(R) in this chapter.

§1. The Whitehead Group K_{1} of a ring

Let R be an associative ring with unit. Identifying each n × n matrix g with the larger matrix ({g \atop 0} {0 \atop 1}) gives an embedding of GL_{n}(R) into GL_{n+1}(R). The union of the resulting sequence

GL_{1}(R) ⊂ GL_{2}(R) ⊂ ⋯ ⊂ GL_{n}(R) ⊂ GL_{n+1}(R) ⊂ ⋯

is called the infinite general linear group GL(R).

Recall that the commutator subgroup [G, G] of a group G is the subgroup generated by its commutators [g, h] = ghg^{−1}h^{−1}. It is always a normal subgroup of G, and has a universal property: the quotient G/[G, G] is an abelian group, and every homomorphism from G to an abelian group factors through G/[G, G].

Definition 1.1. K_{1}(R) is the abelian group GL(R)/[GL(R), GL(R)].

The universal property of K_{1}(R) is this: every homomorphism from GL(R) to an abelian group must factor through the natural quotient GL(R) → K_{1}(R). Depending upon our situation, we will sometimes think of K_{1}(R) as an additive group, and sometimes as a multiplicative group.

A ring map R → S induces a natural map from GL(R) to GL(S), and hence from K_{1}(R) to K_{1}(S). That is, K_{1} is a functor from rings to abelian groups.

Example 1.1.1 (SK_{1}). If R happens to be commutative, the determinant of a matrix provides a group homomorphism from GL(R) onto the group R^{×} of units of R. It is traditional to write SK_{1}(R) for the kernel of the induced surjection det: K_{1}(R) → R^{×}. The special linear group SL_{n}(R) is the subgroup of GL_{n}(R) consisting of matrices with determinant 1, and SL(R) is their union. Since the

Typeset by AMS-TEX

2 III. K_{1} AND K_{2} OF A RING

natural inclusion of the units R^{×} in GL(R) as GL_{1}(R) is split by the homomorphism det: GL(R) → R^{×}, we see that GL(R) is the semidirect product SL(R) ⋊ R^{×}, and there is a direct sum decomposition: K_{1}(R) = R^{×} ⊕ SK_{1}(R).

Example 1.1.2. If F is a field, then K_{1}(F) = F^{×}. We will see this below (see Lemma 1.2.2 and 1.3.1 below), but it is more fun to deduce this from an 1899 theorem of L. E. J. Dickson, that SL_{n}(F) is the commutator subgroup of both GL_{n}(F) and SL_{n}(F), with only two exceptions: GL_{2}(𝔽_{2}) = SL_{2}(𝔽_{2}) ≅ Σ_{3}, which has order 6, and GL_{2}(𝔽_{3}), which has center {±I} and quotient P GL_{2}(𝔽_{3}) = GL_{2}(𝔽_{3})/{±I} isomorphic to Σ_{4}.

Example 1.1.3. If R is the product R′ × R′′ of two rings, then K_{1}(R) = K_{1}(R′) ⊕ K_{1}(R′′). Indeed, GL(R) is the product GL(R′) × GL(R′′), and the commutator subgroup decomposes accordingly.

Example 1.1.4. For all n, the Morita equivalence between R and S = M_{n}(R) (see II.2.7.2) produces an isomorphism between M_{mn}(R) = End_{R}(R^{m} ⊗ R^{n}) and M_{m}(M_{n}(R)) = End_{S}(S^{m}). It is easy to see that the resulting isomorphism of units GL_{mn}(R) ≅ GL_{m}(M_{n}(R)) is compatible with stablization in m, giving an isomorphism GL(R) ≅ GL(M_{n}(R)). Hence K_{1}(R) ≅ K_{1}(M_{n}(R)).

We will show that the commutator subgroup of GL(R) is the subgroup E(R) generated by “elementary” matrices. These are defined as follows.

Definition 1.2. If i ≠ j are distinct positive integers and r ∈ R then the elementary matrix e_{ij}(r) is the matrix in GL(R) which has 1 in every diagonal spot, has r in the (i, j)-spot, and is zero elsewhere.

E_{n}(R) denotes the subgroup of GL_{n}(R) generated by all elementary matrices e_{ij}(r) with 1 ≤ i, j ≤ n, and the union E(R) of the E_{n}(R) is the subgroup of GL(R) generated by all elementary matrices.

Example 1.2.1. A signed permutation matrix is one which permutes the standard basis {e_{i}} up to sign, i.e., it permutes the set {±e_{1}, …, ±e_{n}}. The following signed permutation matrix belongs to E_{2}(R):

w̄_{12} = e_{12}(1)e_{21}(−1)e_{12}(1) = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) .

By changing the subscripts, we see that the signed permutation matrices w̄_{ij} belong to E_{n}(R) for n ≥ i, j. Since the products w̄_{jk}w̄_{ij} correspond to cyclic permutations of 3 basis elements, every matrix corresponding to an even permutation of basis elements belongs to E_{n}(R). Moreover, if g ∈ GL_{n}(R) then we see by Ex. I.1.11 that E_{2n}(R) contains the matrix ({g \atop 0} {0 \atop g^{−1}} ).

1.2.2. If we interpret matrices as linear operators on column vectors, then e_{ij}(r) is the elementary row operation of adding r times row j to row i, and E_{n}(R) is the subset of all matrices in GL_{n}(R) which may be reduced to the identity matrix using only these row operations. The quotient set GL_{n}(R)/E_{n}(R) measures the obstruction to such a reduction.

If F is a field this obstruction is F^{×}, and is measured by the determinant. That is, E_{n}(F) = SL_{n}(F) for all n ≥ 1. Indeed, standard linear algebra shows that every matrix of determinant 1 is a product of elementary matrices.

Remark 1.2.3 (Surjections). If I is an ideal of R, each homomorphism E_{n}(R) → E_{n}(R/I) is onto, because the generators e_{ij}(r) of E_{n}(R) map onto the generators e_{ij}(r̄) of E_{n}(R/I). In contrast, the maps GL_{n}(R) → GL_{n}(R/I) are usually not onto unless I is a radical ideal (Ex. I.1.12(iv)). Indeed, the obstruction is measured by the group K_{0}(I) = K_{0}(R, I); see Proposition 2.3 below.

Division rings 1.2.4. The same linear algebra that we invoked for fields shows that if D is a division ring (a “skew field”) then every invertible matrix may be reduced to a diagonal matrix diag(r, 1, ..., 1), and that E_{n}(D) is a normal subgroup of GL_{n}(D). Thus each GL_{n}(D)/E_{n}(D) is a quotient group of the nonabelian group D^{×}. Dieudonné proved in 1943 that in fact GL_{n}(D)/E_{n}(D) = D^{×}/[D^{×}, D^{×}] for all n > 1 (except for n = 2 when D = 𝔽_{2}). In particular, K_{1}(D) = GL_{n}(D)/E_{n}(D) for all n ≥ 3. A proof of this result is sketched in Exercise 1.2 below.

If D is a d-dimensional division algebra over its center F (which must be a field), then d = n^{2} for some integer n, and n is called the Schur index of D. Indeed, there are (many) field extensions E of F such that D⊗_{F} E ≅ M_{n}(E); such a field is called a splitting field for D. For example, any maximal subfield E ⊂ D has [E : F] = n and is a splitting field.

For any splitting field E, the inclusion of D in M_{n}(E), and M_{r}(D) in M_{nr}(E), induces maps D^{×} ⊂ GL_{n}(E) ⟶^{det} E^{×} and GL_{r}(D) → GL_{nr}(E) ⟶^{det} E^{×} whose image lies in the subgroup F^{×} of E^{×}. (If E/F is Galois, the image is fixed by the Galois group Gal(E/F) and hence lies in F^{×}.) The induced maps D^{×} → F^{×} and GL_{r}(D) → F^{×} are called the reduced norms N_{red} for D, and are independent of E. For example, if D = ℍ is the quaternions then F = ℝ, and N_{red}(t + ix + jy + kz) = t^{2} + x^{2} + y^{2} + z^{2}. It is easy to check here that N_{red} induces K_{1}(ℍ) ≅ ℝ_{+}^{×} ⊂ ℝ^{×}.

Now if A is any central simple F -algebra then A ≅ M_{r}(D) for some D, and M_{m}(A) ≅ M_{mr}(D). The induced maps N_{red} : GL_{m}(A) ≅ GL_{mr}(D) → F^{×} are sometimes called the reduced norm for A, and the kernel of this map is written as SL_{m}(A). We define SK_{1}(A) to be the kernel of the induced map

N_{red}: K_{1}(A) ≅ K_{1}(D) → K_{1}(F) = F^{×}.

In 1950 S. Wang showed that SK_{1}(D) = 1 if F is a number field. For every real embedding σ : F ↪ ℝ, D ⊗_{F} ℝ is a matrix algebra over ℝ, ℂ or ℍ; it is called unramified in case ℍ occurs. The Hasse-Schilling-Maass norm theorem describes the image of the reduced norm, and hence K_{1}(D):

K_{1}(D) ⟶_{N_{red}}^{≃} {x ∈ F^{×} : σ(x) > 0 in ℝ for all ramified σ}.

Wang also showed that SK_{1}(D) = 1 if the Schur index of D is squarefree. In 1976 V. Platanov produced the first examples of a D with SK_{1}(D) ≠ 1, by constructing a map from SK_{1}(D) to a subquotient of the Brauer group Br(F). We will see in 1.7.2 below that the group SK_{1}(D) has exponent n.

Remark 1.2.5. There is no a priori reason to believe that the subgroups E_{n}(R) are normal, except in special cases. For example, we shall show in Ex. 1.3 that if R has stable range d+1 then E_{n}(R) is a normal subgroup of GL_{n}(R) for all n ≥ d+2. Vaserstein proved [V69] that K_{1}(R) = GL_{n}(R)/E_{n}(R) for all n ≥ d + 2.

If R is commutative, we can do better: E_{n}(R) is a normal subgroup of GL_{n}(R) for all n ≥ 3. This theorem was proven by A. Suslin in [Su77]; we give Suslin’s proof in Ex. 1.9. Suslin also gave examples of Dedekind domains for which E_{2}(R) is not normal in GL_{2}(R) in [Su76]. For noncommutative rings, the E_{n}(R) are only known to be normal for large n, and only then when the ring R has finite stable range in the sense of Ex. I.1.5; see Ex. 1.3 below.

Commutators 1.3. Here are some easy-to-check formulas for multiplying elementary matrices. Fixing the indices, we have e_{ij}(r)e_{ij}(s) = e_{ij}(r +s), and e_{ij}(−r) is the inverse of e_{ij}(r). The commutator of two elementary matrices is easy to compute and simple to describe (unless j = k and i = ℓ):

(1.3.1)  [e_{ij}(r), e_{kℓ}(s)] = \{ \begin{matrix}1 & if j ≠ k and i ≠ ℓ \\ e_{iℓ}(rs) & if j = k and i ≠ ℓ \\ e_{kj}(−sr) & if j ≠ k and i = ℓ. \end{matrix}

Recall that a group is called perfect if G = [G, G]. If a subgroup H of G is perfect, then H ⊆ [G, G]. The group E(R) is perfect, as are most of its finite versions:

Lemma 1.3.2. If n ≥ 3 then E_{n}(R) is a perfect group.

Proof. If i, j, k are distinct then e_{ij}(r) = [e_{ik}(r), e_{kj}(1)].

We know from Example 1.1.2 that E_{2}(R) is not always perfect; in fact E_{2}(𝔽_{2}) and E_{2}(𝔽_{3}) are solvable groups.

Rather than become enmeshed in technical issues, it is useful to “stabilize” by increasing the size of the matrices we consider. One technical benefit of stability is given in Ex. 1.4. The following stability result was proven by J.H.C. Whitehead in the 1950 paper [Wh50], and in some sense is the origin of K-theory.

Whitehead’s Lemma 1.3.3. E(R) is the commutator subgroup of GL(R). Hence K_{1}(R) = GL(R)/E(R).

Proof. The commutator subgroup contains E(R) by Lemma 1.3.2. Conversely, every commutator in GL_{n}(R) can be expressed as a product in GL_{2n}(R):

(1.3.4)  [g, h] = ( \begin{matrix}g & 0 \\ 0 & g^{−1} \end{matrix} ) ( \begin{matrix}h & 0 \\ 0 & h^{−1} \end{matrix} ) ( \begin{matrix}(hg)^{−1} & 0 \\ 0 & hg \end{matrix} ) .

But we saw in Example 1.2.1 that each of these terms is in E_{2n}(R).

Example 1.3.5. If F is a field then K_{1}(F) = F^{×}, because we have already seen that E(R) = SL(R). Similarly, if R is a Euclidean domain such as ℤ or F [t] then it is easy to show that SK_{1}(R) = 0 and hence K_{1}(R) = R^{×}; see Ex. 1.5. In particular, K_{1}(ℤ) = ℤ^{×} = {±1} and K_{1}(F [t]) = F^{×}.

To get a feeling for the non-commutative situation, suppose that D is a division ring. Diedonné’s calculation of GL_{n}(D)/E_{n}(D) (described in 1.2.4 and Ex. 1.2) gives an isomorphism K_{1}(D) ≅ D^{×}/[D^{×}, D^{×}].

Example 1.3.6. If F is a finite field extension of ℚ (a number field) and R is an integrally closed subring of F, then Bass, Milnor and Serre proved in [BMS, 4.3] that SK_{1}(R) = 0, so that K_{1}(R) ≅ R^{×}. We mention that if R is finitely generated over ℤ then, by the Dirichlet Unit Theorem, K_{1}(R) = R^{×} is a finitely generated abelian group isomorphic to μ(F) ⊕ ℤ^{s−1}, where μ(F) denotes the cyclic group of all roots of unity in F and s is the number of “places at infinity” for R.

Example 1.3.7. (Vaserstein) If r, s ∈ R are such that 1 + rs is a unit, then so is 1 + sr because (1 + sr)(1 − s(1 + rs)^{−1}r) = 1. The subgroup W (R) of R^{×} generated by the (1 + rs)(1 + sr)^{−1} belongs to E_{2}(R) by Ex. 1.1. For R = M_{2}(𝔽_{2}), W (R) = R^{×} ≅ Σ_{3} and K_{1}(R) = 0 but R^{×} ≠ [R^{×}, R^{×}]. The units of the subring T of upper triangular matrices in M_{2}(𝔽_{2}) is even abelian (T^{×} ≅ ℤ/2), but K_{1}(T) = 0 since ({1 \atop 0} {1 \atop 1}) = (1 + rs)(1 + sr)^{−1} for r = ({1 \atop 0} {0 \atop 0}) and s = ({0 \atop 0} {1 \atop 0}).

Vaserstein has shown [V04] that W (R) = [R^{×}, R^{×}] if Λ = R/rad(R) is a product of matrix rings, none of which is M_{2}(𝔽_{2}), and at most one of the factors in Λ is 𝔽_{2}. In particular, W (R) = [R^{×}, R^{×}] for every local ring R. (See Ex. 1.1.)

Lemma 1.4. If R is a semilocal ring then the natural inclusion of R^{×} = GL_{1}(R) into GL(R) induces an isomorphism K_{1}(R) ≅ R^{×}/W (R),

If R is a commutative semilocal ring, then

SK_{1}(R) = 0 and K_{1}(R) = R^{×}.

Proof. By Example 1.1.1 (1.3.7 and Ex. 1.2 in the noncommutative case), it suffices to prove that R^{×} maps onto K_{1}(R). This will follow by induction on n once we show that GL_{n}(R) = E_{n}(R)GL_{n−1}(R). Let J denote the Jacobson radical of R, so that R/J is a finite product of matrix algebras over division rings. By examples 1.1.3, 1.1.4 and 1.2.4, (R/J)^{×} maps onto K_{1}(R/J); in fact by Exercise 1.3 we know that every ḡ ∈ GL_{n}(R/J) is a product ēḡ_{1}, where ē ∈ E_{n}(R/J) and ḡ_{1} ∈ GL_{1}(R/J).

Given g ∈ GL_{n}(R), its reduction ḡ in GL_{n}(R/J) may be decomposed as above: ḡ = ēḡ_{1}. By Remark 1.2.3, we can lift ē to an element e ∈ E_{n}(R). The matrix e^{−1}g is congruent to the diagonal matrix ḡ_{1} modulo J, so its diagonal entries are all units and its off-diagonal entries lie in J. Using elementary row operations e_{ij}(r) with r ∈ J, it is an easy matter to reduce e^{−1}g to a diagonal matrix, say to D = diag(r_{1}, ..., r_{n}). By Ex. I.1.11, the matrix diag(1, ..., 1, r_{n}, r_{n}^{−1}) is in E_{n}(R). Multiplying D by this matrix yields a matrix in GL_{n−1}(R), finishing the induction and the proof.

Commutative Banach Algebras

Let R be a commutative Banach algebra over the real or complex numbers. For example, R could be the ring ℝ^{X} of continuous real-valued functions of a compact space X. As subspaces of the metric space of n × n matrices over R, the groups SL_{n}(R) and GL_{n}(R) are topological groups.

Proposition 1.5. E_{n}(R) is the path component of the identity matrix in the special linear group SL_{n}(R), n ≥ 2. Hence we may identify the group SK_{1}(R) with the group π_{0}SL(R) of path components of the topological space SL(R).

Proof. To see that E_{n}(R) is path-connected, fix an element g = ∏ e_{i_{α}j_{α}}(r_{α}). The formula t ↦ ∏ e_{i_{α}j_{α}}(r_{α}t), 0 ≤ t ≤ 1 defines a path in E_{n}(R) from the identity to g. To prove that E_{n}(R) is open subset of SL_{n}(R) (and hence a path-component), it suffices to prove that E_{n}(R) contains U_{n−1}, the set of matrices 1+(r_{ij}) in SL_{n}(R) with ||r_{ij}|| < \frac{1}{n−1} for all i, j. We will actually show that each matrix in U_{n−1} can be expressed naturally as a product of n^{2} + 5n − 6 elementary matrices, each of which depends continuously upon the entries r_{ij} ∈ R.

Set u = 1 + r_{11}. Since \frac{n−2}{n−1} < ||u||, u has an inverse v with ||v|| < \frac{n−1}{n−2}. Subtracting vr_{1j} times the first column from the j^{th} we obtain a matrix 1 + r′_{ij} whose first row is (u, 0, ..., 0) and

||r′_{ij}|| < \frac{1}{n − 1} + \frac{n − 1}{n − 2}(\frac{1}{n − 1})^{2} = \frac{1}{n − 2}.

We can continue to clear out entries in this way so that after n(n − 1) elementary operations we have reduced the matrix to diagonal form.

By Ex. I.1.10, any diagonal matrix ({u \atop 0} {0 \atop u^{−1}} ) is the product of 6 elementary matrices. By induction, it follows that any diagonal n × n matrix of determinant 1 can be written naturally as a product of 6(n − 1) elementary matrices.

Let V denote the path component of 1 in the topological group R^{×}, i.e., the kernel of R^{×} → π_{0}R^{×}. By Ex. 1.12, V is a quotient of the additive group R.

Corollary 1.5.1. If R is a commutative Banach algebra, there is a natural surjection from K_{1}(R) onto π_{0}GL(R) = π_{0}(R^{×}) × π_{0}SL(R). The kernel of this map is the divisible subgroup V of R^{×}.

Example 1.5.2. If R = ℝ then K_{1}(ℝ) = ℝ^{×} maps onto π_{0}GL(ℝ) = {±1}, and the kernel is the uniquely divisible multiplicative group V = (0, ∞). If R = ℂ then V = ℂ^{×}, because K_{1}(ℂ) = ℂ^{×} but π_{0}GL(ℂ) = 0.

Example 1.5.3. Let X be a compact space with a nondegenerate basepoint. Then SK_{1}(ℝ^{X}) is the group π_{0}SL(ℝ^{X}) = [X, SL(ℝ)] = [X, SO] of homotopy classes of maps from X to the infinite special orthogonal group SO. By Ex. II.3.11 we have π_{0}GL(ℝ^{X}) = [X, O] = KO^{−1}(X), and there is a short exact sequence

0 → ℝ^{X} ⟶^{exp} K_{1}(ℝ^{X}) → KO^{−1}(X) → 0.

Similarly, SK_{1}(ℂ^{X}) is the group π_{0}SL(ℂ^{X}) = [X, SL(ℂ)] = [X, SU] of homotopy classes of maps from X to the infinite special unitary group SU. Since π_{0}GL(ℂ^{X}) = [X, U] = KU^{−1}(X) by II.3.5.1 and Ex. II.3.11, there is a natural surjection from K_{1}(ℂ^{X}) onto KU^{−1}(X), and the kernel V is the divisible group of all contractible maps X → ℂ^{×}.

Example 1.5.4. When X is the circle S^{1} we have SK_{1}(ℝ^{S^{1}}) = [S^{1}, SO] = π_{1}SO = ℤ/2. On the other hand, we have π_{0}SL_{2}(ℝ^{S^{1}}) = π_{1}SL_{2}(ℝ) = π_{1}SO_{2} = ℤ, generated by the matrix A = ({cos θ \atop − sin θ} {sin θ \atop cos θ} ). Since π_{1}SO_{2}(ℝ) → π_{1}SO is onto, the matrix A represents the nonzero element of SK_{1}(ℝ^{S^{1}}).

The ring R = ℝ[x, y]/(x^{2} + y^{2} − 1) may be embedded in the ring ℝ^{S^{1}} by x ↦ cos(θ), y ↦ sin(θ). Since the matrix ({x \atop −y} {y \atop x}) maps to A, it represents a nontrivial element of SK_{1}(R). In fact it is not difficult to show that SK_{1}(R) ≅ ℤ/2 using Mennicke symbols (Ex. 1.10).

K_{1} and projective modules

Now let P be a finitely generated projective R-module. Choosing an isomorphism P ⊕ Q ≅ R^{n} gives a group homomorphism from Aut(P) to GL_{n}(R). (Send α to α ⊕ 1_{Q}.)

Lemma 1.6. The homomorphism from Aut(P) to GL(R) = ⋃ GL_{n}(R) is welldefined up to inner automorphism of GL(R). Hence there is a well-defined homomorphism Aut(P) → K_{1}(R).

Proof. First suppose that Q and n are fixed. Two different isomorphisms between P ⊕ Q and R^{n} must differ by an automorphism of R^{n}, i.e., by an element g ∈ GL_{n}(R). Thus if α ∈ Aut(P) maps to the matrices A and B, respectively, we must have A = gBg^{−1}. Next we observe that there is no harm in stabilizing, i.e., replacing Q by Q ⊕ R^{m} and P ⊕ Q ≅ R^{n} by P ⊕ (Q ⊕ R^{m}) ≅ R^{n+m}. This is because GL_{n}(R) → GL(R) factors through GL_{n+m}(R). Finally, suppose given a second isomorphism P ⊕ Q′ ≅ R^{m}. Since Q ⊕ R^{m} ≅ R^{n} ⊕ Q′, we may stabilize both Q and Q′ to make them isomorphic, and invoke the above argument.

Corollary 1.6.1. If R and S are rings, there is a natural external product operation K_{0}(R) ⊗ K_{1}(S) → K_{1}(R ⊗ S).

If R is commutative and S is an R-algebra, there is a natural product operation K_{0}(R) ⊗ K_{1}(S) → K_{1}(S), making K_{1}(S) into a module over the ring K_{0}(R).

Proof. For each finitely generated projective R-module P and each m, Lemma 1.6 provides a homomorphism Aut(P ⊗ S^{m}) → K_{1}(R ⊗ S). For each β ∈ GL_{m}(S), let [P] · β denote the image of the automorphism 1_{P} ⊗ β of P ⊗ S^{m} under this map. Fixing β and m, the isomorphism (P ⊕P′)⊗S^{m} ≅ (P ⊗S^{m})⊕(P′ ⊗S^{m}) yields the identity [P ⊕P′]·β = [P]·β +[P′]·β in K_{1}(R⊗S). Hence P ↦ [P]·β is an additive function of P ∈ P(R), so (by definition) it factors through K_{0}(R). Now fix P; the map GL_{m}(S) → K_{1}(R ⊗ S) given by β ↦ [P] · β is compatible with stabilization in m. Thus it factors through a map GL(S) → K_{1}(R ⊗ S), and through a map K_{1}(S) → K_{1}(R ⊗ S). This shows that the product is well-defined and bilinear.

When R is commutative, K_{0}(R) is a ring by II, §2. If S is an R-algebra, there is a ring map R ⊗ S → S. Composing the external product with K_{1}(R ⊗ S) → K_{1}(S) yields a natural product operation K_{0}(R) ⊗ K_{1}(S) → K_{1}(S). The verification that [P ⊗_{R} Q] · β = [P] · ([Q] · β) is routine.

Here is a homological interpretation of K_{1}(R). Recall that the first homology H_{1}(G; ℤ) of any group G is naturally isomorphic to G/[G, G]. (See [WHomo, 6.1.11] for a proof.) For G = GL(R) this yields

(1.6.2)  K_{1}(R) = H_{1}(GL(R); ℤ) = lim_{n→∞} H_{1}(GL_{n}(R); ℤ).

By Lemma 1.6, we also have well-defined compositions

H_{1}(Aut(P); ℤ) → H_{1}(GL_{n}(R); ℤ) → K_{1}(R),

which are independent of the choice of isomorphism P ⊕ Q ≅ R^{n}.

Here is another description of K_{1}(R) in terms of the category P(R) of finitely generated projective R-modules. Consider the translation category tP of P(R): its objects are isomorphism classes of finitely generated projective modules, and the morphisms between P and P′ are the isomorphism classes of Q such that P ⊕ Q ≅ P′. This is a filtering category [WHomo, 2.6.13], and P ↦ H_{1}(Aut(P); ℤ) is a well-defined functor from tP to abelian groups. Hence we can take the filtered direct limit of this functor. Since the free modules are cofinal in tP, we see from (1.6.2) that we have

Corollary 1.6.3 (Bass). K_{1}(R) ≅ \underrightarrow{lim}_{P ∈tP} H_{1}(Aut(P); ℤ).

Recall from II.2.7 that if two rings R and S are Morita equivalent then the categories P(R) and P(S) are equivalent. By Corollary 1.6.3 we have the following:

Proposition 1.6.4 (Morita invariance of K_{1}). The group K_{1}(R) depends only upon the category P(R). That is, if R and S are Morita equivalent rings then K_{1}(R) ≅ K_{1}(S). In particular, the isomorphism of 1.1.4 arises in this way:

K_{1}(R) ≅ K_{1}(M_{n}(R)).

Transfer maps

Let f: R → S be a ring homomorphism. We will see later on that a transfer homomorphism f_{∗}: K_{1}(S) → K_{1}(R) is defined whenever S has a finite R-module resolution by finitely generated projective R-modules. This construction requires a definition of K_{1} for an exact category such as H(R), and is analogous to the transfer map in II(7.9.1) for K_{0}. Without this machinery, we can still construct the transfer map when S is finitely generated projective as an R-module, using the forgetful functor P(S) → P(R); this is the analogue of the method used for the K_{0} transfer map in example II.2.8.1.

Lemma 1.7. Any additive functor T : P(S) → P(R) induces a natural homomorphism K_{1}(T): K_{1}(S) → K_{1}(R), and T_{1} ⊕ T_{2} induces the sum K_{1}(T_{1}) + K_{1}(T_{2}).

Proof. The functor T induces an evident functor tP(S) → tP(R). If P is a finitely generated projective S-module, T also induces a homomorphism Aut_{S}(P) → Aut_{R}(T P) and hence H_{1}(Aut_{S}(P); ℤ) → H_{1}(Aut_{R}(T P); ℤ). As P varies, these assemble to give a natural transformation of functors from the translation category tP(S) to abelian groups. Since K_{1}(S) = \underrightarrow{lim}_{P ∈P(S)} H_{1}(Aut_{S}(P); ℤ) by Corollary 1.6.3, taking the direct limit over tP(S) yields the desired map

K_{1}(S) → \underrightarrow{lim}_{P ∈P(S)} H_{1}(Aut_{R}(P); ℤ) → \underrightarrow{lim}_{Q∈P(R)} H_{1}(Aut_{R}(Q); ℤ) = K_{1}(R).

Corollary 1.7.1. Suppose that S is finitely generated projective as an Rmodule. Then the forgetful functor P(S) → P(R) induces a natural transfer homomorphism f_{∗}: K_{1}(S) → K_{1}(R). If R is commutative, the composite

K_{1}(R) ⟶^{f^{∗}} K_{1}(S) ⟶^{f_{∗}} K_{1}(R)

is multiplication by [S] ∈ K_{0}(R).

Proof. When T is the forgetful map, so that K_{1}(S) → K_{1}(R) is the transfer map f_{∗}, we compute the composite f_{∗}f^{∗}, by computing its effect upon an element α ∈ GL_{n}(R). The matrix f^{∗}(α) = 1_{S} ⊗_{R} α lies in GL_{n}(S). To apply f_{∗} we consider 1_{S} ⊗_{R} α as an element of the group Aut_{R}(S^{n}) = Aut_{R}(S ⊗_{R} R^{n}), which we then map into GL(R). But this is just the product [S] · α of 1.6.1.

When j : F → E is a finite field extension, it is easy to see from 1.1.2 that the transfer map j_{∗} : E^{×} → F^{×} is the classical norm map. For this reason, the transfer map is sometimes called the norm map.

Example 1.7.2. Let D be a division algebra of dimension d = n^{2} over its center F, and recall from 1.2.4 that SK_{1}(D) is the kernel of the reduced norm N_{red}. We will show that SK_{1}(D) has exponent n by showing that i^{∗}N_{red} : K_{1}(D) → K_{1}(D) is multiplication by n.

To see this, choose a maximal subfield E with inclusions F ⟶^{j} E ⟶^{σ} D. By the definition of N_{red}, composing it with j^{∗} : F^{×} ⊂ E^{×} yields the transfer map σ_{∗} : K_{1}(D) → K_{1}(E). Therefore, i^{∗}N_{red} = σ^{∗}j^{∗}N_{red} = σ^{∗}σ_{∗}. Hence it suffices to show that σ^{∗}σ_{∗} : K_{1}(D) → K_{1}(D) is multiplication by n. By 1.7, σ^{∗}σ_{∗} is induced by the additive self-map T : M ↦ M ⊗_{D} (D ⊗_{E} D) of P(D). Since D ⊗_{E} D ≅ D^{n} as a D-bimodule, T (M) ≅ M^{n} and the assertion follows from 1.7.

The transfer map i_{∗} : K_{1}(D) → K_{1}(F) associated to i : F ⊂ D is induced from the classical norm map N_{D/F} : D^{×} ⊂ GL_{d}(F) → F^{×}. In fact, the norm map is n times the reduced norm N_{red} : D^{×} → F^{×} of 1.2.4; see Ex. 1.16 below. Moreover, the composition i^{∗}i_{∗} : K_{1}(D) → K_{1}(D) is multiplication by d since it corresponds to the additive self-map M ↦ M ⊗_{D} (D ⊗_{F} D) of P(D), and D ⊗_{F} D ≅ D^{d} as a D-bimodule (see II.2.8.1).

Corollary 1.7.3. K_{1}(R) = 0 for every flasque ring R.

Proof. Recall from II.2.1.3 that a ring R is flasque if there is an additive selffunctor T (P ↦ P ⊗_{R} M) on P(R) together with a natural transformation θ_{P} : P ⊕T (P) ≅ T (P). By 1.7, the induced self-map on K_{1}(R) satisfies x+T (x) = T (x) (and hence x = 0) for all x ∈ K_{1}(R).

Here is an application of 1.7 that anticipates the higher K-theory groups with coefficients in chapter IV.

Definition 1.7.4. For each natural number m, we define K_{1}(R; ℤ/m) to be the relative group K_{0}(·m) of II.2.10, where ·m is the endo-functor of P(R) sending P to P^{m} = P ⊗_{R} R^{m}. Since the P^{m} are cofinal, we see by Ex. II.2.15 and Ex. 1.14, that it fits into a universal coefficient sequence:

K_{1}(R) ⟶^{m} K_{1}(R) → K_{1}(R; ℤ/m) → K_{0}(R) ⟶^{m} K_{0}(R).

Example 1.8 (Whitehead group W h_{1}). If R is the group ring ℤ[G] of a group G, the (first) Whitehead group W h_{1}(G) is the quotient of K_{1}(ℤ[G]) by the subgroup generated by ±1 and the elements of G, considered as elements of GL_{1}. If G is abelian, then ℤ[G] is a commutative ring and ±G is a subgroup of K_{1}(ℤ[G]), so by 1.3.4 we have W h_{1}(G) = (ℤ[G]^{×})/±G⊕SK_{1}(ℤ[G]). If G is finite then W h_{1}(G) is a finitely generated group whose rank is r − q, where r and q are the number of simple factors in ℝ[G] and ℚ[G], respectively. This and other calculations related to W h_{1}(G) may be found in R. Oliver’s excellent sourcebook [Oliver].

The group W h_{1}(G) arose in Whitehead’s 1950 study [Wh50] of simple homotopy types. Two finite CW complexes have the same simple homotopy type if they are connected by a finite sequence of “elementary expansions and collapses.” Given a homotopy equivalence f: K → L of complexes with fundamental group G, the torsion of f is an element τ(f) ∈ W h_{1}(G). Whitehead proved that τ(f) = 0 if and only if f is a simple homotopy equivalence, and that every element of W h_{1}(G) is the torsion of some f. An excellent source for the geometry behind this is [Cohen].

Example 1.9 (The s-Cobordism Theorem). Here is another area of geometric topology in which Whitehead torsion has played a crucial role, piecewiselinear (“PL”) topology. We say that a triple (W, M, M′) of compact PL manifolds is an h-cobordism if the boundary of W is the disjoint union of M and M′, and both inclusions M ⊂ W, M′ ⊂ W are homotopy equivalences. In this case we can define the torsion τ of M ⊂ W, as an element of W h_{1}(G), G = π_{1}M. The s-cobordism theorem states that if M is fixed with dim(M) ≥ 5 then (W, M, M′) ≅ (M × [0, 1], M × 0, M × 1) if and only if τ = 0. Moreover, every element of W h_{1}(G) arises as the torsion of some h-cobordism (W, M, M′).

Here is an application. Suppose given an h-cobordism (W, M, M′), and let N be the union of W, the cone on M and the cone on M′. Then N is PL homeomorphic to the suspension ΣM of M if and only if (W, M, M′) ≅ (M × [0, 1], M × 0, M × 1) if and only if τ = 0.

This gives a counterexample to the “Hauptvermutung” that two homeomorphic complexes would be PL homeomorphic. Indeed, if (W, M, M′) is an h-cobordism with nonzero torsion, then N and ΣM cannot be PL homeomorphic, yet the theory of “engulfing” implies that they must be homeomorphic manifolds.

Another application, due to Smale, is the Generalized Poincaré Conjecture. Let N be an n-dimensional PL manifold of the homotopy type of the sphere S^{n}, n ≥ 5. Then N is PL homeomorphic to S^{n}. To see this, let W be obtained by removing two small disjoint n-discs D_{1}, D_{2} from N. The boundary of these discs is the boundary of W, and (W, S^{n−1}, S^{n−1}) is an h-cobordism. Its torsion must be zero since π_{1}(S^{n−1}) = 0 and W h_{1}(0) = 0. Hence W is S^{n−1} × [0, 1], and this implies that N = W ∪ D_{1} ∪ D_{2} is S^{n}.

EXERCISES 1.1 If r, s, t ∈ R are such that (1 + rs)t = 1, show that (1 + rs)(1 + sr)^{−1} ∈ E_{2}(R). Hint: Start by calculating e_{12}(r + rsr)e_{21}(st + s)e_{12}(−r)e_{21}(−s).

If r is a unit of R, or if r, s ∈ rad(R), show that (1 + rs)(1 + sr)^{−1} ∈ [R^{×}, R^{×}]. Conclude that if R is a local ring then W (R) = [R^{×}, R^{×}]. Hint: If r, s ∈ rad(R), then t = 1 + s − sr is a unit; compute [t^{−1} + r, t] and (1 + rs)(1 + r). 1.2 Semilocal rings. Let R be a noncommutative semilocal ring (Ex. II.2.6). Show that there exists a unique “determinant” map from GL_{n}(R) onto the abelian group R^{×}/W (R) of Lemma 1.4 with the following properties: (i) det(e) = 1 for every elementary matrix e, and (ii) If ρ = diag(r, 1, ..., 1) and g ∈ GL_{n}(R) then det(ρ·g) = r · det(g). Then show that det is a group homomorphism: det(gh) = det(g) det(h). Conclude that K_{1}(R) ≅ R^{×}/W (R). 1.3 Suppose that a ring R has stable range sr(R) = d + 1 in the sense of Ex. I.1.5. (For example, R could be a d-dimensional commutative noetherian ring.) This condition describes the action of E_{d+2}(R) on unimodular rows in R^{d+2}.

(a) Show that GL_{n}(R) = GL_{d+1}(R)E_{n}(R) for all n > d + 1, and deduce that GL_{d+1}(R) maps onto K_{1}(R).

(b) Show that E_{n}(R) is a normal subgroup of GL_{n}(R) for all n ≥ d + 2. Hint: Conjugate e_{nj}(r) by g ∈ GL_{d+2}(R).

1.4 Let R be the polynomial ring F [x, y] over a field F. P.M. Cohn proved that the

matrix g = ( \begin{matrix}1 + xy & x^{2} \\ −y^{2} & 1 − xy \end{matrix} ) is not in E_{2}(R). Show that g ∈ E_{3}(R) ∩ GL_{2}(R).

1.5 Let R be a Euclidean domain, such as ℤ or the polynomial ring F [t] over a field. Show that E_{n}(R) = SL_{n}(R) for all n, and hence that SK_{1}(R) = 0.

1.6 Here is another interpretation of the group law for K_{1}. For each m, n, let ⊕_{mn} denote the group homomorphism GL_{m}(R) × GL_{n}(R) → GL_{m+n}(R) sending (α, β) to the block diagonal matrix \binom{α 0}{0 β}. Show that in K_{1}(R) we have [α⊕_{mn}β] = [α][β].

1.7 Let E = End_{R}(R^{∞}) be the ring of infinite row-finite matrices over R of Ex. I.1.7. Show that K_{1}(E) = 0. Hint: If α ∈ GL_{n}(E), form the block diagonal matrix α^{∞} = diag(α, α, …) in Aut(V) ≅ GL(E), where V is an infinite sum of copies of (R^{∞})^{n}, and show that α ⊕ α^{∞} is conjugate to α^{∞}.

1.8 In this exercise we show that the center of E(R) is trivial. First show that any matrix in GL_{n}(R) commuting with E_{n}(R) must be a diagonal matrix diag(r, ..., r) with r in the center of R. Conclude that no element in E_{n−1}(R) is in the center of E_{n}(R), and pass to the limit as n → ∞.

1.9 In this exercise we suppose that R is a commutative ring, and give Suslin’s proof that E_{n}(R) is a normal subgroup of GL_{n}(R) when n ≥ 3. Let v = ∑_{i=1}^{n} v_{i}e_{i} be a column vector, and let u, w be row vectors such that u · v = 1 and w · v = 0.

(a) Show that w = ∑_{i<j} r_{ij}(v_{j}e_{i} − v_{i}e_{j}), where r_{ij} = w_{i}u_{j} − w_{j}u_{i}.

(b) Conclude that the matrix I_{n} + (v · w) is in E_{n}(R) if n ≥ 3.

(c) If g ∈ GL_{n}(R) and i < j, let v be the i^{th} column of g and w the j^{th} row of g^{−1}, so that w · v = 0. Show that ge_{ij}(r)g^{−1} = I_{n} + (v · rw) for all r ∈ R. By (b), this proves that E_{n}(R) is normal.

1.10 Mennicke symbols. Let (r, s) be a unimodular row over a commutative ring R. We define the Mennicke symbol [{s \atop r}] to be the class in SK_{1}(R) of the matrix ({r \atop t} {s \atop u}), where t, u ∈ R satisfy ru − st = 1. Show that this Mennicke symbol is independent of the choice of t and u, that [{r \atop s}] = [{s \atop r}], [{s \atop r}][{s′ \atop r} ] = [{ss′ \atop r} ] and [{s \atop r}] = [{s+xr \atop r} ].

If R is noetherian of dimension 1, or more generally has sr(R) ≤ 2, then we know by Ex. 1.3 that GL_{2}(R) maps onto K_{1}(R), and hence SK_{1}(R) is generated by Mennicke symbols.

1.11 Transfer. Suppose that R is a Dedekind domain and 𝔭 is a prime ideal of R. Show that there is a map π_{∗} from K_{1}(R/𝔭) = (R/𝔭)^{×} to SK_{1}(R) sending s̄ ∈ (R/𝔭)^{×} to the Mennicke symbol [{s \atop r}], where s ∈ R maps to s̄ and r ∈ R is an element of 𝔭 − 𝔭^{2} relatively prime to s. Another construction of the transfer map π_{∗} will be given in chapter V.

1.12 If R is a commutative Banach algebra, let exp(R) denote the image of the exponential map R → R^{×}. Show that exp(R) is the path component of 1 in R^{×}.

1.13 If H is a normal subgroup of a group G, then G acts upon H and hence its homology H_{∗}(H; ℤ) by conjugation. Since H always acts trivially upon its homology [WHomo, 6.7.8]), the group G/H acts upon H_{∗}(H; ℤ). Taking H = E(R) and G = GL(R), use Example 1.2.1 to show that GL(R) and K_{1}(R) act trivially upon the homology of E(R). 1.14 (Swan) Let T : P(R) → P(S) be an additive functor, such as the base change f^{∗} associated to a ring map f : R → S. In II.2.10 we constructed a relative group K_{0}(T). Since K_{0}(T) is abelian, we can concatenate the K_{1} map of lemma 1.7 to (II.2.10.2) to get a sequence which is exact at K_{0}(T) and (if T is cofinal) at K_{0}(R):

K_{1}(R) ⟶^{T} K_{1}(S) → K_{0}(T) → K_{0}(R) ⟶^{T} K_{0}(S).

In this exercise, we show that the sequence is also exact at K_{1}(S).

(a) We say that (P, α, Q) ∼ (P′, α′, Q′) if there are N, N′ ∈ P(R) and a commutator γ in Aut_{S} T (Q ⊕ N) so that (P ⊕ N, γ(α ⊕ 1), Q ⊕ N) is isomorphic to (P′ ⊕ N′, α′ ⊕ 1, Q′ ⊕ N′) in P(T). Show that ∼ is an equivalence relation.

(b) Show that the equivalence classes under ∼ form an abelian group under ⊕.

(c) If (P, α, Q) ∼ (P′, α′, Q′), show that [(P, α, Q)] ≅ [(P′, α′, Q′)] in K_{0}(T).

(d) If [(P, α, Q)] ≅ [(P′, α′, Q′)] in K_{0}(T), show that (P, α, Q) ∼ (P′, α′, Q′). Hint: Show that the relations for K_{0}(T) hold in the group of (b). To do so, write P ≅ P′ ⊕ P′′ and Q ≅ Q′ ⊕ Q′′ in the exact sequence II(2.10.1) in P(T).

(e) Use (d) to show that if α ∈ Aut_{S} T (R^{n}) and [(R^{n}, α, R^{n})] = 0 in K_{0}(T) then (after increasing n) there is an isomorphism (p, q) : (R^{n}, α, R^{n}) ≅ (R^{n}, γ, R^{n}) in P(T). Conclude that [α] is the image of [q^{−1}p] ∈ K_{1}(R), proving exactness of the sequence at K_{1}(S).

1.15 Suspension rings. Let R be any ring. Recall from Ex. I.1.8 that the cone ring C(R) is the ring of row-and-column-finite matrices over R. The finite matrices in C(R) form a 2-sided ideal M(R), and the quotient S(R) = C(R)/M(R) is called the suspension ring of R. Use exercise 1.14 and 1.7.3, together with II.2.1.3 and II.2.7.2 to show that K_{1}S(R) ≅ K_{0}(R). 1.16 Let D be a division algebra of dimension d = n^{2} over its center F. Show that the norm (or transfer) map K_{1}(D) → K_{1}(F) is n times the reduced norm N_{red} of 1.2.4. Hint: Choose a maximal subfield E and show that the map K_{1}(D) → K_{1}(E) induced by the norm is induced by the additive map M ↦ M ⊗_{D} (D ⊗_{F} E) from P(D) to P(D). Then show that D ⊗_{F} E ≅ D^{n} as a D-E bimodule. 1.17 Let D be a division algebra, finite dimensional over its center F, and let E be any finite extension of F which is a splitting field of D, i.e., E ⊗_{F} D ≅ M_{n}(E).

(a) Show that the following three maps θ_{E} : K_{1}(E) → K_{1}(D) agree. (i) K_{1}(E) ≅ K_{1}(M_{n}(E)) = K_{1}(E ⊗_{F} D) ⟶^{transfer} K_{1}(D); (ii) K_{1}(E) → K_{1}(M_{r}(D)) ≅ K_{1}(D), where E ⊂ M_{r}(D); (iii) K_{1}(T), where T : P(E) → P(D) is T (M) = M ⊗_{E} V for a simple

E ⊗_{F} D-module V.

(b) If j : E → L is a finite field map over F, show that θ_{E} = θ_{L}j_{∗}.

(c) If σ ∈ Aut(E/F), then θ_{E} = θ_{E}σ.

1.18 If A is any finite-dimensional semisimple algebra over a field with center C, construct a reduced norm A^{×} → C^{×} and define SL_{n}(A) to be the kernel of the reduced norm GL_{n}(A) → C^{×}. Show that the kernel SK_{1}(A) of the induced map K_{1}(A) → C^{×} is isomorphic to SL_{n}(A)/E_{n}(A) for all n ≥ 3.

§2. Relative K_{1}

Let I be an ideal in a ring R. We write GL(I) for the kernel of the natural map GL(R) → GL(R/I); the notation reflects the fact that GL(I) is independent of R (see Ex. I.1.10). In addition, we define E(R, I) to be the smallest normal subgroup of E(R) containing the elementary matrices e_{ij}(x) with x ∈ I. More generally, for each n we define E_{n}(R, I) to be the normal subgroup of E_{n}(R) generated by the matrices e_{ij}(x) with x ∈ I and 1 ≤ i ≠ j ≤ n. Clearly E(R, I) is the union of the subgroups E_{n}(R, I).

Relative Whitehead Lemma 2.1. E(R, I) is a normal subgroup of GL(I), and contains the commutator subgroup of GL(I).

Proof. For any matrix g = 1 + α ∈ GL_{n}(I), the identity

( \begin{matrix}g & 0 \\ 0 & g^{−1} \end{matrix} ) = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 \\ α & 1 \end{matrix} ) ( \begin{matrix}1 & −1 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 & g^{−1}α \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 \\ −gα & 1 \end{matrix} ) .

shows that the matrix ({g \atop 0} {0 \atop g^{−1}} ) is in E_{2n}(R, I). (The product of the first 3 matrices is in E_{2n}(R, I).) Hence if h ∈ E_{n}(R, I) then the conjugate

( \begin{matrix}ghg^{−1} & 0 \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}g & 0 \\ 0 & g^{−1} \end{matrix} ) ( \begin{matrix}h & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}g^{−1} & 0 \\ 0 & g \end{matrix} )

is in E(R, I). Finally, if g, h ∈ GL_{n}(I) then [g, h] is in E_{2n}(R, I) by equation (1.3.4).

Definition 2.2. The relative group K_{1}(R, I) is defined to be the quotient GL(I)/E(R, I). By the Relative Whitehead Lemma, it is an abelian group.

The inclusion of GL(I) in GL(R) induces a map K_{1}(R, I) → K_{1}(R). More generally, if R → S is a ring map sending I into an ideal I′ of S, the natural maps GL(I) → GL(I′) and E(R) → E(S) induce a map K_{1}(R, I) → K_{1}(S, I′).

Remark 2.2.1. Suppose that R → S is a ring map sending an ideal I of R isomorphically onto an ideal of S. The induced map K_{1}(R, I) → K_{1}(S, I) must be a surjection, as both groups are quotients of GL(I). However, Swan discovered that they need not be isomorphic; a simple example is given in Ex. 2.3 below.

Vaserstein proved in [V76, 14.2] that K_{1}(R, I) is independent of R if and only if I = I^{2}. One direction is easy (Ex. 2.10): if I = I^{2} then the commutator subgroup of GL(I) is perfect, and equal to E(R, I). Thus K_{1}(R, I) = GL(I)/[GL(I), GL(I)], a group which is independent of R. (Cf. Ex. 2.6 when R is commutative.)

Proposition 2.3. There is an exact sequence

K_{1}(R, I) → K_{1}(R) → K_{1}(R/I) ⟶^{∂} K_{0}(I) → K_{0}(R) → K_{0}(R/I).

Proof. By Ex. II.2.3 there is an exact sequence

1 → GL(I) → GL(R) → GL(R/I) ⟶^{∂} K_{0}(I) → K_{0}(R) → K_{0}(R/I).

Since the K_{1} groups are quotients of the GL groups, and E(R) maps onto E(R/I), this gives exactness except at K_{1}(R). Suppose g ∈ GL(R) maps to zero under GL(R) → K_{1}(R) → K_{1}(R/I). Then the reduction ḡ of g mod I is in E(R/I). Since E(R) maps onto E(R/I), there is a matrix e ∈ E(R) mapping to ḡ, i.e., ge^{−1} is in the kernel GL(I) of GL(R) → GL(R/I). Hence the class of ge^{−1} in K_{1}(R, I) is defined, and maps to the class of g in K_{1}(R). This proves exactness at the remaining spot.

The relative group SK_{1}(R, I)

If R happens to be commutative, the determinant map K_{1}(R) → R^{×} of Example 1.1.1 induces a relative determinant map det: K_{1}(R, I) → GL_{1}(I), since the determinant of a matrix in GL(I) is congruent to 1 modulo I. It is traditional to write SK_{1}(R, I) for the kernel of det, so the canonical map GL_{1}(I) → K_{1}(R, I) induces a direct sum decomposition K_{1}(R, I) = GL_{1}(I) ⊕ SK_{1}(R, I) compatible with the decomposition K_{1}(R) = R^{×} ⊕ SK_{1}(R) of Example 1.1.1. Here are two important cases in which SK_{1}(R, I) vanishes:

Lemma 2.4. Let I be a radical ideal in R. Then:

(1) K_{1}(R, I) is a quotient of the multiplicative group 1 + I = GL_{1}(I).

(2) If R is a commutative ring, then SK_{1}(R, I) = 0 and K_{1}(R, I) = 1 + I.

Proof. As in the proof of Lemma 1.4, it suffices to show that GL_{n}(I) = E_{n}(R, I)GL_{n−1}(I) for n ≥ 2. If (x_{ij}) is a matrix in GL_{n}(I) then x_{nn} is a unit of R, and for i < n the entries x_{in}, x_{ni} are in I. Multiplying by the diagonal matrix diag(1, …, 1, x_{nn}, x_{nn}^{−1}), we may assume that x_{nn} = 1. Now multiplying on the left by the matrices e_{in}(−x_{in}) and on the right by e_{ni}(−x_{ni}) reduces the matrix to one in GL_{n−1}(I).

The next theorem (and its variant) extends the calculation mentioned in Example 1.3.6 above. We cite them from [BMS, 4.3], mentioning only that their proof involves calculations with Mennicke symbols (see Ex. 1.10 and 2.5) for finitely generated R, i.e., Dedekind rings of arithmetic type.

Bass-Milnor-Serre Theorem 2.5. Let R be an integrally closed subring of a number field F, and I an ideal of R. Then

(1) If F has any embedding into ℝ then SK_{1}(R, I) = 0.

(2) If F is “totally imaginary” (has no embedding into ℝ), then SK_{1}(R, I) ≅ C_{n} is a finite cyclic group whose order n divides the order w_{1} of the group of roots of unity in R. The exponent ord_{p} n of p in the integer n is the minimum over all prime ideals 𝔭 of R containing I of the integer

inf\{ord_{p} w_{1}, sup\{0, [\frac{ord_{𝔭}(I)}{ord_{𝔭}(p)} − \frac{1}{p − 1}]\}

Variant 2.5.1. Let R be the coordinate ring of a smooth affine curve over a finite field. Then SK_{1}(R) = 0.

The Mayer-Vietoris Exact Sequence

Suppose we are given a ring map f: R → S and an ideal I of R mapped isomorphically into an ideal of S. Then we have a Milnor square of rings, as in I.2:

\begin{matrix}R & ⟶^{f} & S \\ ↓ & & ↓ \\ R/I & ⟶^{f̄} & S/I \end{matrix}

Theorem 2.6 (Mayer-Vietoris). Given a Milnor square as above, there is an exact sequence

K_{1}(R)⟶^{Δ} K_{1}(S)⊕K_{1}(R/I)→K_{1}(S/I)⟶^{∂} K_{0}(R)⟶^{Δ} K_{0}(S)⊕K_{0}(R/I)⟶^{±} K_{0}(S/I).

Proof. By Theorem II.2.9 we have an exact sequence

GL(S/I) ⟶^{∂} K_{0}(R) ⟶^{Δ} K_{0}(S) ⊕ K_{0}(R/I) ⟶^{±} K_{0}(S/I).

Since K_{0}(R) is abelian, we may replace GL(S/I) by K_{1}(S/I) in this sequence. This gives the sequence of the Theorem, and exactness at all the K_{0} places. Also by II.2.9, the image of ∂: K_{1}(S/I) → K_{0}(R) is the double coset space

GL(S)\GL(S/I)/GL(R/I).

Note that E(S) → E(S/I) is onto. Therefore the kernel of ∂ is the subgroup of K_{1}(S/I) generated by the images of GL(S) and GL(R/I), and the sequence is exact at K_{1}(S/I). To prove exactness at the final spot, suppose given ḡ ∈ GL_{n}(R/I), h ∈ GL_{n}(S) and an elementary matrix ē ∈ E(S/I) such that f̄(ḡ)ē ≡ h (mod I). Lifting ē to an e ∈ E_{n}(S) (by Remark 1.2.3) yields f̄(ḡ) ≡ he^{−1} (mod I). Since R is the pullback of S and R/I, there is a g ∈ GL_{n}(R), equivalent to ḡ modulo I, such that f(g) = he^{−1}. This establishes exactness at the final spot.

EXERCISES

2.1 Suppose we are given a Milnor square in which R and S are commutative rings. Using the Units-Pic sequence (I.3.10), conclude that there are exact sequences

SK_{1}(R, I) → SK_{1}(R) → SK_{1}(R/I) ⟶^{∂} SK_{0}(I) → SK_{0}(R) → SK_{0}(R/I),

SK_{1}(R) → SK_{1}(S)⊕SK_{1}(R/I) ⟶^{∂} SK_{0}(R) → SK_{0}(S)⊕SK_{0}(R/I) → SK_{0}(S/I).

2.2 Rim Squares. Let C_{p} be a cyclic group of prime order p with generator t, and let ζ = e^{2πi/p}. The ring ℤ[ζ] is the integral closure of ℤ in the number field ℚ(ζ). Let f: ℤC_{p} → ℤ[ζ] be the ring surjection sending t to ζ, and let I denote the kernel of the augmentation ℤC_{p} → ℤ.

(a) Show that I is isomorphic to the ideal of ℤ[ζ] generated by ζ − 1, so that we have a Milnor square with the rings ℤC_{p}, ℤ[ζ], ℤ and 𝔽_{p}.

(b) Show that for each k = 1, ..., p−1 the element (ζ^{k} −1)/(ζ −1) = 1+⋯+ζ^{k−1} is a unit of ℤ[ζ], mapping onto k ∈ 𝔽_{p}^{×}.

These elements are often called cyclotomic units, and generate a subgroup of finite index in ℤ[ζ]^{×}. If p ≥ 3, the Dirichlet Unit Theorem says that the units of ℤ[ζ] split as the direct sum of the finite group {±ζ^{k}} of order 2p (p ≠ 2) and a free abelian group of rank (p − 3)/2.

(c) Conclude that if p > 3 then both K_{1}(ℤC_{p}) and W h_{1}(C_{p}) are nonzero. In fact, SK_{1}(ℤC_{p}) = 0.

2.3 Failure of Excision for K_{1}. Here is Swan’s simple example to show that K_{1}(R, I) depends upon R. Let F be a field and let R be the algebra of all upper triangular

matrices r = ( \begin{matrix}x & y \\ 0 & z \end{matrix} ) in M_{2}(F). Let I be the ideal of all such matrices with x =

z = 0, and let R_{0} be the commutative subalgebra F ⊕ I. Show that K_{1}(R_{0}, I) ≅ F but that K_{1}(R, I) = 0. Hint: Calculate e_{21}(r)e_{12}(y)e_{21}(−r).

2.4 (Vaserstein) If I is an ideal of R, and x ∈ I and r ∈ R are such that (1 + rx) is a unit, modify Ex. 1.1 to show that (1 + rx)(1 + xr)^{−1} is in E_{2}(R, I).

If I is a radical ideal and W = W (R, I) denotes the subgroup of units generated by the (1 + rx)(1 + xr)^{−1}, use Lemma 2.4 to conclude that (1 + I)/W maps onto K_{1}(R, I). Vaserstein proved in [V69] that K_{1}(R, I) ≅ (1 + I)/W for every radical ideal.

2.5 Mennicke symbols. If I is an ideal of a commutative ring R, r ∈ (1 + I) and s ∈ I, we define the Mennicke symbol [{s \atop r}] to be the class in SK_{1}(R, I) of the matrix ({r \atop t} {s \atop u}), where t ∈ I and u ∈ (1 + I) satisfy ru − st = 1. Show that this Mennicke symbol is independent of the choice of t and u, with [{s \atop r}][{s \atop r′} ] = [{s \atop rr′} ], [{s \atop r}][{s′ \atop r} ] = [{ss′ \atop r} ]. (Hint: Use Ex. 1.10.) Finally, show that if t ∈ I then

[{s \atop r}] = [{s + rt \atop r}] = [{s \atop r + st}].

2.6 The obstruction to excision. Let R → S be a map of commutative rings, sending an ideal I of R isomorphically onto an ideal of S. Given x ∈ I and s ∈ S, let ψ(x, s) denote the Mennicke symbol [{x \atop 1−sx}] in SK_{1}(R, I).

(a) Verify that ψ(x, s) vanishes in SK_{1}(S, I).

(b) Prove that ψ is bilinear, and that ψ(x, s) = 1 if either x ∈ I^{2} or s ∈ R. Thus ψ induces a map from (I/I^{2}) ⊗ (S/R) to SK_{1}(R, I).

(c) Prove that the Leibniz rule holds: ψ(x, ss′) = ψ(sx, s′)ψ(s′x, s). For every map R → S, the S-module Ω_{S/R} of relative Kähler differentials is presented with generators ds, s ∈ S, subject to the following relations: d(s + s′) = ds + ds′, d(ss′) = s ds′ + s′ ds, and if r ∈ R then dr = 0. (See [WHomo].)

(d) (Vorst) Show that Ω_{S/R} ⊗_{S} I/I^{2} is the quotient of (S/R) ⊗ (I/I^{2}) by the subgroup generated by the elements s ⊗ s′x + s′ ⊗ sx − ss′ ⊗ x. Then conclude that ψ induces a map Ω_{S/R} ⊗_{S} I/I^{2} → SK_{1}(R, I).

Swan proved in [Swan71] that the resulting sequence is exact:

Ω_{S/R} ⊗_{S} I/I^{2} ⟶^{ψ} SK_{1}(R, I) → SK_{1}(S, I) → 1.

2.7 Suppose that the ring map R → R/I is split by a map R/I → R. Show that K_{1}(R) ≅ K_{1}(R/I) ⊕ K_{1}(R, I). The corresponding decomposition of K_{0}(R) follows from the ideal sequence 2.3, or from the definition of K_{0}(I), since R ≅ R/I ⊕ I; see Ex. II.2.4. 2.8 Suppose that p^{r} = 0 in R for some prime p. Show that K_{1}(R, pR) is a p-group. Conclude that the kernel of the surjection K_{1}(R) → K_{1}(R/pR) is also a p-group. 2.9 If I is a nilpotent ideal in a ℚ-algebra R, or even a complete radical ideal, show that K_{1}(R, I) ≅ I/[R, I], where [R, I] is the subgroup spanned by all elements [r, x] = rx − xr, r ∈ R and x ∈ I. In particular, this proves that K_{1}(R, I) is uniquely divisible. Hint: If [R, I] = 0, ln : 1 + I → I is a bijection. If not, use Ex. 2.4 and the Campbell-Hausdorff formula. 2.10 Suppose that I is an ideal satisfying I = I^{2}. Show that [GL(I), GL(I)] is a perfect group. Conclude that E(R, I) = [GL(I), GL(I)] and hence that K_{1}(R, I) is independent of R. Hint: Use the commutator formulas (1.3.1).

§3. The Fundamental Theorems for K_{1} and K_{0}

The Fundamental Theorem for K_{1} is a calculation of K_{1}(R[t, t^{−1}]), and describes one of the many relationships between K_{1} and K_{0}. The core of this calculation depends upon the construction of an exact sequence (see 3.2 below and II.7.8.1):

K_{1}(R[t]) → K_{1}(R[t, t^{−1}]) ⟶^{∂} K_{0}H_{{t^{n}}}(R[t]) → 0

We will construct a localization sequence connecting K_{1} and K_{0} in somewhat greater generality first. Recall from chapter II, Theorem 9.8 that for any multiplicatively closed set S of central elements in a ring R there is an exact sequence K_{0}(R on S) → K_{0}(R) → K_{0}(S^{−1}R), where K_{0}(R on S) denotes K_{0} of the Waldhausen category Ch_{S}^{b}P(R). If S consists of nonzerodivisors, K_{0}(R on S) also equals K_{0}H_{S}(R) by Ex. II.9.13; see Corollary II.7.7.4.

Our first goal is to extend this sequence to the left using K_{1}, and we begin by constructing the boundary map ∂.

Let α be an endomorphism of R^{n}. We say that α is an S-isomorphism if S^{−1} ker(α) = S^{−1}coker(α) = 0, or equivalently, α/1 ∈ GL_{n}(S^{−1}R). Write cone(α) for the mapping cone of α, which is the chain complex R^{n} ⟶^{−α} R^{n} concentrated in degrees 0 and 1; see [WHomo, 1.5.1]. It is clear that α is an S-isomorphism if and only if cone(α) ∈ Ch_{S}^{b}P(R).

Lemma 3.1. Let S be a multiplicatively closed set of central elements in a ring R. Then there is a group homomorphism

K_{1}(S^{−1}R) ⟶^{∂} K_{0}(R on S)

sending each S-isomorphism α to the class [cone(α)] of the mapping cone of α. In particular, each s ∈ S is an endomorphism of R so ∂(s) is the class of the chain complex cone(s) : R ⟶^{−s} R.

Before proving this lemma, we give one important special case. When S consists of nonzerodivisors, every S-isomorphism α must be an injection, and coker(α) is a module of projective dimension one, i.e., an object of H_{S}(R). Moreover, under the isomorphism K_{0}Ch_{S}^{b}P(R) ≅ K_{0}H_{S}(R) of Ex. II.9.13, the class of cone(α) in K_{0}Ch_{S}^{b}P(R) corresponds to the element [coker(α)] ∈ K_{0}H_{S}(R). Thus we immediately have:

Corollary 3.1.1. If S consists of nonzerodivisors then there is a homomorphism K_{1}(S^{−1}R) ⟶^{∂} K_{0}H_{S}(R) sending each S-isomorphism α to [coker(α)], and sending s ∈ S to [R/sR].

Proof of 3.1. If β ∈ End(R^{m}) is also an S-isomorphism, then the diagram

0  ↣  R^{n}  ↠^{=} R^{n}
↓  ↓ (1, β)  ↓ β
R^{n}  ↣^{(10)}  R^{n} ⊕ R^{n}  ↠^{(01)} R^{n}
αβ ↓  ↓ \binom{αβ}{−α}  ↓
R^{n}  ↣^{=}  R^{n}  ↠  0

is a short exact sequence in Ch_{S}^{b}P(R), where we regard the columns as chain complexes. Since the middle column of the diagram is quasi-isomorphic to its subcomplex 0 → 0 ⊕ R^{n} ⟶^{−α} R^{n}, we get the relation

[cone(α)] − [cone(αβ)] = [cone(β)[−1]] = −[cone(β)],

or

[cone(α)] + [cone(β)] = [cone(αβ)]  (3.1.2)

in K_{0}Ch_{S}^{b}P(R). In particular, if β is the diagonal matrix diag(t, ..., t) then cone(β) is the direct sum of n copies of cone(t), so we have

[cone(αt)] = [cone(α)] + n[cone(t)].  (3.1.3)

Every g ∈ GL_{n}(S^{−1}R) can be represented as α/s for some S-isomorphism α and some s ∈ S, and we define ∂(g) = ∂(α/s) to be the element [cone(α)] − n[cone(s)] of K_{0}Ch_{S}^{b}P(R). By (3.1.3) we have ∂(α/s) = ∂(αt/st), which implies that ∂(g) is independent of the choice of α and s. By (3.1.2) this implies that ∂ is a well-defined homomorphism from each GL_{n}(S^{−1}R) to K_{0}Ch_{S}^{b}P(R). Finally, the maps ∂ are compatible with the inclusions GL_{n} ⊂ GL_{n+1}, because

∂({α/s \atop 0} {0 \atop 1}) = ∂(({α \atop 0} {0 \atop 1})/s) = [cone({α \atop 0} {0 \atop s})] − (n + 1)[cone(s)]
= [cone(α)] + [cone(s)] − (n + 1)[cone(s)] = ∂(α/s).

Hence ∂ extends to GL(S^{−1}R), and hence must factor through the universal map to K_{1}(S^{−1}R).

Key Example 3.1.4. For the Fundamental Theorem, we shall need the following special case of this construction. Let T be the multiplicative set {t^{n}} in the polynomial ring R[t]. Then the map ∂ goes from K_{1}(R[t, t^{−1}]) to K_{0}H_{T} (R[t]). If ν is a nilpotent endomorphism of R^{n} then t − ν is a T -isomorphism, because its inverse is the polynomial t^{−1}(1 + νt^{−1} + ν^{2}t^{−2} + …). If (R^{n}, ν) denotes the R[t]-module R^{n} on which t acts as ν,

∂(t − ν) = [R[t]^{n}/(t − ν)] = [(R^{n}, ν)],
∂(1 − νt^{−1}) = ∂(t − ν) − ∂(t · id_{n}) = [(R^{n}, ν)] − n[(R, 0)].

We can also compose ∂ with the product K_{0}(R) ⊗ K_{1}(ℤ[t, t^{−1}]) ⟶^{·} K_{1}(R[t, t^{−1}]) of Corollary 1.6.1. Given a finitely generated projective R-module P, the product [P] · t is the image of t · id_{P [t,t^{−1}]} under the map Aut(P [t, t^{−1}]) → K_{1}(R[t, t^{−1}]) of Lemma 1.6. To compute ∂([P] · t), choose Q such that P ⊕ Q ≅ R^{n}. Since the cokernel of t·id_{P [t]}: P [t] → P [t] is the R[t]-module (P, 0), we have an exact sequence of R[t]-modules:

0 → R[t]^{n} ⟶^{t·id_{P [t]}⊕1·id_{Q[t]}} R[t]^{n} → (P, 0) → 0.

Therefore we have the formula ∂([P] · t) = [(P, 0)].

Lemma 3.1.5. K_{0}Ch_{S}^{b}P(R) is generated by the classes [Q_{·}] of chain complexes concentrated in degrees 0 and 1, i.e., by complexes Q_{·} of the form Q_{1} → Q_{0}.

The kernel of K_{0}Ch_{S}^{b}P(R) → K_{0}(R) is generated by the complexes R^{n} ⟶^{α} R^{n} associated to S-isomorphisms, i.e., by the classes ∂(α) = [cone(α)].

Proof. By the Shifting Lemma II.9.2.1, K_{0} is generated by bounded complexes of the form 0 → P_{n} → ⋯ → P_{0} → 0. If n ≥ 2, choose a free R-module F = R^{N} mapping onto H_{0}(P_{·}). By assumption, we have sH_{0}(P_{·}) = 0 for some s ∈ S. By the projective lifting property, there are maps f_{0}, f_{1} making the diagram

F ⟶^{s} F ⟶ F/sF ⟶ 0
f_{1}↓ ^{f_{0}}↓  ↓
P_{1} ⟶ P_{0} ⟶ H_{0}(P) ⟶ 0

commute. Thus if Q_{·} denotes the complex F ⟶^{s} F we have a chain map Q_{·} ⟶^{f} P_{·} inducing a surjection on H_{0}. The mapping cone of f fits into a cofibration sequence P_{·} ↣ cone(f) ↠ Q_{·}[−1] in Ch_{S}^{b}P(R), so we have [P_{·}] = [Q_{·}] + [cone(f)] in K_{0}(R on S). Moreover, H_{0}(cone(f)) = 0, so there is a decomposition P_{1} ⊕ F ≅ P_{0} ⊕ P′_{1} so that the mapping cone is the direct sum of an exact complex P_{0} ⟶^{≅} P_{0} and a complex P′_{·} of the form 0 → P_{n} → ⋯ → P_{3} → P_{2} ⊕ F → P′_{1} → 0. Since P′_{·} has length n − 1, induction on n implies that [cone(f)] = [P′_{·}] is a sum of terms of the form [Q_{1} → Q_{0}].

Hence every element of K_{0} has the form x = [P_{1} ⟶^{α} P_{0}] − [Q_{1} ⟶^{β} Q_{0}]. Choose s ∈ S so that sβ^{−1} is represented by an S-isomorphism Q_{0} ⟶^{γ} Q_{1}; adding γ to both terms of x, as well as the appropriate zero term Q′ ⟶^{=} Q′, we may assume that Q_{1} = Q_{0} = R^{n}, i.e., that the second term of x is the mapping cone of some S-isomorphism β ∈ End(R^{n}). With this reduction, the map to K_{0}(R) sends x to [P_{1}] − [P_{0}]. If this vanishes, then P_{1} and P_{0} are stably isomorphic. Adding the appropriate P′ ⟶^{=} P′ makes P_{1} = P_{0} = R^{m} for some m, and writes x in the form

x = cone(α) − cone(β) = ∂(α) − ∂(β).

Theorem 3.2. Let S be a multiplicatively closed set of central elements in a ring R. Then the map ∂ of Lemma 3.1 fits into an exact sequence

K_{1}(R) → K_{1}(S^{−1}R) ⟶^{∂} K_{0}(R on S) → K_{0}(R) → K_{0}(S^{−1}R).

Proof. We have proven exactness at K_{0}(R) in Theorem II.9.8, and the composition of any two consecutive maps is zero by inspection. Exactness at K_{0}(R on S) was proven in Lemma 3.1.5. Hence it suffices to establish exactness at K_{1}(S^{−1}R).

For reasons of exposition, we shall give the proof when S consists of nonzerodivisors, relegating the general proof (which is similar but more technical) to Exercise 3.5. The point of this simplification is that we can work with the exact category H_{S}(R). In particular, for every S-isomorphism α the class of the module coker(α) is simpler to manipulate than the class of the mapping cone.

Recall from the proof of Lemma 3.1 that every element of GL_{n}(S^{−1}R) can be represented as α/s for some S-isomorphism α ∈ End(R^{n}) and some s ∈ S, and that ∂(α/s) is defined to be [coker(α)] − [R^{n}/sR^{n}]. If ∂(α/s) = 0, then from Ex. II.7.2 there are short exact sequences in H_{S}(R)

0 → C′ → C_{1} → C′′ → 0, 0 → C′ → C_{2} → C′′ → 0

such that coker(α)⊕C_{1} ≅ (R^{n}/sR^{n})⊕C_{2}. By Ex. 3.4 we may add terms to C′, C′′ to assume that C′ = coker(α′) and C′′ = coker(α′′) for appropriate S-isomorphisms of some R^{m}. By the Horseshoe Lemma ([WHomo, 2.2.8]) we can construct two exact sequences of projective resolutions

0  0  0
↓  ↓  ↓
0 ⟶ R^{m} ⟶ R^{2m} ⟶ R^{m} ⟶ 0
α′↓  ^{α_{i}}↓  _{α′′}↓
0 ⟶ R^{m} ⟶ R^{2m} ⟶ R^{m} ⟶ 0
↓  ↓  ↓
0 ⟶ C′ ⟶ C_{i} ⟶ C′′ ⟶ 0
↓  ↓  ↓
0  0  0.

Inverting S makes each α_{i} an isomorphism conjugate to ({α′ \atop 0} {0 \atop α′′} ). Thus in K_{1}(S^{−1}R) we have [α_{1}] = [α′]+[α′′] = [α_{2}]. On the other hand, the two endomorphisms α⊕α_{1} and s · id_{n} ⊕ α_{2} of R^{2m+n} have isomorphic cokernels by construction. Lemma 3.2.1 below implies that in K_{1}(S^{−1}R) we have

[α/s] = [α ⊕ α_{1}] − [s · id_{n} ⊕ α_{2}] = g for some g ∈ GL(R).

This completes the proof of Theorem 3.2.

Lemma 3.2.1. Suppose that S consists of nonzerodivisors. If α, β ∈ End_{R}(R^{n}) are S-isomorphisms with R^{n}/αR^{n} isomorphic to R^{n}/βR^{n}, then there is a g ∈ GL_{4n}(R) such that [α] = [g][β] in K_{1}(S^{−1}R).

Proof. Put M = coker(α) ⊕ coker(β), and let γ: R^{n}/αR^{n} ≅ R^{n}/βR^{n} be an automorphism. By Ex. 3.3(b) with Q = R^{2n} we can lift the automorphism ({0 \atop γ} {γ^{−1} \atop 0} ) of M to an automorphism γ_{0} of R^{4n}. If π_{1} and π_{2} denote the projections

R^{4n} ⟶^{(pr,0,0,0)} coker(α), and R^{4n} ⟶^{(0,pr,0,0)} coker(β), respectively, then we have γπ_{1} = π_{2}γ_{0}. This yields a commutative diagram

0 ⟶ R^{4n} ⟶^{(α,1,1,1)} R^{4n} ⟶^{π_{1}} R^{n}/αR^{n} ⟶ 0
↓^{γ_{1}}  ≅↓^{γ_{0}}  ≅↓^{γ}
0 ⟶ R^{4n} ⟶^{(1,β,1,1)} R^{4n} ⟶^{π_{2}} R^{n}/βR^{n} ⟶ 0

in which γ_{1} is the induced map. Since γ and γ_{0} are isomorphisms, so is γ_{1}. Because γ_{0}(α, 1, 1, 1) = (1, β, 1, 1)γ_{1} in GL_{4n}(S^{−1}R), we have [γ_{0}] + [α] = [β] + [γ_{1}], or [α] = [γ_{1}γ_{0}^{−1}][β] in K_{1}(S^{−1}R).

NK_{1} and the group Nil_{0}

Definition 3.3 (NF). If F is any functor from rings to abelian groups, we write NF (R) for the cokernel of the natural map F (R) → F (R[t]); NF is also a functor on rings. Moreover, the ring map R[t] ⟶^{t=1} R provides a splitting F (R[t]) → F (R) of the natural map, so we have a decomposition F (R[t]) ≅ F (R) ⊕ NF (R).

In particular, when F is K_{i} (i = 0, 1) we have functors NK_{i} and a decomposition K_{i}(R[t]) ≅ K_{i}(R) ⊕ NK_{i}(R). Since the ring maps R[t] ⟶^{t=r} R are split surjections for every r ∈ R, we see by Proposition 2.3 and Ex. 2.7 that for every r we also have

NK_{0}(R) ≅ K_{0}(R[t], (t − r)) and NK_{1}(R) ≅ K_{1}(R[t], (t − r)).

We will sometimes speak about NF for functors F defined on any category of rings closed under polynomial extensions and containing the map “t = 1,” such as k-algebras or commutative rings. For example, the functors NU and N Pic were discussed briefly for commutative rings in chapter I, Ex. 3.17 and 3.19.

Example 3.3.1. (Chase) Suppose that A is an algebra over ℤ/p. Then the group NK_{1}(A) is a p-group. To see this, first observe that it is true for the algebras A_{n} = ℤ/p[x]/(x^{n}) by 2.4, since (1 + tf(x, t))^{p} = 1 + t^{p}f(x^{p}, t^{p}). Then observe that by Higman’s trick (3.5.1 below) every element of NK_{1}(A) is the image of 1 − xt ∈ NK_{1}(A_{n}) (for some n) under a map A_{n} → M_{n}(A), x ↦ ν.

By Ex. 2.8, NK_{1}(A) is also a p-group for every ℤ/p^{r}-algebra A.

Example 3.3.2. If A is an algebra over a field k of characeteristic zero, then NK_{1}(A) is a uniquely divisible abelian group. In fact, NK_{1}(A) has the structure of a k-vector space; see Ex. 3.7.

Definition 3.4 (F -regular rings). We say that a ring R is F -regular if F (R) = F (R[t_{1}, …, t_{n}]) for all n. Since NF (R[t]) = NF (R) ⊕ N^{2}F (R), we see by induction on p that R is F -regular if and only if N^{p}F (R) = 0 for all p ≥ 1.

For example, Traverso’s theorem (I.3.11) states that a commutative ring R is Picregular if and only if R_{red} is seminormal. We also saw in I.3.12 that commutative rings are U-regular (U=units) if and only if R is reduced.

We saw in II.6.5 that any regular ring is K_{0}-regular. We will see in theorem 3.8 below that regular rings are also K_{1}-regular, and we sill see in chapter Vthat they are K_{m}-regular for every m. Rosenberg has also shown that commutative C^{∗}- algebras are K_{m}-regular for all m; see [Ro96].

Lemma 3.4.1. Let R = R_{0} ⊕ R_{1} ⊕ ⋯ be a graded ring. Then the kernel of F (R) → F (R_{0}) embeds in NF (R) and even in the kernel of NF (R) → NF (R_{0}).

In particular, if NF (R) = 0 then F (R) ≅ F (R_{0}).

Proof. Let f denote the ring map R → R[t] defined by f(r_{n}) = r_{n}t^{n} for every r_{n} ∈ R_{n}. Since the composition of f and “t = 1” is the identity on R, F (f) is an injection. Let Q denote the kernel of F (R) → F (R_{0}), so that F (R) = F (R_{0}) ⊕ Q. Since the composition of f and “t = 0” is the projection R → R_{0} → R, Q embeds into the kernel NF (R) of the evaluation F (R[t]) → F (R) at t = 0. Similarly, since the composition of f and R[t] → R_{0}[t] is projection R → R_{0} → R_{0}[t], Q embeds into the kernel of NF (R) → NF (R_{0}).

A typical application of this result is that if R is a graded seminormal algebra with R_{0} a field then Pic(R) = 0.

Application 3.4.2. It follows that NF (R) is a summand of N^{2}F (R) and hence N^{p}F (R) for all p > 0. Indeed, the first part is the application of 3.4.1 to R[s], and the second part is obtained by induction, replacing F by N^{p−2}F.

If s ∈ S is central, the algebra map S[x] → S[x], x → sx, induces an operation [s] : NK_{0}(S) → NK_{0}(S). (It is the multiplication by 1 − st ∈ W (S) in Ex. 3.7.) Write S_{s} for S[1/s].

Theorem 3.4.3 (Vorst). NK_{0}(S_{s}) is the “localization” of NK_{0}(S) along [s],

i.e., NK_{0}(S_{s}) ≅ \underrightarrow{lim}(NK_{0}(S) ⟶^{[s]} NK_{0}(S) ⟶^{[s]} ⋯). In particular, if S is K_{0}- regular, so is S_{s}.

Proof. Write I for the ideal (x) of S_{s}[x] and set R = S + I. Then NK_{0}(S_{s}) = K_{0}(I) by Ex. II.2.3. But R = \underrightarrow{lim}(S[x] → S[x] → ⋯) and I is the direct limit of xS[x] → xS[x] → ⋯, so K_{0}(I) = \underrightarrow{lim}(K_{0}(xS[x]) → ⋯) as claimed.

Corollary 3.4.4. If A is K_{0}-regular then so is A[s, s^{−1}].

3.5. We are going to describe the group NK_{1}(R) in terms of nilpotent matrices. For this, we need the following trick, which was published by Graham Higman in 1940. For clarity, if I = fA is an ideal in A we write GL(A, f) for GL(I).

Higman’s Trick 3.5.1. For every g ∈ GL(R[t], t) there is a nilpotent matrix ν over R such that [g] = [1 − νt] in K_{1}(R[t]).

Similarly, for every g ∈ GL(R[t, t^{−1}], t − 1) there is a nilpotent matrix ν over R such that [g] = [1 − ν(t − 1)] in K_{1}(R[t, t^{−1}], t − 1).

Proof. Every invertible p × p matrix over R[t] can be written as a polynomial g = γ_{0} + γ_{1}t + γ_{2}t^{2} + ⋯ + γ_{n}t^{n} with the γ_{i} in M_{p}(R). If g is congruent to the identity modulo t, then γ_{0} = 1. If n ≥ 2 and we write g = 1 − ht + γ_{n}t^{n}, then modulo E_{2p}(R[t], t) we have

( \begin{matrix}g & 0 \\ 0 & 1 \end{matrix} ) ∼ ( \begin{matrix}g & γ_{n}t^{n−1} \\ 0 & 1 \end{matrix} ) ∼ ( \begin{matrix}1 − ht & γ_{n}t^{n−1} \\ −t & 1 \end{matrix} ) = 1 − ( \begin{matrix}h & −γ_{n}t^{n−2} \\ 1 & 0 \end{matrix} ) t.

By induction on n, [g] is represented by a matrix of the form 1 − νt. The matrix ν is nilpotent by Ex. 3.1.

Over R[t, t^{−1}] we can use a similar argument. After multiplying by a power of t, we may write g as a polynomial in t. Such a polynomial may be rewritten as a polynomial ∑ γ_{i}x^{i} in x = (t − 1). If g is congruent to the identity modulo (t − 1) then again we have γ_{0} = 1. By Higman’s trick (applied to x), we may reduce g to a matrix of the form 1 − νx, and again ν must be nilpotent by Ex. 3.1.

We will also need the category Nil(R) of II.7.4.4. Recall that the objects of this category are pairs (P, ν), where P is a finitely generated projective R-module and ν is a nilpotent endomorphism of P. Let T denote the multiplicative set {t^{n}} in R[t]. From II.7.8.4 we have

K_{0}(R[t] on T) ≅ K_{0}Nil(R) ≅ K_{0}(R) ⊕ Nil_{0}(R),

where Nil_{0}(R) is the subgroup generated by elements of the form [(R^{n}, ν)] − n[(R, 0)] for some n and some nilpotent matrix ν.

Lemma 3.5.2. For every ring R, the product with t ∈ K_{1}(ℤ[t, t^{−1}]) induces a split injection K_{0}(R) ⟶^{·t} K_{1}(R[t, t^{−1}]).

Proof. Since the forgetful map K_{0}Nil(R) → K_{0}(R) sends [(P, ν)] to [P], the calculation in Example 3.1.4 shows that the composition

K_{0}(R) ⟶^{·t} K_{1}(R[t, t^{−1}]) ⟶^{∂} K_{0}Nil(R) → K_{0}(R)

is the identity map. Hence the first map is a split injection.

Momentarily changing variables from t to s, we now define an additive function τ from Nil(R) to K_{1}(R[s]). Given an object (P, ν), let τ(P, ν) be the image of the automorphism 1 − νs of P [s] under the natural map Aut(P [s]) → K_{1}(R[s]) of Lemma 1.6. Given a short exact sequence

0 → (P′, ν′) → (P, ν) → (P′′, ν′′) → 0

in Nil(R), a choice of a splitting P ≅ P′ ⊕ P′′ allows us to write

(1 − νs) = ( \begin{matrix}1 − ν′s & γs \\ 0 & 1 − ν′′s \end{matrix} ) = ( \begin{matrix}1 − ν′s & 0 \\ 0 & 1 − ν′′s \end{matrix} ) ( \begin{matrix}1 & γ′s \\ 0 & 1 \end{matrix} )

in Aut(P [s]). Hence in K_{1}(R[s]) we have [1 − νs] = [1 − ν′s][1 − ν′′s]. Therefore τ is an additive function, and induces a homomorphism τ: K_{0}Nil(R) → K_{1}(R[s]). Since τ(P, 0) = 1 for all P and 1 − νs is congruent to 1 modulo s, we see that τ is actually a map from Nil_{0}(R) to K_{1}(R[s], s).

Proposition 3.5.3. Nil_{0}(R) ≅ NK_{1}(R), and K_{0}Nil(R) ≅ K_{0}(R) ⊕ NK_{1}(R).

Proof. For convenience, we identify s with t^{−1}, so that R[s, s^{−1}] = R[t, t^{−1}]. Applying Lemma 3.1 to R[t] and T = {1, t, t^{2}, … }, we form the composition

δ : K_{1}(R[s], s) → K_{1}(R[s]) →K_{1}(R[s, s^{−1}])
(3.5.4)  =K_{1}(R[t, t^{−1}]) ⟶^{∂} K_{0}(R on T) → Nil_{0}(R).

Let us call this composition δ. We claim that τ is the inverse of δ. By Higman’s Trick, every element of K_{1}(R[s], s) is represented by a matrix 1−νs with ν nilpotent. In Example 3.1.4 we saw that δ(1−νs) = [(R^{n}, ν)]−n[(R, 0)]. By the construction of τ we have the desired equations: τδ(1 − νs) = τ[(R^{n}, ν)] = (1 − νs) and

δτ([(R^{n}, ν)] − n[(R, 0)]) = δ(1 − νs) = [(R^{n}, ν)] − n[(R, 0)].

Corollary 3.5.5. K_{1}(R[s]) → K_{1}(R[s, s^{−1}]) is an injection for every ring R.

Proof. By Ex. 2.7, we have K_{1}(R[s]) ≅ K_{1}(R) ⊕ K_{1}(R[s], s). Since K_{1}(R) is a summand of K_{1}(R[s, s^{−1}]), the isomorphism δ: K_{1}(R[s], s) ≅ Nil_{0}(R) of (3.5.4) factors through K_{1}(R[s], s) → K_{1}(R[s, s^{−1}])/K_{1}(R). This quotient map must then be an injection. The result follows.

The Fundamental Theorems for K_{1} and K_{0}

Fundamental Theorem for K_{1} 3.6. For every ring R, there is a split surjection K_{1}(R[t, t^{−1}]) ⟶^{∂} K_{0}(R), with inverse [P] ↦ [P] · t. This map fits into a naturally split exact sequence:

0 → K_{1}(R) ⟶^{Δ} K_{1}(R[t]) ⊕ K_{1}(R[t^{−1}]) ⟶^{±} K_{1}(R[t, t^{−1}]) ⟶^{∂} K_{0}(R) → 0.

Consequently, we have a natural direct sum decomposition:

K_{1}(R[t, t^{−1}]) ≅ K_{1}(R) ⊕ K_{0}(R) ⊕ NK_{1}(R) ⊕ NK_{1}(R).

Proof. We merely assemble the pieces of the proof from §3.5. The first assertion is just Lemma 3.5.2. The natural maps from K_{1}(R) into K_{1}(R[t]), K_{1}(R[t^{−1}]) and K_{1}(R[t, t^{−1}]) are injections, split by “t = 1” (as in 3.5), so the obviously exact sequence

(3.6.1)  0 → K_{1}(R) ⟶^{Δ} K_{1}(R) ⊕ K_{1}(R) ⟶^{±} K_{1}(R) → 0

is a summand of the sequence we want to prove exact. From Proposition II.7.8.1, Theorem 3.2 and Corollary 3.5.5, we have an exact sequence

(3.6.2)  0 → K_{1}(R[t]) → K_{1}(R[t, t^{−1}]) ⟶^{∂} K_{0}Nil(R) → 0.

Since K_{0}Nil(R) ≅ K_{0}(R) ⊕ Nil_{0}(R), the map ∂ in (3.6.2) is split by the maps of 3.5.2 and 3.5.3. The sequence in the Fundamental Theorem for K_{1} is obtained by rearranging the terms in sequences (3.6.1) and (3.6.2).

In order to formulate the corresponding Fundamental Theorem for K_{0}, we define K_{−1}(R) to be the cokernel of the map K_{0}(R[t]) ⊕ K_{0}(R[t^{−1}]) → K_{0}(R[t, t^{−1}]). We will reprove the following result more formally in the next section.

Fundamental Theorem for K_{0} 3.7. For every ring R, there is a naturally split exact sequence:

0 → K_{0}(R) ⟶^{Δ} K_{0}(R[t]) ⊕ K_{0}(R[t^{−1}]) ⟶^{±} K_{0}(R[t, t^{−1}] ⟶^{∂} K_{−1}(R) → 0.

Consequently, we have a natural direct sum decomposition:

K_{0}(R[t, t^{−1}] ≅ K_{0}(R) ⊕ K_{−1}(R) ⊕ NK_{0}(R) ⊕ NK_{0}(R).

Proof. Let s be a second indeterminate. The Fundamental Theorem for K_{1}, applied to the variable t, gives a natural decomposition

K_{1}(R[s, t, t^{−1}]) ≅ K_{1}(R[s]) ⊕ NK_{1}(R[s]) ⊕ NK_{1}(R[s]) ⊕ K_{0}(R[s]), and similar decompositions for the other terms in the map

K_{1}(R[s, t, t^{−1}]) ⊕ K_{1}(R[s^{−1}, t, t^{−1}]) → K_{1}(R[s, s^{−1}, t, t^{−1}]).

Therefore the cokernel of this map also has a natural splitting. But the cokernel is K_{0}(R[t, t^{−1}]), as we see by applying the Fundamental Theorem for K_{1} to the variable s.

Theorem 3.8. If R is a regular ring, K_{1}(R[t]) ≅ K_{1}(R) and there is a natural isomorphism K_{1}(R[t, t^{−1}]) ≅ K_{1}(R) ⊕ K_{0}(R).

Proof. Consider the category M_{t}(R[t]) of finitely generated t-torsion R[t]- modules; by Devissage II.6.3.3, K_{0}M_{t}(R[t]) ≅ K_{0}(R). Since R is regular, every such module has a finite resolution by finitely generated projective R[t]-modules, i.e., M_{t}(R[t]) is the same as the category H_{t}(R[t]) of II.7.7. By II.7.8.4,

K_{0}Nil(R) ≅ K_{0}H_{t}(R[t]) ≅ K_{0}(R).

Hence Nil_{0}(R) = 0. By 3.5.3, NK_{1}(R) = 0 and K_{1}(R[t]) ≅ K_{1}(R). The description of K_{1}(R[t, t^{−1}]) now comes from the Fundamental Theorem 3.6.

Example 3.8.1. If R is a commutative regular ring, and A = R[x]/(x^{N}), it follows from 2.4 and 3.8 that SK_{1}(A[t]) = SK_{1}(A) and hence (by 3.5.3 and I.3.12) Nil_{0}(A) ≅ NK_{1}(A) ≅ (1 + tA[t])^{×} = (1 + xtA[t])^{×}. This isomorphism sends [(P, ν)] ∈ Nil_{0}(A) to det(1−νt) ∈ (1+tA[t])^{×}. By inspection, this is the restriction of the canonical End_{0}(A)-module map Nil_{0}(A) → End_{0}(A) of II.7.4.4, followed by the inclusion End_{0}(A) ⊂ W (A) of II.7.4.3. It follows that Nil_{0}(A) is an ideal of the ring End_{0}(A).

EXERCISES

3.1 Let A be a ring and a ∈ A, show that the following are equivalent: (i) a is nilpotent; (ii) 1 − at is a unit of A[t]; (iii) 1 − a(t − 1) is a unit of A[t, t^{−1}]. 3.2 Let α, β: P → Q be two maps between finitely generated projective R-modules. If S is a central multiplicatively closed set in R and S^{−1}α, S^{−1}β are isomorphisms, then g = β^{−1}α is an automorphism of S^{−1}P. Show that ∂(g) = [cone(α)] − [cone(β)]. In particular, if S consists of nonzerodivisors then ∂(g) = [coker(α)] − [coker(β)]. 3.3 (Bass) Prove that every module M in H(R) has a projective resolution P_{·} → M such that every automorphism α of M lifts to an automorphism of the chain complex P_{·}. To do so, proceed as follows.

(a) Fix a surjection π: Q → M, and use Ex. I.1.11 to lift the automorphism α ⊕ α^{−1} of M ⊕ M to an automorphism β of Q ⊕ Q.

(b) Defining e: Q ⊕ Q → M to be e(x, y) = π(x), show that every automorphism of M can be lifted to an automorphism of Q ⊕ Q.

(c) Set P_{0} = Q ⊕ Q, and repeat the construction on Z_{0} = ker(e) to get a finite resolution P_{·} of M with the desired property.

3.4 Suppose that S consists of nonzerodivisors, and that M is a module in H_{S}(R).

(a) Prove that there is a module M′ and an S-isomorphism α ∈ End(R^{m}) so that coker(α) = M ⊕ M′. Hint: Modify the proof of Lemma 3.1.5, where M is the cokernel of a map P_{0} ⟶^{β} P_{1}.

(b) Given S-isomorphisms α′, α′′ ∈ End(R^{m}) and a short exact sequence of Storsion modules 0 → coker(α′) → M → coker(α′′) → 0, show that there is an S-isomorphism α ∈ R^{2m} with M ≅ coker(α).

3.5 Modify the proofs of the previous two exercises to prove Theorem 3.2 when S contains zerodivisors.

3.6 Noncommutative localization. By definition, a multiplicatively closed set S in a ring R is called a right denominator set if it satisfies the following two conditions:

(i) For any s ∈ S and r ∈ R there exists an s′ ∈ S and r′ ∈ R such that sr′ = rs′; (ii) if sr = 0 for any r ∈ R, s ∈ S then rs′ = 0 for some s′ ∈ S. This is the most general condition under which a (right) ring of fractions S^{−1}R exists, in which every element of S^{−1}R has the form r/s = rs^{−1}, and if r/1 = 0 then some rs = 0 in R.

Prove Theorem 3.2 when S is a right denominator set consisting of nonzerdivisors. To do this, proceed as follows.

(a) Show that for any finite set of elements x_{i} in S^{−1}R there is an s ∈ S and r_{i} ∈ R so that x_{i} = r_{i}/s for all i.

(b) Reprove II.7.7.3 and II.9.8 for denominator sets, using (a); this yields exactness at K_{0}(R).

(c) Modify the proof of Lemma 3.1 and 3.1.5 to construct the map ∂ and prove exactness at K_{0}H_{S}(R).

(d) Modify the proof of Theorem 3.2 to prove exactness at K_{1}(S^{−1}R).

3.7 Let A be an algebra over a commutative ring R. Recall from Ex. II.7.18 that NK_{1}(A) = Nil_{0}(A) is a module over the ring W (R) = 1 + tR[[t]] of Witt vectors II.4.3. In this exercise we develop a little of the structure of W (R), which yields information about the structure of NK_{1}(A) and hence (by theorem 3.7) the structure of NK_{0}(R).

(a) If 1/p ∈ R for some prime integer p, show that W (R) is an algebra over ℤ[1/p]. Conclude that NK_{1}(A) and NK_{0}(A) are uniquely p-divisible abelian groups. Hint: use the fact that the coefficients in the power series expansion for r(t) = (1 + t)^{1/p} only involve powers of p.

(b) If ℚ ⊆ R, consider the exponential map ∏_{i=1}^{∞} R → W (R), sending (r_{1}, ...) to ∏_{i=1}^{∞} exp(−r_{i}t^{i}/i). This is an isomorphism of abelian groups, whose inverse (the “ghost map”) is given by the coefficients of f ↦ −t d/dt(ln f). Show that this is a ring isomorphism. Conclude that NK_{1}(A) and NK_{0}(A) have the structure of R-modules.

(c) If n ∈ ℤ is nonzero, Stienstra showed that NK_{1}(A)[1/n] ≅ NK_{1}(A[1/n]). Use this to show that if G is a finite group of order n then NK_{1}(ℤ[G]) is annihilated by some power of n.

3.8 If I is a nilpotent ideal in a ℚ-algebra A, show that NK_{1}(A, I) → K_{1}(A, I) is onto. Thus Ex. 3.7 gives another proof that K_{1}(A, I) is divisible (Ex. 2.9).

3.9 If s ∈ S is central, show that NK_{1}(S_{s}) is a localization of NK_{1}(S). Conclude that if S is K_{1}-regular then S is K_{0}-regular. Hint: Use the sequence of Ex. 2.6

3.10 (Karoubi) Let S be a multiplicatively closed set of central nonzerodivisors in a ring A. We say that a ring homomorphism f : A → B is an analytic isomorphism along S if f(S) consists of central nonzerodivisors in B, and if A/sA ≅ B/sB for every s ∈ S. (This implies that the s-adic completions of A and B are isomorphism, whence the name.)

If f is an analytic isomorphism along S, show that M ↦ M ⊗_{A} B defines an equivalence of categories H_{S}(A) ≅ H_{S}(B). (One proof is given in V.7.5 below.) Using Theorem 3.2 and Ex. II.9.13, this shows that we have an exact sequence

K_{1}(S^{−1}A) ⊕ K_{1}(B) → K_{1}(S^{−1}B) → K_{0}(A) → K_{0}(S^{−1}A) ⊕ K_{0}(B) → K_{0}(S^{−1}B).

28 III. K_{1} AND K_{2} OF A RING

Hint: For M in H_{S}^{1}(A), show that Tor_{1}^{A}(M, B) = 0, so that H_{S}^{1}(A) → H_{S}^{1}(B) is exact. Then use Lemma II.7.7.1 to show that H_{S}(A) is the category of modules having a finite resolution by modules in H_{S}^{1}(A), and similarly for H_{S}(B).

§4. Negative K-theory

In the last section, we defined K_{−1}(R) to be the cokernel of the map K_{0}(R[t]) ⊕ K_{0}(R[t^{−1}]) → K_{0}(R[t, t^{−1}]). Of course we can keep going, and define all the negative K-groups by induction on n:

Definition 4.1. For n > 0, we inductively define K_{−n}(R) to be the cokernel of the map

K_{−n+1}(R[t]) ⊕ K_{−n+1}(R[t^{−1}]) → K_{−n+1}(R[t, t^{−1}]).

Clearly, each K_{−n} is a functor from rings to abelian groups. It follows from Theorem II.7.8 that if R is regular noetherian then K_{n}(R) = 0 for all n < 0.

To describe the properties of these negative K-groups, it is convenient to cast the Fundamental Theorems above in terms of Bass’ notion of contracted functors. With this in mind, we make the following definitions.

Definition 4.1.1 (Contracted functors). Let F be a functor from rings to abelian groups. For each R, we define LF (R) to be the cokernel of the map F (R[t])⊕F (R[t^{−1}]) → F (R[t, t^{−1}]). We write Seq(F, R) for the following sequence, where Δ(a) = (a, a) and ±(b, c) = b − c:

0 → F (R) ⟶^{Δ} F (R[t]) ⊕ F (R[t^{−1}]) ⟶^{±} F (R[t, t^{−1}]) → LF (R) → 0.

We say that F is acyclic if Seq(F, R) is exact for all R. We say that F is a contracted functor if F is acyclic and in addition there is a splitting h = h_{t,R} of the defining map F (R[t, t^{−1}]) → LF (R), a splitting which is natural in both t and R. The notation F_{−1} is sometimes used for LF.

By iterating this definition, we can speak about the functors NLF, L^{2}F, etc. For example, Definition 4.1 states that K_{−n} = L^{n}(K_{0}).

As with the definition of NF (3.3), it will occasionally be useful to define LF etc. on a more restricted class of rings, such as commutative algebras. Suppose that R is a category of rings such that if R is in R then so are R[t], R[t, t^{−1}] and the maps R → R[t] ⇉ R[t, t^{−1}]. Then the definitions of LF, L^{n}F and Seq(F, R) in 4.1.1 make sense for any functor F from R to any abelian category.

Example 4.1.2 (Fundamental Theorem for K_{−n}). The Fundamental Theorems for K_{1} and K_{0} may be restated as the assertions that these are contracted functors. It follows from Proposition 4.2 below that each K_{−n} is a contracted functor; by Definition 4.1, this means that there is a naturally split exact sequence:

0 → K_{−n}(R) ⟶^{Δ} K_{−n}(R[t]) ⊕ K_{−n}(R[t^{−1}]) ⟶^{±} K_{−n}(R[t, t^{−1}]) ⟶^{∂} K_{−n−1}(R) → 0.

Example 4.1.3 (Units). Let U(R) = R^{×} denote the group of units in a commutative ring R. By Ex. I.3.17, U is a contracted functor with contraction LU(R) = [Spec(R), ℤ]; the splitting map LU(R) → U(R[t, t^{−1}]) sends a continuous function f: Spec(R) → ℤ to the unit t^{f} of R[t, t^{−1}]. From Ex. 4.2 below we see that the functors L^{2}U and NLU are zero. Thus we can write a simple formula for the units of any extension R[t_{1}, t_{1}^{−1}, …, t_{n}, t_{n}^{−1}]. If R is reduced, so that NU(R) vanishes (Ex. I.3.17), then we just have

U(R[t_{1}, t_{1}^{−1}, …, t_{n}, t_{n}^{−1}]) = U(R) × ∐_{i=1}^{n}[Spec(R), ℤ] · t_{i}.

Example 4.1.4 (Pic). Recall from chapter I, §3 that the Picard group Pic(R) of a commutative ring is a functor, and that N Pic(R) = 0 exactly when R_{red} is seminormal. By Ex. I.3.18 the sequence Seq(Pic, R) is exact. In fact Pic is a contracted functor with NL Pic = L^{2} Pic = 0; see [We91]. The group L Pic(R) is the étale cohomology group H_{et}^{1}(Spec(R), ℤ).

A morphism of contracted functors is a natural transformation η: F ⇒ F′ between two contracted functors such that the following square commutes for all R.

LF (R) ⟶^{h} F (R[t, t^{−1}])
(Lη)_{R}↓  ↓^{η_{R[t,t^{−1}]}}
LF′(R) ⟶^{h′} F′(R[t, t^{−1}])

Proposition 4.2. Let η: F ⇒ F′ be a morphism of contracted functors. Then both ker(η) and coker(η) are also contracted functors.

In particular, if F is contracted, then NF and LF are also contracted functors. Moreover, there is a natural isomorphism of contracted functors NLF ≅ LNF.

Proof. If C ⟶^{ϕ} D is a morphism between split exact sequences, which have compatible spittings, then the sequences ker(ϕ) and coker(ϕ) are always split exact, with splittings induced from the splittings of C and D. Applying this remark to Seq(F, R) → Seq(F′, R) shows that ker(η) and coker(η) are contracted functors: both Seq(ker(η), R) and Seq(coker(η), R) are split exact. It also shows that

0 → ker(η)(R) → F (R) ⟶^{η_{R}} F′(R) → coker(η)(R) → 0

is an exact sequence of contracted functors.

Since NF (R) is the cokernel of the morphism F (R) → F′(R) = F (R[t]) and LF (R) is the cokernel of the morphism ± in Seq(F, R), both NF and LF are contracted functors. Finally, the natural isomorphism NLF (R) ≅ LNF (R) arises from inspecting one corner of the large commutative diagram represented by

0 → Seq(F, (R[s], s)) → Seq(F, R[s]) → Seq(F, R) → 0.

Example 4.2.1 (SK_{1}). If R is a commutative ring, it follows from Examples 1.1.1 and 4.1.3 that det: K_{1}(R) → U(R) is a morphism of contracted functors. Hence SK_{1} is a contracted functor. The contracted map L det is the map rank: K_{0}(R) → H_{0}(R) = [Spec(R), ℤ] of II.2.3; it follows that L(SK_{1})(R) = K̃_{0}(R). From Ex. 4.2 we also have L^{2}(SK_{1})(R) = LK̃_{0}(R) = K_{−1}(R).

We can give an elegant formula for F (R[t_{1}, t_{1}^{−1}, …, t_{n}, t_{n}^{−1}]), using the following notation. If p(N, L) = ∑ m_{ij}N^{i}L^{j} is any formal polynomial in N and L with integer coefficients m_{ij} > 0, and F is a functor from rings to abelian groups, we set p(N, L)F equal to the direct sum of m_{ij} copies of each group N^{i}L^{j}F (R).

Corollary 4.2.2. F (R[t_{1}, …, t_{n}]) ≅ (1 + N)^{n}F (R) for every F. If F is a contracted functor, then F (R[t_{1}, t_{1}^{−1}, …, t_{n}, t_{n}^{−1}]) ≅ (1 + 2N + L)^{n}F (R).

Proof. The case n = 1 follows from the definitions; the general case follows by induction. □

For example, if L^{2}F = 0 and R is F -regular, then (1 + 2N + L)^{n}F (R) stands for F (R) ⊕ n LF (R). In particular, the formula for units in Example 4.1.3 is just the case F = U of 4.2.2.

Example 4.2.3. Since L^{j}K_{0} = K_{−j}, K_{0}(R[t_{1}, t_{1}^{−1}, …, t_{n}, t_{n}^{−1}]) is the direct sum of many pieces N^{i}K_{−j}(R), including K_{−n}(R) and \binom{n}{j} copies of K_{−j}(R).

From 3.4.4 we see that if R is K_{n}-regular for some n ≤ 0 then R is also K_{n−1}- regular. In particular, if R is K_{0}-regular then R is also K_{n}-regular for all n < 0.

Conjecture 4.2.4. Let R be a commutative noetherian ring of Krull dimension d. It is conjectured that K_{−j}(R) vanishes for all j > d, and that R is K_{−d}-regular; see [We80]. This is so for d = 0, 1 by exercises 4.3 and 4.4, and Example 4.3.1 below shows that the bound is best possible. It was recently shown to be true for ℚ-algebras in [CHSW].

The Mayer-Vietoris sequence

Suppose that f: R → S is a ring map, and I is an ideal of R mapped isomorphically into an ideal of S. By Theorem 2.6 there is an exact “Mayer-Vietoris” sequence:

K_{1}(R)⟶^{Δ} K_{1}(S)⊕K_{1}(R/I)→K_{1}(S/I)⟶^{∂} K_{0}(R)⟶^{Δ} K_{0}(S)⊕K_{0}(R/I)⟶^{±} K_{0}(S/I).

Applying the contraction operation L to this sequence gives a sequence relating K_{0} to K_{−1}, whose first three terms are identical to the last three terms of the displayed sequence. Splicing these together yields a longer sequence. Repeatedly applying L and splicing sequences leads to the following result.

Theorem 4.3 (Mayer-Vietoris). Suppose we are given a ring map f: R → S and an ideal I of R mapped isomorphically into an ideal of S. Then the Mayer-Vietoris sequence of Theorem 2.6 continues as a long exact Mayer-Vietoris sequence of negative K-groups.

⟶^{Δ}[{K_{0}(S) ⊕ \atop K_{0}(R/I)}]⟶^{±}K_{0}(S/I)⟶^{∂} K_{−1}(R)⟶^{Δ} [{K_{−1}(S) ⊕ \atop K_{−1}(R/I)}]⟶^{±} K_{−1}(S/I)⟶^{∂} K_{−2}(R) →
⋯ → K_{−n+1}(S/I)⟶^{∂} K_{−n}(R)⟶^{Δ} [{K_{−n}(S) ⊕ \atop K_{−n}(R/I)}]⟶^{±} K_{−n}(S/I)⟶^{∂} K_{−n−1}(R) ⟶ ⋯

Example 4.3.1 (B. Dayton). Fix a regular ring R, and let Δ^{n}(R) denote the coordinate ring R[t_{0}, …, t_{n}]/(f), f = t_{0} ⋯ t_{n}(1 − ∑ t_{i}) of the n-dimensional tetrahedron over R. Using I = (1 − ∑ t_{i})Δ^{n}(R) and Δ^{n}(R)/I ≅ R[t_{1}, …, t_{n}] via t_{0} ↦ 1 − (t_{1} + ⋯ + t_{n}), we have a Milnor square

Δ^{n}(R) → A_{n}
↓  ↓
R[t_{1}, …, t_{n}] → Δ^{n−1}(R)

where A_{n} = R[t_{0}, …, t_{n}]/(t_{0} ⋯ t_{n}). By Ex. 4.8, the negative K-groups of A_{n} vanish and K_{i}(A_{n}) =K_{i}(R) for i = 0, 1. Thus K_{0}(Δ^{n}(R)) ≅K_{0}(R)⊕K_{1}(Δ^{n−1}(R))/K_{1}(R) for n > 0, and K_{−j}(Δ^{n}(R)) ≅ K_{1−j}(Δ^{n−1}(R)) for j > 0. These groups vanish for j > n, with K_{−n}(Δ^{n}(R)) ≅ K_{0}(R). In particular, if F is a field then Δ^{n}(F) is an n-dimensional noetherian ring with K_{−n}(Δ^{n}(F)) ≅ ℤ; see Conjecture 4.2.4.

When we have introduced higher K-theory, we will see that in fact K_{0}(Δ^{n}(R)) ≅ K_{n}(R) and K_{1}(Δ^{n}(R)) ≅ K_{n+1}(R). (See IV, Ex. 12.1.) This is just one way in which higher K-theory appears in classical K-theory.

Theories of Negative K-theory

Here is an alternative approach to defining negative K-theory, due to Karoubi and Villamayor [KV71].

Definition 4.4. A theory of negative K-theory for (nonunital) rings consists of a sequence of functions K_{n} (n ≤ 0) from nonunital rings to abelian groups, together with natural boundary maps ∂ : K_{n}(R/I) → K_{n−1}(I) for every 2-sided ideal I ⊂ R, satisfying the following axioms.

(1) K_{0}(R) is the Grothendieck group of chapter II;

(2) K_{n}(I) → K_{n}(R) → K_{n}(R/I) ⟶^{∂} K_{n−1}(I) → K_{n−1}(R) is exact for every I ⊂ R;

(3) If Λ is a flasque ring (II.2.1.3), then K_{n}(Λ) = 0 for all n ≤ 0;

(4) The inclusion R ⊂ M(R) = ∪M_{m}(R) induces an isomorphism K_{n}(R) ≅ K_{n}M(R) for each n ≤ 0.

Example 4.4.1. Bass’ negative K-groups (4.1) form a theory of negative Ktheory for rings. This follows from the contraction of 2.3 (see Ex. 4.5), Ex. 4.9 and the contraction of Morita Invariance 1.6.4.

Example 4.4.2. Embedding M(R) as an ideal in a flasque ring Λ, axiom (2) shows that K_{−1}R ≅ K_{0}(Λ/M(R)). This was the approach used by Karoubi and Villamayor in [KV71] to inductively define a theory of negative K-theory; see Ex. 4.10.

Example 4.4.3. If A is a hensel local ring then K_{−1}(A) = 0. This was proven by Drinfeld in [Drin], using a Calkin category model for negative K-theory.

Theorem 4.5. Every theory of negative K-theory for rings is canonically isomorphic to the negative K-theory.

Proof. Suppose that {K′_{n}} is another theory of negative K-theory for rings. We will show that there are natural isomorphisms h_{n}(A) : K_{n}(A) → K′_{n}(A) commuting with the boundary operators. By induction, we may assume that h_{n} is given. Since C(R) is flasque (II.2.1.3), and S(R) = C(R)/M(R), the axioms yield isomorphisms ∂ : K_{n}S(R) ≅ K_{n−1}(R) and ∂′ : K′_{n}S(R) ≅ K′_{n−1}(R). We define h_{n−1}(R) : K_{n−1}(R) → K′_{n−1}(R) to be ∂′ ∘ h_{n}(SR) ∘ ∂^{−1}.

It remains to check that the h_{n} commute with the boundary maps associated to an ideal I ⊂ R. Since M(I) is an ideal in C(I), C(R) and M(R), the axioms yield K_{n}S(I) ≅ K_{n}C(R)/M(I) and similarly for K′_{n}. The naturality of ∂ and ∂′ relative to M(R/I) = M(R)/M(I) → C(R)/M(I) yield the diagram

K_{n}(R/I) ⟶^{≅} K_{n}M(R/I) ⟶ K_{n}C(R)/M(I) ⟵^{≅} K_{n}S(I) ⟶^{∂} K_{n−1}(I)
h_{n}(R/I)↓≅  ↓≅  ≅↓  ≅↓  ↓^{h_{n−1}(I)}
K′_{n}(R/I) ⟶^{≅} K′_{n}M(R/I) ⟶ K′_{n}C(R)/M(I) ⟵^{≅} K′_{n}S(I) ⟶^{∂′} K′_{n−1}(I).

Since the horizontal composites are the given maps ∂ : K_{n}(R/I) → K_{n−1}(I) and ∂′ : K′_{n}(R/I) → K′_{n−1}(I), we have the desired relation: ∂′h_{n}(R/I) ≅ h_{n−1}(I) ∂.

EXERCISES

4.1 Suppose that 0 → F′ → F → F′′ → 0 is an exact sequence of functors, with F′ and F′′ contracted. Show that F is acyclic, but need not be contracted. 4.2 For a commutative ring R, let H_{0}(R) denote the group [Spec(R), ℤ] of all continuous functions from Spec(R) to ℤ. Show that NH_{0} = LH_{0} = 0, i.e., that H_{0}(R) = H_{0}(R[t]) = H_{0}(R[t, t^{−1}]). 4.3 Let R be an Artinian ring. Show that R is K_{0}-regular, and that K_{−n}(R) = 0 for all n > 0. 4.4 (Bass-Murthy) Let R be a 1-dimensional commutative noetherian ring with finite normalization R̃ and conductor ideal I. Show that R is K_{−1}-regular, and that K_{−n}(R) = 0 for all n ≥ 2. If h_{0}(R) denotes the rank of the free abelian group H_{0}(R) = [Spec(R), ℤ], show that K_{−1}(R) ≅ L Pic(R) ≅ ℤ^{r}, where r = h_{0}(R) − h_{0}(R̃) + h_{0}(R̃/I) − h_{0}(R/I).

Now suppose that R is any 1-dimensional commutative noetherian ring. Even if its normalization is not finitely generated over R, show that R is K_{−1}-regular, and that K_{−n}(R) = 0 for all n ≥ 2. 4.5 (Carter) Let f : R → R′ be a ring homomorphism. In II.2.10 we defined a group K_{0}(f) and showed in Ex. 1.14 that it fits into an exact sequence

K_{1}(R) ⟶^{f^{∗}} K_{1}(R′) → K_{0}(f) → K_{0}(R) ⟶^{f^{∗}} K_{0}(R′).

Show that A ↦ K_{0}(f ⊗ A) defines a functor on commutative rings A, and define K_{−n}(f) to be L^{n}K_{0}(f ⊗−). Show that each K_{−n}(f) is an acyclic functor, and that the above sequence continues into negative K-theory as:

⋯ → K_{0}(R) → K_{0}(R′) ⟶^{∂} K_{−1}(f) → K_{−1}(R) →
K_{−1}(R′) ⟶^{∂} K_{−2}(f) → K_{−2}(R) → ⋯

With the help of higher K-theory to define K_{1}(f) and to construct the product “·t”, it will follow that K_{0}(f) and hence every K_{−n}(f) is a contracted functor. 4.6 Let T : P(R) → P(R′) be any cofinal additive functor. Show that the functor K_{0}(T) of II.2.10 and its contractions K_{−n}(T) are acyclic, and that they extend the sequence of Ex. 1.14 into a long exact sequence, as in the previous exercise.

When T is the endofunctor ·m of 1.7.4, we write K_{0}(R; ℤ/m) for LK_{0}(·m) and K_{−n}(R; ℤ/m) for L^{n+1}K_{0}(·m). Show that the sequence of 1.7.4 extends to a long exact sequence

K_{0}(R) ⟶^{m} K_{0}(R) → K_{0}(R; ℤ/m) → K_{−1}(R) ⟶^{m} K_{−1}(R) → K_{−1}(R; ℤ/m) → · · · .

4.7 Let G be a finite group of order n, and let R̃ be a “maximal order” in ℚ[G]. It is well known that R̃ is a regular ring containing ℤ[G], and that I = nR̃ is an ideal of ℤ[G]; see [Bass, p. 560]. Show that K_{−n}ℤ[G] = 0 for n ≥ 2, and that K_{−1} has the following resolution by free abelian groups:

0 → ℤ → H_{0}(R̃) ⊕ H_{0}(ℤ/n[G]) → H_{0}(R̃/nR̃) → K_{−1}(ℤ[G]) → 0.

D. Carter has shown in [Carter] that K_{−1}ℤ[G] ≅ ℤ^{r} ⊕ (ℤ/2ℤ)^{s}, where s equals the number of simple components M_{n_{i}}(D_{i}) of the semisimple ring ℚ[G] such that the Schur index of D is even (see 1.2.4), but the Schur index of D_{p} is odd at each prime p dividing n. In particular, if G is abelian then K_{−1}ℤ[G] is torsionfree (see [Bass, p. 695]). 4.8 Coordinate hyperplanes. Let R be a regular ring. By induction on n, show that the graded rings A_{n} = R[t_{0}, …, t_{n}]/(t_{0} ⋯ t_{n}) are K_{i}-regular for all i ≤ 1. Conclude that K_{1}(A_{n}) = K_{1}(R), K_{0}(A_{n}) = K_{0}(R) and K_{i}(A_{n}) = 0 for all i < 0.

Show that the rings Δ^{n}(R) of 4.4.3 are also K_{1}-regular. 4.9 Let Λ be a flasque ring. Show that Λ[t, t^{−1}] is also flasque, and conclude that K_{n}(Λ) = 0 for all n ≤ 0. 4.10 (Karoubi) Recall from Ex. 1.15 that the suspension ring S(R) satisfies ∂ : K_{1}(S(R)) ≅ K_{0}(R). For each n ≥ 0, set K_{0}S^{n}(R) = K_{0}(S^{n}(R)). Show that the functors {K′_{n} = K_{0}S^{−n}} form a theory of negative K-theory for rings, and conclude that K_{n}(R) ≅ K_{0}(S^{n}(R)). 4.11 (Karoubi) Let f : A → B be an analytic isomorphism along S in the sense of Ex. 3.10. Using Ex. 4.5, show that there is an exact sequence for all n ≤ 0, continuing the sequence of Ex. 3.10:

⋯ → K_{n+1}(S^{−1}A) ⊕ K_{n+1}(B) → K_{n+1}(S^{−1}B) →
K_{n}(A) → K_{n}(S^{−1}A) ⊕ K_{n}(B) → K_{n}(S^{−1}B) → · · · .

4.12 (Reid) Let f = y^{2} − x^{3} + x^{2} in k[x, y] and set B = k[x, y]/(f). Using Theorem 4.3, show that K_{−1}(B) ≅ K_{−1}(B_{(x,y)}) ≅ ℤ. Let A be the subring k + 𝔪 of k[x, y], where 𝔪 = fk[X, y]; show that K_{−2}(A) ≅ K_{−2}(A_{𝔪}) ≅ ℤ. Writing the integrally closed ring A as the union of finitely generated normal subrings k[f, xf, yf, ...], conclude that there is a 2-dimensional normal ring A_{0}, finitely generated over k, with K_{−2}(A_{0}) ≠ 0. 4.13 (Reid) We saw in Example 4.4.3 that K_{−1}(A) = 0 for every hensel local ring. In this exercise we construct a complete local 2-dimensional ring with K_{−2}(Â) ≠ 0. Let A be the ring of Exercise 4.12, and Â its completion at the maximal ideal 𝔪. Let Â_{f} denote the completion of A at the ideal Af. Using Ex. 4.10, show that K_{−2}(A) ≅ K_{−2}(Â_{f}) ≅ K_{−2}(Â), and hence that K_{−2}(Â) ≠ 0.

§5. K_{2} of a ring

The group K_{2} of a ring was defined by J. Milnor in 1967, following a 1962 paper by R. Steinberg on Universal Central Extensions of Chevalley groups. Milnor’s 1971 book [Milnor] is still the best source for the fundamental theorems about it. In this section we will give an introduction to the subject, but we will not prove the harder theorems.

Following Steinberg, we define a group in terms of generators and relations designed to imitate the behavior of the elementary matrices, as described in (1.2.1). To avoid technical complications, we shall avoid any definition of St_{2}(R).

Definition 5.1. For n ≥ 3 the Steinberg group St_{n}(R) of a ring R is the group defined by generators x_{ij}(r), with i, j a pair of distinct integers between 1 and n and r ∈ R, subject to the following “Steinberg relations” (5.1.1) x_{ij}(r)x_{ij}(s) = x_{ij}(r + s)

(5.1.2) [x_{ij}(r), x_{kℓ}(s)] = \{ \begin{matrix}1 & if j ≠ k and i ≠ ℓ \\ x_{iℓ}(rs) & if j = k and i ≠ ℓ \\ x_{kj}(−sr) & if j ≠ k and i = ℓ. \end{matrix}

As observed in (1.3.1), the Steinberg relations are also satisfied by the elementary matrices e_{ij}(r) which generate the subgroup E_{n}(R) of GL_{n}(R). Hence there is a canonical group surjection ϕ_{n}: St_{n}(R) → E_{n}(R) sending x_{ij}(r) to e_{ij}(r).

The Steinberg relations for n+1 include the Steinberg relations for n, so there is an obvious map St_{n}(R) → St_{n+1}(R). We write St(R) for \underrightarrow{lim} St_{n}(R), and observe that by stabilizing the ϕ_{n} induce a surjection ϕ: St(R) → E(R).

Definition 5.2. The group K_{2}(R) is the kernel of ϕ: St(R) → E(R). Thus there is an exact sequence of groups

1 → K_{2}(R) → St(R) ⟶^{ϕ} GL(R) → K_{1}(R) → 1.

It will follow from Theorem 5.3 below that K_{2}(R) is an abelian group. Moreover, it is clear that St and K_{2} are both covariant functors from rings to groups, just as GL and K_{1} are.

Theorem 5.2.1 (Steinberg). K_{2}(R) is an abelian group. In fact it is precisely the center of St(R).

Proof. If x ∈ St(R) commutes with every element of St(R), then ϕ(x) must commute with all of E(R). But the center of E(R) is trivial (by Ex. 1.8) so ϕ(x) = 1, i.e., x ∈ K_{2}(R). Thus the center of St(R) is contained in K_{2}(R).

Conversely, suppose that y ∈ St(R) satisfies ϕ(y) = 1. Then in E(R) we have

ϕ([y, p]) = ϕ(y)ϕ(p)ϕ(y)^{−1}ϕ(p)^{−1} = ϕ(p)ϕ(p)^{−1} = 1

for every p ∈ St(R). Choose an integer n large enough that y can be expressed as a word in the symbols x_{ij}(r) with i, j < n. For each element p = x_{kn}(s) with k < n and s ∈ R, the Steinberg relations imply that the commutator [y, p] is an element of the subgroup P_{n} of St(R) generated by the symbols x_{in}(r) with i < n. On the other hand, we know by Ex. 5.2 that ϕ maps P_{n} injectively into E(R). Since ϕ([y, p]) = 1 this implies that [y, p] = 1. Hence y commutes with every generator x_{kn}(s) with k < n.

By symmetry, this proves that y also commutes with every generator x_{nk}(s) with k < n. Hence y commutes with all of St_{n}(R), since it commutes with every x_{kl}(s) = [x_{kn}(s), x_{nl}(1)] with k, l < n. Since n can be arbitrarily large, this proves that y is in the center of St(R).

Example 5.2.2. The group K_{2}(ℤ) is cyclic of order 2. This calculation uses the Euclidean Algorithm to rewrite elements of St(ℤ), and is given in §10 of [Milnor]. In fact, Milnor proves that the symbol {−1, −1} = {x_{12}(1)x_{21}(−1)x_{12}(1)}^{4} is the only nontrivial element of ker(ϕ_{n}) for all n ≥ 3. It is easy to see that {−1, −1} is in the kernel of each ϕ_{n}, because the 2 × 2 matrix e_{12}(1)e_{21}(−1)e_{12}(1) = ({0 \atop −1} {1 \atop 0}) has order 4 in GL_{n}(ℤ). We will see in Example 6.2.1 below that {−1, −1} is still nonzero in K_{2}(ℝ).

Tate has used the same Euclidean Algorithm type techniques to show that K_{2}(ℤ[\sqrt{−7}]) and K_{2}(ℤ[\sqrt{−15}]) are also cyclic of order 2, generated by the symbol \{−1, −1\}, while K_{2}(R) = 1 for the imaginary quadratic rings R = ℤ[i], ℤ[\sqrt{−3}], ℤ[\sqrt{−2}] and ℤ[\sqrt{−11}]. See the appendix to [BT] for details.

Example 5.2.3. For every field F we have K_{2}(F [t]) = K_{2}(F). This was originally proven by R. K. Dennis and J. Sylvester using the same Euclidean Algorithm type techniques as in the previous example. We shall not describe the details, because we shall see in chapter V that K_{2}(R[t]) = K_{2}(R) for every regular ring.

Universal Central Extensions

The Steinberg group St(R) can be described in terms of universal central extensions, and the best exposition of this is [Milnor, §5]. Properly speaking, this is a subject in pure group theory; see [Suz, 2.9]. However, since extensions of a group G are classified by the cohomology group H^{2}(G), the theory of universal central extensions is also a part of homological algebra; see [WHomo, §6.9]. Here are the relevant definitions.

Let G be a group and A an abelian group. A central extension of G by A is a short exact sequence of groups 1 → A → X ⟶^{π} G → 1 such that A is in the center of X. We say that a central extension is split if it is isomorphic to an extension of the form 1 → A → A × G ⟶^{pr} G → 1, where pr(a, g) = g.

If A or π is clear from the context, we may omit it from the notation. For example, 1 → K_{2}(R) → St(R) → E(R) → 1 is a central extension by Steinberg’s Theorem 5.2.1, but we usually just say that St(R) is a central extension of E(R).

Two extensions X and Y of G by A are said to be equivalent if there is an isomorphism f: X → Y which is the identity on A and which induces the identity map on G. It is well-known that the equivalence classes of central extensions of G by a fixed group A are in 1–1 correspondence with the elements of the cohomology group H^{2}(G; A); see [WHomo, §6.6].

More generally, by a homomorphism over G from X ⟶^{π} G to another central extension 1 → B → Y ⟶^{τ} G → 1 we mean a group map f: X → Y such that π = τf.

Definition 5.3.1. A universal central extension of G is a central extension X ⟶^{π} G such that for every other central extension Y ⟶^{τ} G there is a unique homomorphism f over G from X to Y. Clearly a universal central extension is unique up to isomorphism over G, provided it exists.

Lemma 5.3.2. If G has a universal central extension X, then both G and X must be perfect groups.

Proof. Otherwise B = X/[X, X] is nontrivial, and there would be two homomorphisms over G from X to the central extension 1 → B → B × G → G → 1, namely the maps (0, π) and (pr, π), where pr is the natural projection X → B.

Lemma 5.3.3. If X and Y are central extensions of G, and X is a perfect group, there is at most one homomorphism over G from X to Y.

Proof. If f and f_{1} are two such homomorphisms, then for any x and x′ in X we can write f_{1}(x) = f(x)c, f_{1}(x′) = f(x′)c′ for elements c and c′ in the center of Y. Therefore f_{1}(xx′x^{−1}(x′)^{−1}) = f(xx′x^{−1}(x′)^{−1}). Since the commutators [x, x′] = xx′x^{−1}(x′)^{−1} generate X we must have f_{1} = f.

Example 5.3.4. Every presentation of G gives rise to two natural central extensions as follows. A presentation corresponds to the choice of a free group F mapping onto G, and a description of the kernel R ⊂ F. Since [R, F] is a normal subgroup of F, we may form the following central extensions:

1 → R/[R, F] →F/[R, F] → G → 1,
(5.3.5)  1 → (R ∩ [F, F])/[R, F] →[F, F]/[R, F] → [G, G] → 1.

The group (R ∩ [F, F])/[R, F] in (5.3.5) is the homology group H_{2}(G; ℤ); this identity was discovered in 1941 by Hopf [WHomo, 6.8.8]. If G = [G, G], both are extensions of G, and (5.3.5) is the universal central extension by the following theorem.

Recognition Theorem 5.4. Every perfect group G has a universal central extension, namely the extension (5.3.5):

1 → H_{2}(G; ℤ) → [F, F]/[R, F] → G → 1.

Let X be any central extension of G, the following are equivalent: (1) X is a universal central extension; (2) X is perfect, and every central extension of X splits; (3) H_{1}(X; ℤ) = H_{2}(X; ℤ) = 0.

Proof. Given any central extension X of G, the map F → G lifts to a map h: F → X because F is free. Since h(R) is in the center of X, h([R, F]) = 1. Thus h induces a map from [F, F]/[R, F] to X over G. This map is unique by Lemma 5.3.3. This proves that (5.3.5) is a universal central extension, and proves the equivalence of (1) and (3). The implication (1) ⇒ (2) is Lemma 5.3.2 and Ex. 5.7, and (2) ⇒ (1) is immediate.

Theorem 5.5 (Kervaire, Steinberg). The Steinberg group St(R) is the universal central extension of E(R). Hence

K_{2}(R) ≅ H_{2}(E(R); ℤ).

This theorem follows immediately from the Recognition Theorem 5.4, and the following splitting result:

Proposition 5.5.1. If n ≥ 5, every central extension Y ⟶^{π} St_{n}(R) is split. Hence St_{n}(R) is the universal central extension of E_{n}(R).

Proof. We first show that if j ≠ k and l ≠ i then every two elements y, z ∈ Y with π(y) = x_{ij}(r) and π(z) = x_{kl}(s) must commute in Y. Pick t distinct from i, j, k, l and choose y′, y′′ ∈ Y with π(y′) = x_{it}(1) and π(y′′) = x_{tj}(r). The Steinberg relations imply that both [y′, z] and [y′′, z] are in the center of Y, and since π(y) = π[y′, y′′] this implies that z commutes with [y′, y′′] and y.

We now choose distinct indices i, j, k, l and elements u, v, w ∈ Y with

π(u) = x_{ij}(1), π(v) = x_{jk}(s) and π(w) = x_{kl}(r).

If G denotes the subgroup of Y generated by u, v, w then its commutator subgroup [G, G] is generated by elements mapping under π to x_{ik}(s), x_{jl}(sr) or x_{il}(sr). From the first paragraph of this proof it follows that [u, w] = 1 and that [G, G] is abelian. By Ex. 5.3 we have [[u, v], w] = [u, [v, w]]. Therefore if π(y) = x_{ik}(s) and π(z) = x_{jl}(sr) we have [y, w] = [u, z]. Taking s = 1, this identity proves that the element

y_{il}(r) = [u, z], where π(u) = x_{ij}(1), π(z) = x_{jl}(r)

doesn’t depend upon the choice of j, nor upon the lifts u and z of x_{ij}(1) and x_{jl}(r).

We claim that the elements y_{ij}(r) satisfy the Steinberg relations, so that there is a group homomorphism St_{n}(R) → Y sending x_{ij}(r) to y_{ij}(r). Such a homomorphism will provide the desired splitting of the extension π. The first paragraph of this proof implies that if j ≠ k and l ≠ i then y_{ij}(r) and y_{kl}(s) commute. The identity [y, w] = [u, z] above may be rewritten as

[y_{ik}(r), y_{kl}(s)] = y_{il}(rs) for i, k, l distinct.

The final relation y_{ij}(r)y_{ij}(s) = y_{ij}(r+s) is a routine calculation with commutators left to the reader.

Remark 5.5.2 (Stability for K_{2}). The kernel of St_{n}(R) → E_{n}(R) is written as K_{2}(n, R), and there are natural maps K_{2}(n, R) → K_{2}(R). If R is noetherian of dimension d, or more generally has sr(R) = d+1, then the following stability result holds: K_{2}(n, R) ≅ K_{2}(R) for all n ≥ d + 3. This result evolved in the mid-1970’s as a sequence of results by Dennis, Vaserstein, van der Kallen and Suslin-Tulenbaev. We refer the reader to section 19C25 of Math Reviews for more details.

Transfer maps on K_{2}

Here is a description of K_{2}(R) in terms of the translation category tP(R) of finitely generated projective R-modules, analogous to the description given for K_{1} in Corollary 1.6.3.

Proposition 5.6 (Bass). K_{2}(R) ≅ \underrightarrow{lim}_{P ∈tP} H_{2}([Aut(P), Aut(P)]; ℤ).

Proof. If G is a group, then G acts by conjugation upon [G, G] and hence upon the homology H_{2}([G, G]; ℤ). Taking coinvariants, we obtain the functor H′_{2} from groups to abelian groups defined by H′_{2}(G) = H_{0}(G; H_{2}([G, G]; ℤ)). By construction, G acts trivially upon H′_{2}(G) and commutes with direct limits of groups.

Note that if G acts trivially upon H_{2}([G, G]; ℤ) then H′_{2}(G) = H_{2}([G, G]; ℤ). For example, GL(R) acts trivially upon the homology of E(R) = [GL(R), GL(R)] by Ex. 1.13. By Theorem 5.5 this implies that H′_{2}(GL(R)) = H_{2}(E(R); ℤ) = K_{2}(R).

Since morphisms in the translation category tP(R) are well-defined up to isomorphism, it follows that P ↦ H′_{2}(Aut(P)) is a well-defined functor from tP(R) to abelian groups. Hence we can take the filtered colimit of this functor, as we did in 1.6.3. Since the free modules are cofinal in tP(R), the result follows from the identification of the colimit as

lim_{n→∞} H′_{2}(GL_{n}(R)) ≅ H′_{2}(GL(R)) = K_{2}(R).

Corollary 5.6.1 (Morita Invariance of K_{2}). The group K_{2}(R) depends only upon the category P(R). That is, if R and S are Morita equivalent rings (see II.2.7) then K_{2}(R) ≅ K_{2}(S). In particular, the maps R → M_{n}(R) induce isomorphisms on K_{2}:

K_{2}(R) ≅ K_{2}(M_{n}(R)).

Corollary 5.6.2. Any additive functor T : P(S) → P(R) induces a homomorphism K_{2}(T): K_{2}(S) → K_{2}(R), and T_{1} ⊕ T_{2} induces the sum K_{2}(T_{1}) + K_{2}(T_{2}).

Proof. The proof of 1.7 goes through, replacing H_{1}(Aut P) by H′_{2}(Aut P).

Corollary 5.6.3 (Finite transfer). Let f: R → S be a ring homomorphism such that S is finitely generated projective as an R-module. Then the forgetful functor P(S) → P(R) induces a natural transfer homomorphism f_{∗}: K_{2}(S) → K_{2}(R).

If R is commutative, so that K_{2}(R) is a K_{0}(R)-module by Ex. 5.4, the composition f_{∗}f^{∗}: K_{2}(R) → K_{2}(S) → K_{2}(R) is multiplication by [S] ∈ K_{0}(R). In particular, if S is free of rank n, then f_{∗}f^{∗} is multiplication by n.

Proof. The composite f_{∗}f^{∗} is obtained from the self-map T (P) = P ⊗_{R} S of P(S). It induces the self-map ⊗_{R}S on tP(R) giving rise to multiplication by [S] on K_{2}(R) in Ex. 5.5.

We will see in chapter Vthat we can also define a transfer map K_{2}(S) → K_{2}(R) when S is a finite R-algebra of finite projective dimension over R.

Example 5.6.4. Let D be a division algebra of dimension d = n^{2} over its center F. As in Example 1.7.2, the transfer i_{∗} : K_{2}(D) → K_{2}(F) has a kernel of exponent n^{2}, since i^{∗}i_{∗} is induced by the functor T (M) = M ⊗_{D} (D ⊗_{F} D) ≅ M^{d} and hence is multiplication by n^{2}.

If E is a splitting field for D, the construction of Ex. 1.17 yields a natural map θ_{E} : K_{2}(E) → K_{2}(D). If n is squarefree, Merkurjev and Suslin construct a reduced norm N_{red} : K_{2}(D) → K_{2}(F) such that N_{red}θ_{E} = N_{E/F}; see [MS]. If K_{2}(F) → K_{2}(F) is injective, it is induced by the norm map K_{2}(D) → K_{2}(E), as in 1.2.4.

Relative K_{2} and relative Steinberg groups

Given an ideal I in a ring R, we may construct the augmented ring R ⊕ I, with multiplication (r, x)(s, y) = (rs, ry + xs + xy). This ring is equipped with two natural maps pr, add: R ⊕ I → R, defined by pr(r, x) = r and add(r, x) = r + x. This “double” ring was used to define the relative group K_{0}(I) in Ex. II.2.3.

Let St′(R, I) denote the normal subgroup of St(R ⊕ I) generated by all x_{ij}(0, v) with v ∈ I. Clearly there is a map from St′(R, I) to the subgroup E(R ⊕ I, 0 ⊕ I) of GL(R ⊕ I) (see Lemma 2.1), and an exact sequence

1 → St′(R, I) → St(R ⊕ I) ⟶^{pr} St(R) → 1.

The following definition is taken from [Keu78] and [Lo78], and modifies [Milnor].

Definition 5.7. The relative Steinberg group St(R, I) is defined to be the quotient of St′(R, I) by the normal subgroup generated by all “cross-commutators” [x_{ij}(0, u), x_{kl}(v, −v)] with u, v ∈ I.

Clearly the homomorphism St(R⊕I) ⟶^{add} St(R) sends these cross-commutators to 1, so it induces a homomorphism St(R, I) ⟶^{add} St(R) whose image is the normal subgroup generated by the x_{ij}(v), v ∈ I. By the definition of E(R, I), the surjection St(R) → E(R) maps St(R, I) onto E(R, I). We define K_{2}(R, I) to be the kernel of the map St(R, I) → E(R, I).

Theorem 5.7.1. If I is an ideal of a ring R, then the exact sequence of Proposition 2.3 extends to an exact sequence

K_{2}(R, I) → K_{2}(R) → K_{2}(R/I) → K_{1}(R, I) → K_{1}(R) → K_{1}(R/I) → K_{0}(I) ⋯

Proof. We have a commutative diagram with exact rows:

K_{2}(R, I) ⟶ St(R, I) ⟶ GL(I) ⟶ K_{1}(R, I)
↓  ^{add}↓  ^{into}↓  ↓
K_{2}(R) ⟶ St(R) ⟶ GL(R) ⟶ K_{1}(R)
↓  ^{onto}↓  ↓  ↓
K_{2}(R/I) ⟶ St(R/I) ⟶ GL(R/I) ⟶ K_{1}(R/I)

The exact sequence now follows from the Snake Lemma and Ex. 5.1.

If I and J are ideals in a ring R with I ∩ J = 0, we may also consider I as an ideal of R/J. As in §1, these rings form a Milnor square:

R ⟶ R/J
↓  ↓
R/I ⟶ R/(I + J).

Theorem 5.8 (Mayer-Vietoris). If I and J are ideals of R with I ∩ J = 0, then the Mayer-Vietoris sequence of Theorem 2.6 can be extended to K_{2}:

\begin{matrix}K_{2}(R) ⟶^{Δ} & K_{2}(R/I) ⊕ K_{2}(R/J) ⟶^{±} K_{2}(R/I + J) ⟶^{∂} \\ K_{1}(R) ⟶^{Δ} & K_{1}(R/I) ⊕ K_{1}(R/J) ⟶^{±} K_{1}(R/I + J) ⟶^{∂} K_{0}(R) ⟶ ⋯ \end{matrix}

Proof. By Ex. 5.10, we have the following commutative diagram:

K_{2}(R, I) ⟶ K_{2}(R) ⟶ K_{2}(R/I) ⟶ K_{1}(R, I) ⟶ K_{1}(R) ⟶ K_{1}(R/I)
onto↓  ↓  ↓  ∥  ↓  ↓
K_{2}(R/J, I) ⟶ K_{2}(R/J) ⟶ K_{2}(\frac{R}{I+J}) ⟶ K_{1}(R/J, I) ⟶ K_{1}(R/J) ⟶ K_{1}(\frac{R}{I+J})

By chasing this diagram, we obtain the exact Mayer-Vietoris sequence.

Commutative Banach Algebras

Let R be a commutative Banach algebra over the real or complex numbers. Just as SK_{1}(R) = π_{0}SL(R) and K_{1}(R) surjects onto π_{0}GL(R) (by 1.5 and 1.5.1), there is a relation between K_{2}(R) and π_{1}GL(R).

Proposition 5.9. Let R be a commutative Banach algebra. Then there is a surjection from K_{2}(R) onto π_{1}SL(R) = π_{1}E(R).

Proof. ([Milnor, p.59]) By Proposition 1.5, we know that E_{n}(R) is the path component of the identity in the topological group SL_{n}(R), so π_{1}SL(R) = π_{1}E(R). Using the exponential map M_{n}(R) → GL_{n}(R), we see that E_{n}(R) is locally contractible, so it has a universal covering space Ẽ_{n}. The group map Ẽ_{n} → E_{n}(R) is a central extension with kernel π_{1}E_{n}(R). Taking the direct limit as n → ∞, we get a central extension 1 → π_{1}E(R) → Ẽ → E(R) → 1. By universality, there is a unique homomorphism ϕ̃: St(R) → Ẽ over E(R), and hence a unique map K_{2}(R) → π_{1}E(R). Thus it suffices to show that ϕ̃ is onto.

The map ϕ̃ may be constructed explicitly as follows. Let ẽ_{ij}(r) ∈ Ẽ be the endpoint of the path which starts at 1 and lifts the path t ↦ e_{ij}(tr) in E(R). We claim that the map ϕ̃ sends x_{ij}(r) to ẽ_{ij}(r). To see this, it suffices to show that the Steinberg relations (5.1) are satisfied. But the paths ẽ_{ij}(tr)ẽ_{ij}(ts) and [ẽ_{ij}(tr), ẽ_{kl}(s)] cover the two paths e_{ij}(tr)e_{ij}(s) and [e_{ij}(tr), e_{kl}(s)] in E(R). Evaluating at t = 1 yields the Steinberg relations.

By Proposition 1.5 there is a neighborhood U_{n} of 1 in SL_{n}(R) in which we may express every matrix g as a product of elementary matrices e_{ij}(r), where r depends continuously upon g. Replacing each e_{ij}(r) with ẽ_{ij}(r) defines a continuous lifting of U_{n} to Ẽ_{n}. Therefore the image of each map ϕ̃: St_{n}(R) → Ẽ_{n} contains a neighborhood Ũ_{n} of 1. Since any open subset of a connected group (such as Ẽ_{n}) generates the entire group, this proves that each ϕ̃_{n} is surjective. Passing to the limit as n → ∞, we see that ϕ̃: St(R) → Ẽ is also surjective.

Example 5.9.1. If R = ℝ then π_{1}SL(ℝ) ≅ π_{1}SO is cyclic of order 2. It follows that K_{2}(ℝ) has at least one nontrivial element. In fact, the symbol {−1, −1} of Example 5.1.1 maps to the nonzero element of π_{1}SO. We will see in 6.8.3 below that the kernel of K_{2}(ℝ) → π_{1}SO is a uniquely divisible abelian group with uncountably many elements.

Example 5.9.2. Let X be a compact space with a nondegenerate basepoint. By Ex. II.3.11, we have KO^{−2}(X) ≅ [X, ΩSO] = π_{1}SL(ℝ^{X}), so K_{2}(ℝ^{X}) maps onto the group KO^{−2}(X).

Similarly, since ΩU ≃ ℤ × ΩSU, we see by Ex. II.3.11 that KU^{−2}(X) ≅ [X, ΩU] = [X, ℤ] × [X, ΩSU]. Since π_{1}SL(ℂ^{X}) = π_{1}(SU^{X}) = [X, ΩSU] and [X, ℤ] is a subgroup of ℂ^{X}, we can combine Proposition 5.9 with Example 1.5.3 to obtain the exact sequence

K_{2}(ℂ^{X}) → KU^{−2}(X) → ℂ^{X} ⟶^{exp} K_{1}(ℂ^{X}) → KU^{−1}(X) → 0.

Steinberg symbols

If two matrices A, B ∈ E(R) commute, we can construct an element in K_{2}(R) by lifting their commutator to St(R). To do this, choose a, b ∈ St(R) with ϕ(a) = A, ϕ(b) = B and define A⋆B = [a, b] ∈ K_{2}(R). This definition is independent of the choice of a and b because any other lift will equal ac, bc′ for central elements c, c′, and [ac, bc′] = [a, b].

If P ∈ GL(R) then (P AP^{−1})⋆(P BP^{−1}) = A⋆B. To see this, suppose that A, B, P ∈ GL_{n}(R) and let g ∈ St_{2n}(R) be a lift of the block diagonal matrix D = diag(P, P^{−1}). Since gag^{−1} and gbg^{−1} lift P AP^{−1} and P BP^{−1} and [a, b] is central we have the desired relation: [gag^{−1}, gbg^{−1}] = g[a, b]g^{−1} = [a, b].

The ⋆ symbol is also skew-symmetric and bilinear: (A⋆B)(B⋆A) = 1 and (A_{1}A_{2})⋆B = (A_{1}⋆B)(A_{2}⋆B). These relations are immediate from the commutator identies [a, b][b, a] = 1 and [a_{1}a_{2}, b] = [a_{1}, [a_{2}, b]][a_{2}, b][a_{1}, b].

Definition 5.10. If r, s are commuting units in a ring R, we define the Steinberg symbol {r, s} ∈ K_{2}(R) to be

\{r, s\} = ( \begin{matrix}r \\ & r^{−1} \\ & & 1 \end{matrix} ) ⋆ ( \begin{matrix}s \\ & 1 \\ & & s^{−1} \end{matrix} ) = ( \begin{matrix}r \\ & 1 \\ & & r^{−1} \end{matrix} ) ⋆ ( \begin{matrix}s \\ & s^{−1} \\ & & 1 \end{matrix} ) .

Because the ⋆ symbols are skew-symmetric and bilinear, so are the Steinberg symbols: {r, s}{s, r} = 1 and {r_{1}r_{2}, s} = {r_{1}, s}{r_{2}, s}.

Example 5.10.1. For any unit r of R we set w_{ij}(r) = x_{ij}(r)x_{ji}(−r^{−1})x_{ij}(r) and h_{ij}(r) = w_{ij}(r)w_{ij}(−1). In GL(R), ϕw_{ij}(r) is the monomial matrix with r and −r^{−1} in the (i, j) and (j, i) places, while ϕh_{ij}(r) is the diagonal matrix with r and r^{−1} in the i^{th} and j^{th} diagonal spots. By definition we then have:

{r, s} = [h_{12}(r), h_{13}(s)] = [h_{ij}(r), h_{ik}(s)].

Lemma 5.10.2. If both r and 1 − r are units of R, then in K_{2}(R) we have:

{r, 1 − r} = 1  and  {r, −r} = 1.

Proof. By Ex. 5.8, w_{12}(−1) = w_{21}(1) = x_{21}(1)x_{12}(−1)x_{21}(1), w_{12}(r)x_{21}(1) = x_{12}(−r^{2})w_{12}(r) and x_{21}(1)w_{12}(s) = w_{12}(s)x_{12}(−s^{2}). If s = 1−r we can successively use the identities r − r^{2} = rs, r + s = 1, s − s^{2} = rs and \frac{1}{r} + \frac{1}{s} = \frac{1}{rs} to obtain:

w_{12}(r)w_{12}(−1)w_{12}(s) = x_{12}(−r^{2})w_{12}(r)x_{12}(−1) w_{12}(s)x_{12}(−s^{2})

= x_{12}(rs)x_{21}(−r^{−1})x_{12}(0)x_{21}(−s^{−1})x_{12}(rs)

= x_{12}(rs)x_{21}(\frac{−1}{rs})x_{12}(rs)

= w_{12}(rs).

Multiplying by w_{12}(−1) yields h_{12}(r)h_{12}(s) = h_{12}(rs) when r + s = 1. By Ex. 5.9, this yields the first equation {r, s} = 1. Since −r = (1 − r)/(1 − r^{−1}), the first equation implies the second equation:

(5.10.3)  {r, −r} = {r, 1 − r}{r, 1 − r^{−1}}^{−1} = {r^{−1}, 1 − r^{−1}} = 1.

Remark 5.10.4. The equation {r, −r} = 1 holds more generally for every unit r, even if 1 − r is not a unit. This follows from the fact that K_{2}(ℤ[r, \frac{1}{r}]) injects into K_{2}(ℤ[r, \frac{1}{r}, \frac{1}{1−r}]), a fact we shall establish in chapter V, 6.1.3. For a direct proof, see [Milnor, 9.8].

The following useful result was proven for fields and division rings in §9 of [Milnor]. It was extended to commutative semilocal rings by Dennis and Stein [DS], and we cite it here for completeness.

Theorem 5.10.5. If R is a field, division ring, local ring, or even a semilocal ring, then K_{2}(R) is generated by the Steinberg symbols {r, s}.

Definition 5.11 (Dennis-Stein symbols). If r, s ∈ R commute and 1 − rs is a unit then the element

⟨r, s⟩ = x_{ji}(−s(1 − rs)^{−1})x_{ij}(−r)x_{ji}(s)x_{ij}((1 − rs)^{−1}r)h_{ij}(1 − rs)^{−1}

of St(R) belongs to K_{2}(R), because ϕ⟨r, s⟩ = 1. By Ex. 5.11, it is independent of the choice of i ≠ j, and if r is a unit of R then ⟨r, s⟩ = {r, 1 − rs}. If I is an ideal of R and s ∈ I then we can even consider ⟨r, s⟩ as an element of K_{2}(R, I); see 5.7.

These elements are called Dennis-Stein symbols because they were first studied in [DS], where the following identities were established. (D1) ⟨r, s⟩⟨s, r⟩ = 1 (D2) ⟨r, s⟩⟨r, t⟩ = ⟨r, s + t − rst⟩ (D3) ⟨r, st⟩ = ⟨rs, t⟩⟨tr, s⟩ (this holds in K_{2}(R, I) if any of r, s, or t are in I.)

We warn the reader that the meaning of the symbol ⟨r, s⟩ changed circa 1980. We use the modern definition of this symbol, which equals ⟨−r, s⟩^{−1} in the old literature, including that of loc. cit. By (D3) of our definition, ⟨r, 1⟩=0 for all r.

The following result is essentially due to Maazen, Stienstra and van der Kallen. However, their work preceded the correct definition of K_{2}(R, I) so the correct historical reference is [Keune].

Theorem 5.11.1. (a) Let R be a commutative local ring, or a field. Then K_{2}(R) may be presented as the abelian group generated by the symbols ⟨r, s⟩ with r, s ∈ R such that 1 − rs is a unit, subject only to the relations (D1), (D2) and (D3).

(b) Let I be a radical ideal, contained in a commutative ring R. Then K_{2}(R, I) may be presented as the abelian group generated by the symbols ⟨r, s⟩ with either r ∈ R and s ∈ I, or else r ∈ I and s ∈ R. These generators are subject only to the relations (D1), (D2), and the relation (D3) whenever r, s, or t is in I.

The product K_{1}(R) ⊗ K_{1}(R) → K_{2}(R)

Let R be a commutative ring, and suppose given two invertible matrices g ∈ GL_{m}(R), h ∈ GL_{n}(R). Identifying the tensor product R^{m} ⊗ R^{n} with R^{m+n}, then g⊗ 1_{n} and 1_{m}⊗ h are commuting automorphisms of R^{m}⊗R^{n}. Hence there is a ring homomorphism from A = ℤ[x, x^{−1}, y, y^{−1}] to E = End_{R}(R^{m} ⊗ R^{n}) ≅ M_{m+n}(R) sending x and y to g ⊗ 1_{n} and 1_{m} ⊗ h. Recall that by Morita Invariance 5.6.1 the natural map K_{2}(R) → K_{2}(E) is an isomorphism.

Definition 5.12. The element {g, h} of K_{2}(R) is defined to be the image of the Steinberg symbol {x, y} under the homomorphism K_{2}(A) → K_{2}(E) ≅ K_{2}(R).

Note that if m = n = 1 this agrees with the definition of the usual Steinberg symbol in 5.10, because R = E.

Lemma 5.12.1. The symbol {g, h} is independent of the choice of m and n, and is skew-symmetric. Moreover, for each α ∈ GL_{m}(R) we have {g, h} = {αgα^{−1}, h}.

Proof. If we embed GL_{m}(R) and GL_{n}(R) in GL_{m′}(R) and GL_{n′}(R), respectively, then we embed E into the larger ring E′ = End_{R}(R^{m′} ⊗ R^{n′}), which is also Morita equivalent to R. Since the natural maps K_{2}(R) → K_{2}(E) → K_{2}(E′) are isomorphisms, and K_{2}(A) → K_{2}(E) → K_{2}(E′) ≅ K_{2}(R) defines the symbol with respect to the larger embedding, the symbol is independent of m and n.

Any linear automorphism of R^{m+n} induces an inner automorphism of E. Since the composition of R → E with such an automorphism is still R → E, the symbol {g, h} is unchanged by such an operation. Applying this to α ⊗ 1_{n}, the map A → E → E sends x and y to αgα^{−1} ⊗ 1_{n} and 1_{m} ⊗ h, so {g, h} must equal {αgα^{−1}, h}.

As another application, note that if m = n the inner automorphism of E induced by R^{m} ⊗R^{n} ≅ R^{n} ⊗R^{m} sends {h, g} to the image of {y, x} under K_{2}(A) → K_{2}(E). This proves skew-symmetry, since {y, x} = {x, y}^{−1}.

Theorem 5.12.2. For every commutative ring R, there is a skew-symmetric bilinear pairing K_{1}(R) ⊗ K_{1}(R) → K_{2}(R) induced by the symbol {g, h}.

Proof. We first show that the symbol is bimultiplicative when g and g′ commute in GL_{m}(R). Mapping A[z, z^{−1}] into E by z ↦ g′ ⊗ 1_{n} allows us to deduce {gg′, h} = {g, h}{g′, h} from the corresponding property of Steinberg symbols. If g and g′ do not commute, the following trick establishes bimultiplicativity:

\{gg′, h\} = \{({g \atop 0} {0 \atop 1})({1 \atop 0} {0 \atop g′} ), h\} = \{({g \atop 0} {0 \atop 1}), h\}\{({1 \atop 0} {0 \atop g′} ), h\} = \{g, h\}\{g′, h\}.

If either g or h is a commutator, this implies that the symbol {g, h} vanishes in the abelian group K_{2}(R). Since the symbol {g, h} is compatible with stabilization, it describes a function K_{1}(R)×K_{1}(R) → K_{2}(R) which is multiplicative in each entry: {gg′, h} = {g, h}{g′, h}. If we write K_{1} and K_{2} additively the function is additive in each entry, i.e., bilinear.

EXERCISES

5.1 Relative Steinberg groups. Let I be an ideal in a ring R. Show that there is an exact sequence St(R, I) ⟶^{add} St(R) → St(R/I) → 1. 5.2 Consider the function ρ_{n}: R^{n−1} → St_{n}(R) sending (r_{1}, ..., r_{n−1}) to the product x_{1n}(r_{1})x_{2n}(r_{2}) ⋯ x_{n−1,n}(r_{n−1}). The Steinberg relations show that this is a group homomorphism.

Show that ρ is an injection by showing that the composite ϕρ: R^{n−1} → St_{n}(R) → GL_{n}(R) is an injection. Then show that the elements x_{ij}(r) with i, j < n normalize the subgroup P_{n} = ρ(R^{n}) of St_{n}(R), i.e., that x_{ij}(r)P_{n}x_{ij}(−r) = P_{n}.

Use this and induction to show that the subgroup T_{n} of St_{n}(R) generated by the x_{ij}(r) with i < j maps isomorphically onto the subgroup of lower triangular matrices in GL_{n}(R). 5.3 Let G be a group whose commutator group [G, G] is abelian. Prove that the Jacobi identity holds for every u, v, w ∈ G:

[u, [v, w]][v, [w, u]][w, [u, v]] = 1.

If in addition [u, w] = 1 this implies that [[u, v], w] = [u, [v, w]]. 5.4 Product with K_{0}. Construct a product operation K_{0}(R) ⊗ K_{2}(A) → K_{2}(A), assuming that R is commutative and A is an associative R-algebra. To do this, fix a finitely generated projective R-module P. Each isomorphism P ⊕ Q = R^{n} gives rise to a homomorphism h^{P}: GL_{m}(A) → GL_{mn}(A) ⊂ GL(A) sending α to α ⊗ 1 and E_{m}(A) to E(A). Show that h^{P} is well-defined up to conjugation by an element of E(A). Since conjugation acts trivially on homology, this implies that the induced map h^{P}_{∗}: H_{2}(E_{m}(A); ℤ) → H_{2}(E(A); ℤ) = K_{2}(A) is well-defined. Then show that h^{P ⊕Q}_{∗} = h^{P}_{∗} ⊕ h^{Q}_{∗} and pass to the limit as m → ∞ to obtain the required endomorphism [P]· of K_{2}(A). 5.5 If R is commutative and P ∈ P(R), show that Q ↦ Q ⊗_{R} P defines a functor from the translation category tP(A) to itself for every R-algebra A, and that the resulting endomorphism of K_{2}(A) = \underrightarrow{lim} H_{2}([Aut(Q), Aut(Q)]) is the map h_{∗}^{P} of the previous exercise. Use this description to show that the product makes K_{2}(A) into a module over the ring K_{0}(R).

5.6 Projection Formula. Suppose that f: R → S is a finite map of commutative rings, with S ∈ P(R). Show that for all r ∈ K_{i}(R) and s ∈ K_{j}(S) with i + j = 2 we have f_{∗}(f^{∗}(r) · s) = r · f_{∗}(s) in K_{2}(R). The case i = 0 states that the transfer f_{∗}: K_{2}(S) → K_{2}(R) is K_{0}(R)-linear, while the case i = 1 yields the useful formula f_{∗}{r, s} = {r, Ns} for Steinberg symbols in K_{2}(R), where r ∈ R^{×}, s ∈ S^{×} and Ns = f_{∗}(s) ∈ R^{×} is the norm of s.

5.7 If Y ⟶^{ρ} X and X ⟶^{π} G are central extensions, show that the “composition” Y ⟶^{πρ} G is also a central extension. If X is a universal central extension of G, conclude that every central extension Y ⟶^{ρ} X splits.

5.8 Show that the following identies hold in St(R) (for i, j and k distinct).

(a) w_{ij}(r)w_{ij}(−r) = 1;

(b) w_{ik}(r)x_{ij}(s)w_{ik}(−r) = x_{kj}(−r^{−1}s);

(c) w_{ij}(r)x_{ij}(s)w_{ij}(−r) = x_{ji}(−r^{−1}sr^{−1});

(d) w_{ij}(r)x_{ji}(s)w_{ij}(−r) = x_{ij}(−rsr);

(e) w_{ij}(r)w_{ji}(r^{−1}) = 1;

5.9 Use the previous exercise to show that {r, s} = h_{ij}(rs)h_{ij}(s)^{−1}h_{ij}(r)^{−1}. Hint: Conjugate h_{ij}(s) by w_{ik}(r)w_{ik}(−1).

5.10 Excision. If I and J are ideals in a ring R with I ∩ J = 0, we may also consider I as an ideal of R/J. Show that St(R, I) surjects onto St(R/J, I), while the subgroups E(R, I) and E(R/J, I) of GL(I) are equal. Use the 5-lemma to conclude that K_{1}(R, I) ≅ K_{1}(R/J, I) and that K_{2}(R, I) → K_{2}(R/J, I) is onto.

In fact, the sequence I/I^{2} ⊗_{R⊗R} J/J^{2} → K_{2}(R, I) → K_{2}(R/J, I) → 0 is exact, where the first map sends x ⊗ y to ⟨x, y⟩; see [Swan71].

5.11 Dennis-Stein symbols. Let ⟨r, s⟩_{ij} denote the element of St(R) given in Definition 5.11. Show that this element is in K_{2}(R). Then use Ex. 5.8 to show that if w = w_{ik}(1)w_{jℓ}(1)w_{kℓ}^{2}(1) (so that ϕ(w) is the permutation matrix sending i, j to k, ℓ) the permutation matrix send i, j to k, ℓ then w ⟨r, s⟩_{ij} w^{−1} = ⟨r, s⟩_{kℓ}. This shows that the Dennis-Stein symbol is independent of the choice of indices i, j.

5.12 Let A be an abelian group and F a field. Show that, for all n ≥ 5, homomorphisms K_{2}(F) ⟶^{c} A are in 1–1 correspondence with central extensions of SL_{n}(F) having kernel A.

5.13 If p is an odd prime, use Theorem 5.11.1 to show that K_{2}(ℤ/p^{n}) = 1. If n ≥ 2, show that K_{2}(ℤ/2^{n}) ≅ K_{2}(ℤ/4) ≅ {±1} on {−1, −1} = ⟨−1, −2⟩ = ⟨2, 2⟩.

Using the Mayer-Vietoris sequence 5.8, conclude that K_{1}(R) = R^{×} = {±1} for the ring R = ℤ[x]/(x^{2} − p^{2n}). Note that R/(x ± p^{n})R = ℤ.

5.14 Let R be a commutative ring, and let Ω_{R} denote the module of Kähler differentials of R over ℤ, as in Ex. 2.6.

(a) If I is a radical ideal of R, show that there is a surjection from K_{2}(R, I) onto I ⊗_{R} Ω_{R/I}, sending ⟨x, r⟩ to x ⊗ dr (r ∈ R, x ∈ I).

(b) If I^{2} = 0, show that the kernel of the map in (a) is generated by the Dennis-Stein symbols ⟨x, y⟩ with x, y ∈ I.

(c) (Van der Kallen) The dual numbers over R is the ring R[ε] with ε^{2} = 0. If \frac{1}{2} ∈ R, show that the map K_{2}(R[ε], ε) → Ω_{R} of part (a) is an isomorphism.

(d) Let k be a field. Show that the group K_{2}(k[[t]], t) is ℓ-divisible for every ℓ invertible in k. If char(K) = p > 0, show that this group is not p-divisible.

5.15 Assume the fact that ℤ/2 is K_{2}-regular (see 3.4 and chapter V). Show that:

(a) K_{2}(ℤ/4[x]) is an elementary abelian 2-group with basis ⟨2, 2⟩, ⟨2x^{n}, x⟩, and ⟨2x^{2n+1}, 2⟩, n ≥ 0. Hint: Split the map K_{2}(A, 2) → Ω_{A/2} ≅ ℤ/2[x] of Ex. 5.14 and use 0 = ⟨2f, 1⟩ = ⟨2(f + f^{2}), 2⟩.

(b) The group K_{2}(ℤ/4[x, y]) is an elementary abelian 2-group with basis ⟨2, 2⟩, ⟨2x^{m}y^{n}, x⟩, ⟨2x^{m}y^{n}, y⟩ (m, n ≥ 0) and ⟨2x^{m}y^{n}, x⟩ (one of m, n odd).

(c) Consider the maps ∂_{1} : NK_{2}(ℤ/4) → K_{2}(ℤ/4) and ∂_{2} : N^{2}K_{2}(ℤ/4) → NK_{2}(ℤ/4) induced by the maps ℤ/4[x] → ℤ/4 sending x to 1, and ℤ/4[x, y] → ℤ/4[x] sending y to 1 − x, respectively. Show that the following sequence is exact: N^{2}K_{2}(ℤ/4) → NK_{2}(ℤ/4) → K_{2}(ℤ/4) → 0.

§6. K_{2} of fields

The following theorem was proven by Hideya Matsumoto in 1969. We refer the reader to [Milnor, §12] for a self-contained proof.

Matsumoto’s Theorem 6.1. If F is a field then K_{2}(F) is the abelian group generated by the set of Steinberg symbols {x, y} with x, y ∈ F^{×}, subject only to the relations:

(1) (Bilinearity) {xx′, y} = {x, y}{x′, y} and {x, yy′} = {x, y}{x, y′};

(2) (Steinberg Identity) {x, 1 − x} = 1 for all x ≠ 0, 1.

In other words, K_{2}(F) is the quotient of F^{×} ⊗ F^{×} by the subgroup generated by the elements x ⊗ (1 − x). Note that the calculation (5.10.3) implies that {x, −x} = 1 for all x, and this implies that the Steinberg symbols are skewsymmetric: {x, y}{y, x} = {x, −xy}{y, −xy} = {xy, −xy} = 1.

Corollary 6.1.1. K_{2}(𝔽_{q}) = 1 for every finite field 𝔽_{q}.

Proof. If x generates the cyclic group 𝔽_{q}^{×}, we must show that the generator x⊗x of the cyclic group 𝔽_{q}^{×} ⊗ 𝔽_{q}^{×} vanishes in K_{2}. If q is even, then {x, x} = {x, −x} = 1, so we may suppose that q is odd. Since {x, x}^{2} = 1 by skew-symmetry, we have {x, x} = {x, x}^{mn} = {x^{m}, x^{n}} for every odd m and n. Since odd powers of x are the same as non-squares, it suffices to find a non-square u such that 1 − u is also a non-square. But such a u exists because u ↦ (1 − u) is an involution on the set 𝔽_{q} −{0, 1}, and this set consists of (q −1)/2 non-squares but only (q −3)/2 squares.

Example 6.1.2. Let F (t) be a rational function field in one variable t over F. Then K_{2}(F) is a direct summand of K_{2}F (t).

To see this, we construct a map λ: K_{2}F (t) → K_{2}(F) inverse to the natural map K_{2}(F) → K_{2}F (t). To this end, we define the leading coefficient of the rational function f(t) = (a_{0}t^{n} + ⋯ + a_{n})/(b_{0}t^{m} + ⋯ + b_{m}) to be lead(f) = a_{0}/b_{0} and set λ({f, g}) = {lead(f), lead(g)}. To see that this defines a homomorphism K_{2}F (t) → K_{2}(F), we check the presentation in Matsumoto’s Theorem. Bilinearity is clear from lead(f_{1}f_{2}) = lead(f_{1})lead(f_{2}), and {lead(f), lead(1 − f)} = 1 holds in K_{2}(F) because lead(1−f) is either 1, 1−lead(f) or −lead(f), according to whether m > n, m = n or m < n.

Because K_{2} commutes with filtered colimits, it follows that K_{2}(F) injects into K_{2}F (T) for every purely transcendental extension F (T) of F.

Lemma 6.1.3. For every field extension F ⊂ E, the kernel of K_{2}(F) → K_{2}(E) is a torsion subgroup.

Proof. E is an algebraic extension of some purely transcendental extension F (X) of F, and K_{2}(F) injects into K_{2}F (X) by Example 6.1.2. Thus we may assume that E is algebraic over F. Since E is the filtered union of finite extensions, we may even assume that E/F is a finite field extension. But in this case the result holds because (by 5.6.3) the composite K_{2}(F) → K_{2}(E) → K_{2}(F) is multiplication by the integer [E : F].

The next result is useful for manipulations with symbols.

Lemma 6.1.4 (Bass-Tate). If E = F (u) is a field extension of F, then every symbol of the form {b_{1}u − a_{1}, b_{2}u − a_{2}} (a_{i}, b_{i} ∈ F) is a product of symbols {c_{i}, d_{i}} and {c_{i}, u − d_{i}} with c_{i}, d_{i} ∈ F.

Proof. Bilinearity allows us to assume that b_{1} = b_{2} = 1. Set x = u − a_{1}, y = u − a_{2} and a = a_{2} − a_{1}, so x = a + y. Then 1 = \frac{a}{x} + \frac{y}{x} yields the relation 1 = \{\frac{a}{x}, \frac{y}{x}\}. Using \{x, x\} = \{−1, x\}, this expands to the desired expression: \{x, y\} = {a, y}{−1, x}{a^{−1}, x}.

Together with the Projection Formula (Ex. 5.6), this yields:

Corollary 6.1.5. If E = F (u) is a quadratic field extension of F, then K_{2}(E) is generated by elements coming from K_{2}(F), together with elements of the form {c, u − d}. Thus the transfer map N_{E/F}: K_{2}(E) → K_{2}(F) is completely determined by the formulas N_{E/F} {c, d} = {c, d}^{2}, N_{E/F} {c, u − d} = {c, N(u − d)} (c, d ∈ F)

Example 6.1.6. Since ℂ is a quadratic extension of ℝ, every element of K_{2}(ℂ) is a product of symbols {r, s} and {r, e^{iθ}} with r, s, θ ∈ ℝ. Moreover, N{r, e^{iθ}} = 1 in K_{2}(ℝ). Under the automorphism of K_{2}(ℂ) induced by complex conjugation, the symbols of the first kind are fixed and the symbols of the second kind are sent to their inverses. We will see in Theorem 6.4 below that K_{2}(ℂ) is uniquely divisible, i.e., a vector space over ℚ, and the decomposition of K_{2}(ℂ) into eigenspaces for ±1 corresponds to symbols of the first and second kind.

Example 6.1.7. Let F be an algebraically closed field. By Lemma 6.1.4, K_{2}F (t) is generated by linear symbols of the form {a, b} and {t − a, b}. It will follow from 6.5.2 below that every element u of K_{2}F (t) uniquely determines finitely many elements a_{i} ∈ F, b_{i} ∈ F^{×} so that u = λ(u) ∏{t − a_{i}, b_{i}}, where λ(u) ∈ K_{2}(F) was described in Example 6.1.2.

Steinberg symbols

Definition 6.2. A Steinberg symbol on a field F with values in a multiplicative abelian group A is a bilinear map c: F^{×} ⊗ F^{×} → A satisfying c(r, 1 − r) = 1. By Matsumoto’s Theorem, these are in 1–1 correspondence with homomorphisms

K_{2}(F) ⟶^{c} A.

Example 6.2.1. There is a Steinberg symbol (x, y)_{∞} on the field ℝ with values in the group {±1}. Define (x, y)_{∞} to be: −1 if both x and y are negative, and +1 otherwise. The Steinberg identity (x, 1−x)_{∞} = +1 holds because x and 1−x cannot be negative at the same time. The resulting map K_{2}(ℝ) → {±1} is onto because (−1, −1)_{∞} = −1. This shows that the symbol {−1, −1} in K_{2}(ℤ) is nontrivial, as promised in 5.2.2, and even shows that K_{2}(ℤ) is a direct summand in K_{2}(ℝ).

For our next two examples, recall that a local field is a field F which is complete under a discrete valuation v, and whose residue field k_{v} is finite. Classically, every local field is either a finite extension of the p-adic rationals ℚ̂_{p} or of 𝔽_{p}((t)).

Example 6.2.2 (Hilbert symbols). Let F be a local field containing \frac{1}{2}. The Hilbert (quadratic residue) symbol on F is defined by setting c_{F} (r, s) ∈ {±1} equal to +1 or −1, depending on whether or not the equation rx^{2} +sy^{2} = 1 has a solution in F. Bilinearity is classical when F is local; see [OMeara, p.164]. The Steinberg identity is trivial, because x = y = 1 is always a solution when r + s = 1.

Of course, the definition of c_{F} (r, s) makes sense for any field of characteristic ≠ 2, but it will not always be a Steinberg symbol because it can fail to be bilinear in r. It is a Steinberg symbol when F = ℝ, because the Hilbert symbol c_{ℝ}(r, s) is the same as the symbol (r, s)_{∞} of the previous example.

Example 6.2.3 (norm residue symbols). The roots of unity in a local field F form a finite cyclic group μ, equal to the group μ_{m} of all m^{th} roots of unity for some integer m with \frac{1}{m} ∈ F. The classical m^{th} power norm residue symbol is a map K_{2}(F) → μ_{m} defined as follows (see [S-LF] for more details).

Because F^{×m} has finite index in F^{×}, there is a finite “Kummer” extension K containing the m^{th} roots of every element of F. The Galois group G_{F} = Gal(K/F) is canonically isomorphic to Hom(F^{×}, μ_{m}), with the automorphism g of K corresponding to the homomorphism ζ: F^{×} → μ_{m} sending a ∈ F^{×} to ζ(a) = g(x)/x, where x^{m} = a. In addition, the cokernel of the norm map K^{×} ⟶^{N} F^{×} is isomorphic to G_{F} by the “norm residue” isomorphism of local class field theory. The composite F^{×} ⟶ F^{×}/NK^{×} ≅ G_{F} ≅ Hom(F^{×}, μ_{m}), written as x ↦ (x, −)_{F}, is adjoint to a nondegenerate bilinear map ( , )_{F}: F^{×} ⊗ F^{×} → μ_{m}.

The Steinberg identity (a, 1 − a)_{F} = 1 is proven by noting that (1 − a) is a norm from the intermediate field E = F (x), x^{m} = a. Since G_{E} ⊂ G_{F} corresponds to the norm map E^{×}/N_{K/E}K^{×} ↪ F^{×}/N_{K/F} K^{×}, the element g of G_{F} = Gal(K/F) corresponding to the map ζ(a) = (a, 1 − a)_{F} from F^{×} to μ_{m} must belong to G_{E}, i.e., ζ must extend to a map E^{×} → μ_{m}. But then (a, 1 − a)_{F} = ζ(a) = ζ(x)^{m} = 1.

The name “norm residue” comes from the fact that for each x, the map y ↦ {x, y} is trivial if and only if x ∈ NK^{×}. Since a primitive m^{th} root of unity ζ is not a norm from K, it follows that there is an x ∈ F such that (ζ, x)_{F} ≠ 1. Therefore the norm residue symbol is a split surjection with inverse ζ^{i} ↦ {ζ^{i}, x}.

The role of the norm residue symbol is explained by the following structural result, whose proof we cite from the literature.

Moore’s Theorem 6.2.4. If F is a local field, then K_{2}(F) is the direct sum of a uniquely divisible abelian group U and a finite cyclic group, isomorphic under the norm residue symbol to the group μ = μ_{m} of roots of unity in F.

Proof. We have seen that the norm residue symbol is a split surjection. A proof that its kernel U is divisible, due to C. Moore, is given in the Appendix to [Milnor]. The fact that U is torsionfree (hence uniquely divisible) was proven by Tate [Tate] when char(F) = p, and by Merkurjev [Merk] when char(F) = 0.

Example 6.2.5 (2-adic rationals). The group K_{2}(ℚ̂_{2}) is the direct sum of the cyclic group of order 2 generated by {−1, −1} and a uniquely divisible group. Since x^{2} + y^{2} = −1 has no solution in F = ℚ̂_{2} we see from definition (6.2.2) that the Hilbert symbol c_{F} (−1, −1) = −1.

Tame symbols

Every discrete valuation v on a field F provides a Steinberg symbol. Recall that v is a homomorphism F^{×} → ℤ such that v(r+s) ≥ min{v(r), v(s)}. By convention, v(0) = ∞, so that the ring R of all r with v(r) ≥ 0 is a discrete valuation ring (DVR). The units R^{×} form the set v^{−1}(0), and the maximal ideal of R is generated by any π ∈ R with v(π) = 1. The residue field k_{v} is defined to be R/(π). If u ∈ R, we write ū for the image of u under R → k_{v}.

Lemma 6.3. For every discrete valuation v on F there is a Steinberg symbol K_{2}(F) ⟶^{∂_{v}} k_{v}^{×}, defined by

∂_{v}(\{r, s\}) = (−1)^{v(r)v(s)}\overline{(\frac{s^{v(r)}}{r^{v(s)}} )}.

This symbol is called the tame symbol of the valuation v. The tame symbol is onto, because if u ∈ R^{×} then v(u) = 0 and ∂_{v}(π, u) = ū.

Proof. Writing r = u_{1}π^{v_{1}} and s = u_{2}π^{v_{2}} with u_{1}, u_{2} ∈ R^{×}, we must show that ∂_{v}(r, s) = (−1)^{v_{1}v_{2}} \begin{matrix} & ū_{2}^{v_{1}} \\ & ū_{1}^{v_{2}} \end{matrix} is a Steinberg symbol. By inspection, ∂_{v}(r, s) is an element of k_{v}^{×}, and ∂_{v} is bilinear. To see that ∂_{v}(r, s) = 1 when r + s = 1 we consider several cases. If v_{1} > 0 then r is in the maximal ideal, so s = 1 − r is a unit and ∂_{v}(r, s) = s̄^{v_{1}} = 1. The proof when v_{2} > 0 is the same, and the case v_{1} = v_{2} = 0 is trivial. If v_{1} < 0 then v(\frac{1}{r}) > 0 and \frac{1−r}{r} = −1 + \frac{1}{r} is congruent to −1 (mod π). Since v(r) = v(1 − r), we have

∂_{v}(r, 1 − r) = (−1)^{v_{1}}(\frac{1 − r}{r} )^{v_{1}} = (−1)^{v_{1}}(−1)^{v_{1}} = 1.

Ramification 6.3.1. Suppose that E is a finite extension of F, and that w is a valuation on E over the valuation v on F. Then there is an integer e, called the ramification index, such that w(r) = e · v(r) for every r ∈ F. The natural map K_{2}(F) → K_{2}(E) is compatible with the tame symbols in the sense that for every r_{1}, r_{2} ∈ F^{×} we have ∂_{w}(r_{1}, r_{2}) = ∂_{v}(r_{1}, r_{2})^{e} in k_{w}^{×}.

K_{2}(F) ⟶^{∂_{v}} k_{v}^{×}
↓  ^{e}↓_{x↦x^{e}}
K_{2}(E) ⟶^{∂_{w}} k_{w}^{×}

Let S denote the integral closure of R in E. Then S has finitely many prime ideals 𝔭_{1}, ..., 𝔭_{n} lying over 𝔭, with corresponding valuations w_{1}, ..., w_{n} on E. We say that S is unramified over R if the ramification indices e_{1}, ..., e_{n} are all 1; in this case the diagonal inclusion Δ: k_{v}^{×} ↪ ∏_{i} k_{w_{i}}^{×} is compatible with the tame symbols in the sense that Δ∂_{v}(r_{1}, r_{2}) is the product of the ∂_{w_{i}}(r_{1}, r_{2}).

Corollary 6.3.2. If F contains the rational function field ℚ(t) or 𝔽_{p}(t_{1}, t_{2}), then K_{2}(F) has the same cardinality as F. In particular, if F is uncountable then so is K_{2}(F).

Proof. By hypothesis, F contains a transcendental element t. Choose a subset X = {x_{α}} of F so that X∪{t} is a transcendence basis for F over its ground field F_{0}, and set k = F_{0}(X). Then the subfield k(t) of F has a t-adic valuation with residue class field k. Hence K_{2}(k(t)) contains a subgroup {t, k^{×}} mapped isomorphically under the tame symbol to k^{×}. By Lemma 6.1.3, the kernel of k^{×} → K_{2}(k(t)) → K_{2}(F) is contained in the torsion subgroup μ(k) of roots of unity in k. Thus the cardinality of K_{2}(F) is bounded below by the cardinality of k^{×}/μ(k). Since F is an algebraic extension of k(t), and k contains either ℚ or 𝔽_{p}(t_{2}), we have the inequality |F | = |k| = |k^{×}/μ(k)| ≤ |K_{2}(F)|. The other inequality |K_{2}(F)| ≤ |F | is immediate from Matsumoto’s Theorem, since F is infinite.

Theorem 6.4 (Bass-Tate). When F is an algebraically closed field, K_{2}(F) is a uniquely divisible abelian group.

Theorem 6.4 is an immediate consequence of proposition 6.4.1 below. To see this, recall that an abelian group is uniquely divisible when it is uniquely p-divisible for each prime p; a group is said to be uniquely p-divisible if it is p-divisible and has no p-torsion.

Proposition 6.4.1 (Bass-Tate). Let p be a prime number such that each polynomial t^{p} − a (a ∈ F) splits in F [t] into linear factors. Then K_{2}(F) is uniquely p-divisible.

Proof. The hypothesis implies that F^{×} is p-divisible. Since the tensor product of p-divisible abelian groups is always uniquely p-divisible, F^{×} ⊗ F^{×} is uniquely p-divisible. Let R denote the kernel of the natural surjection F^{×} ⊗ F^{×} → K_{2}(F). By inspection (or by the Snake Lemma), K_{2}(F) is p-divisible and the p-torsion subgroup of K_{2}(F) is isomorphic to R/pR.

Therefore it suffices to prove that R is p-divisible. Now R is generated by the elements ψ(a) = (a)⊗(1−a) of F^{×} ⊗F^{×} (a ∈ F −{0, 1}), so it suffices to show that each ψ(a) is in pR. By hypothesis, there are b_{i} ∈ F such that t^{p} − a = ∏(t − b_{i}) in F [t], so 1 − a = ∏(1 − b_{i}) and b_{i}^{p} = a for each i. But then we compute in F^{×} ⊗ F^{×}:

ψ(a) = (a) ⊗ (1 − a) = ∑(a) ⊗ (1 − b_{i}) = ∑(b_{i})^{p} ⊗ (1 − b_{i}) = p ∑ ψ(b_{i}).

Corollary 6.4.2. If F is a perfect field of characteristic p, then K_{2}(F) is uniquely p-divisible.

The Localization Sequence for K_{2}

The following result will be proven in chapter V, but we find it useful to quote this result now. If 𝔭 is a nonzero prime ideal of a Dedekind domain R, the local ring R_{𝔭} is a discrete valuation ring, and hence determines a tame symbol.

Localization Theorem 6.5. Let R be a Dedekind domain with field of fractions F. Then the tame symbols K_{2}(F) ⟶^{∂_{𝔭}} (R/𝔭)^{×} associated to the prime ideals of R fit into a long exact sequence

∐_{𝔭} K_{2}(R/𝔭) → K_{2}(R) → K_{2}(F) \underrightarrow{∂=∐ ∂_{𝔭}} ∐_{𝔭}(R/𝔭)^{×} → SK_{1}(R) → 1

where the coproducts are over all nonzero prime ideals 𝔭 of R, and the maps from (R/𝔭)^{×} = K_{1}(R/𝔭) to SK_{1}(R) are the transfer maps of Ex. 1.11. The transfer maps K_{2}(R/𝔭) → K_{2}(R) will be defined in chapter V.

Application 6.5.1 (K_{2}ℚ). If R = ℤ then, since K_{2}(ℤ/p) = 1 and SK_{1}(ℤ) = 1, we have an exact sequence 1 → K_{2}(ℤ) → K_{2}(ℚ) ⟶^{∂} ∐ 𝔽_{p}^{×} → 1. As noted in Example 6.2.1, this sequence is split by the symbol (r, s)_{∞}, so we have K_{2}(ℚ) ≅ K_{2}(ℤ) ⊕ ∐ 𝔽_{p}^{×}.

Application 6.5.2 (Function fields). If R is the polynomial ring F [t] for some field F, we know that K_{2}(F [t]) = K_{2}(F) (see 5.2.3). Moreover, the natural map K_{2}(F) → K_{2}F (t) is split by the leading coefficient symbol λ of Example 6.1.2. Therefore we have a split exact sequence

1 → K_{2}(F) → K_{2}F (t) ⟶^{∂} ∐_{𝔭}(F [t]/𝔭)^{×} → 1.

Weil’s Reciprocity Formula 6.5.3. Just as in the case R = ℤ, there is a valuation on F (t) not arising from a prime ideal of F [t]. In this case, it is the valuation v_{∞}(f) = −deg(f) associated with the point at infinity, i.e., with parameter t^{−1}. Since the symbol (f, g)_{∞} vanishes on K_{2}(F), it must be expressable in terms of the tame symbols ∂_{𝔭}(f, g) = (f, g)_{𝔭}. The appropriate reciprocity formula first appeared in Weil’s 1940 paper on the Riemann Hypothesis for curves:

(f, g)_{∞} · ∏_{𝔭} N_{𝔭}(f, g)_{𝔭} = 1 in F^{×}.

In Weil’s formula “N_{𝔭}” denotes the usual norm map (F [t]/𝔭)^{×} → F^{×}. To establish this reciprocity formula, we observe that K_{2}F (t)/K_{2}F = ∐(F [t]/𝔭)^{×} injects into K_{2}F̄(t)/K_{2}F̄, where F̄ is the algebraic closure of F. Thus we may assume that F is algebraically closed. By Example 6.1.7, K_{2}F (t) is generated by linear symbols of the form {a, t − b}. But (a, t − b)_{∞} = a and ∂_{t−b}(a, t − b) = a^{−1}, so the formula is clear.

Our next structural result was discovered by Merkurjev and Suslin in 1981, and published in their landmark paper [MS]; see [GSz, 8.4]. Recall that an automorphism σ of a field E induces an automorphism of K_{2}(E) sending {x, y} to {σx, σy}.

Theorem 6.6 (Hilbert’s Theorem 90 for K_{2}). Let E/F be a cyclic Galois field extension of prime degree p, and let σ be a generator of Gal(E/F). Then the following sequence is exact, where N denotes the transfer map on K_{2}:

K_{2}(E) ⟶^{1−σ} K_{2}(E) ⟶^{N} K_{2}(F).

Merkurjev and Suslin gave this result the suggestive name “Hilbert’s Theorem 90 for K_{2},” because of its formal similarity to the following result, which is universally called “Hilbert’s Theorem 90 (for units)” because it was the 90^{th} theorem in Hibert’s classical 1897 survey of algebraic number theory, Theorie der Algebraische Zahlkörper.

Theorem 6.6.1 (Hilbert’s Theorem 90 for units). Let E/F be a cyclic Galois field extension, and let σ be a generator of Gal(E/F). If 1 − σ denotes the map a ↦ a/σ(a), then the following sequence is exact:

1 → F^{×} → E^{×} ⟶^{1−σ} E^{×} ⟶^{N} F^{×}.

We omit the proof of Hilbert’s Theorem 90 for K_{2} (and for K_{n}^{M}; see 7.8.4 below), since the proof does not involve K-theory, contenting ourselves with two special cases: when n = char(F) (7.8.3) and the following special case.

Proposition 6.6.2. Let F be a field containing a primitive n^{th} root of unity ζ, and let E be a cyclic field extension of degree n, with σ a generator of Gal(E/F).

Suppose in addition that the norm map E^{×} ⟶^{N} F^{×} is onto, and that F has no extension fields of degree < n. Then the following sequence is exact:

K_{2}(E) ⟶^{1−σ} K_{2}(E) ⟶^{N} K_{2}(F) → 1.

Proof. Since Nζ = 1, Hilbert’s Theorem 90 gives an r ∈ E with σ(r) = ζr. Setting c = N(r) ∈ F, it is well-known and easy to see that E = F (r), r^{n} = c.

Again by Hilbert’s Theorem 90 for units and our assumption about norms, E^{×} ⟶^{1−σ} E^{×} ⟶^{N} F^{×} → 1 is an exact sequence of abelian groups. Applying the right exact functor ⊗F^{×} retains exactness. Therefore we have a commutative diagram with exact rows

E^{×} ⊗ F^{×} ⟶^{(1−σ)⊗1} E^{×} ⊗ F^{×} ⟶^{N⊗1} F^{×} ⊗ F^{×} ⟶ 1
↓  ↓  ↓^{γ}
K_{2}(E) ⟶^{1−σ} K_{2}(E) ⟶  C  ⟶ 1

in which C denotes the cokernel of 1 − σ.

Now every element of E is a polynomial f(r) in r of degree < n, and f(t) is a product of linear terms b_{i}t − a_{i} by our assumption. By Lemma 6.1.4, every element of K_{2}(E) is a product of symbols of the form {a, b} and {a, r − b}. Therefore the vertical maps F^{×} ⊗ E^{×} → K_{2}(E) are onto in the above diagram. Hence γ is onto.

If a ∈ F^{×} and x ∈ E^{×} then the projection formula (Ex. 5.6) yields

N(1 − σ){a, x} = N{a, x/(σx)} = {a, Nx/N(σx)} = 1.

Hence the transfer map K_{2}(E) → K_{2}(F) factors through C. A diagram chase shows that it suffices to show that γ is a Steinberg symbol, so that it factors through K_{2}(F). For this we must show that for all y ∈ E we have γ(Ny ⊗ (1 − Ny)) = 1, i.e., that {y, 1 − Ny} ∈ (1 − σ)K_{2}(E).

Fix y ∈ E and set z = N_{E/F} (y) ∈ F. Factor t^{n} − z = ∏ f_{i} in F [t], with the f_{i} irreducible, and let F_{i} denote the field F (x_{i}), where f_{i}(x_{i}) = 0 and x_{i}^{n} = z. Setting t = 1, 1 − z = ∏ f_{i}(1) = ∏ N_{F_{i}/F} (1 − x_{i}). Setting E_{i} = E ⊗_{F} F_{i}, so that N_{F_{i}/F} (1 − x_{i}) = N_{E_{i}/E}(1 − x_{i}) and σ(x_{i}) = x_{i}, the projection formula (Ex. 5.6) gives

{y, 1 − z} = ∏ N_{E_{i}/E}{y, 1 − x_{i}} = ∏ N_{E_{i}/E}{y/x_{i}, 1 − x_{i}}.

Thus it suffices to show that each N_{E_{i}/E}{y/x_{i}, 1−x_{i}} is in (1−σ)K_{2}(E). Now E_{i}/F_{i} is a cyclic extension whose norm N = N_{E_{i}/F_{i}} satisfies N(y/x_{i}) = N(y)/x_{i}^{n} = 1. By Hilbert’s Theorem 90 for units, y/x_{i} = v_{i}/σv_{i} for some v_{i} ∈ E_{i}. We now compute:

N_{E_{i}/E}{y/x_{i}, 1 − x_{i}} = N_{E_{i}/E}{v_{i}/σv_{i}, 1 − x_{i}} = (1 − σ)N_{E_{i}/E}{v_{i}, 1 − x_{i}}. □

Here are three pretty applications of Hilbert’s Theorem 90 for K_{2}. When F is a perfect field, the first of these has already been proven in Proposition 6.4.1.

Theorem 6.7. If char(F) = p ≠ 0, then the group K_{2}(F) has no p-torsion.

Proof. Let x be an indeterminate and y = x^{p}−x; the field extension F (x)/F (y) is an Artin-Schrier extension, and its Galois group is generated by an automorphism σ satisfying σ(x) = x + 1. By 6.5.2, K_{2}(F) is a subgroup of both K_{2}F (x) and K_{2}F (y), and the projection formula shows that the norm N: K_{2}F (x) → K_{2}F (y) sends u ∈ K_{2}(F) to u^{p}.

Now fix u ∈ K_{2}(F) satisfying u^{p} = 1; we shall prove that u = 1. By Hilbert’s Theorem 90 for K_{2}, u = (1 − σ)v = v(σv)^{−1} for some v ∈ K_{2}F (x).

Every prime ideal 𝔭 of F [x] is unramified over 𝔭_{y} = 𝔭 ∩ F [y], because F [x]/𝔭 is either equal to, or an Artin-Schrier extension of, F [y]/𝔭_{y}. By 6.3.1 and 6.5.2, we have a commutative diagram in which the vertical maps ∂ are surjective:

K_{2}F (y)  ⟶^{i^{∗}} K_{2}F (x) ⟶^{1−σ} K_{2}F (x)
∂↓  ^{∂}↓  ^{∂}↓
∐_{𝔭_{y}}(F [y]/𝔭_{y})^{×} ⟶^{Δ} ∐_{𝔭}(F [x]/𝔭)^{×} ⟶^{1−σ} ∐_{𝔭}(F [x]/𝔭)^{×}

We claim that the bottom row is exact. By decomposing the row into subsequences invariant under σ, we see that there are two cases to consider. If a prime 𝔭 is not fixed by σ, then the fields F [x]/σ^{i}𝔭 are all isomorphic to E = F [y]/𝔭_{y}, and for a_{i} ∈ E^{×} we have

(1 − σ)(a_{0}, a_{1}, …, a_{p−1}) = (a_{0}a_{p−1}^{−1}, a_{1}a_{0}^{−1}, …, a_{p−1}a_{p−2}^{−1})

in ∏_{i=0}^{p−1}(F [x]/σ^{i}𝔭)^{×}. This vanishes if and only if the a_{i} agree, in which case (a_{0}, …, a_{p−1}) is the image of a ∈ E^{×}. On the other hand, if σ fixes 𝔭 then F [x]/𝔭 is a cyclic Galois extension of E = F [y]/𝔭_{y}. Therefore if a ∈ F [x]/𝔭 and (1 − σ)a = a/(σa)^{−1} equals 1, then a = σ(a), i.e., a ∈ E. This establishes the claim.

A diagram chase shows that since 1 = ∂u = ∂(1 − σ)v, there is a v_{0} in K_{2}F (y) with ∂(v) = ∂(i^{∗}v_{0}). Since i^{∗} = σi^{∗}, we have (1 − σ)i^{∗}v_{0} = 1. Replacing v by v(i^{∗}v_{0})^{−1}, we may assume that ∂(v) = 1, i.e., that v is in the subgroup K_{2}(F) of K_{2}F (x). Therefore we have u = v(σv)^{−1} = 1. As u was any element of K_{2}(F) satisfying u^{p} = 1, K_{2}(F) has no p-torsion.

Example 6.7.1. If F = 𝔽_{q}(t), q = p^{r}, we have K_{2}(F) = ∐(𝔽_{q}[t]/𝔭)^{×}. Since the units of each finite field 𝔽_{q}[t]/𝔭 form a cyclic group, and its order can be arbitrarily large (yet prime to p), K_{2}𝔽_{q}(t) is a very large torsion group.

Theorem 6.8. If F contains a primitive n^{th} root of unity ζ, then every element of K_{2}(F) of exponent n has the form {ζ, x} for some x ∈ F^{×}.

Proof. We first suppose that n is a prime number p. Let x be an indeterminate and y = x^{p}; the Galois group of the field extension F (x)/F (y) is generated by an automorphism σ satisfying σ(x) = ζx. By Application 6.5.2, K_{2}(F) is a subgroup of K_{2}F (x), and by the projection formula the norm N: K_{2}F (x) → K_{2}F (y) sends u ∈ K_{2}(F) to u^{p}.

Fix u ∈ K_{2}(F) satisfying u^{p} = 1. By Hilbert’s Theorem 90 for K_{2}, if u^{p} = 1 then u = (1 − σ)v = v(σv)^{−1} for some v ∈ K_{2}F (x).

Now the extension F [y] ⊂ F [x] is unramified at every prime ideal except 𝔭 = (x). As in the proof of Theorem 6.7, we have a commutative diagram whose bottom row is exact:

K_{2}F (y)  ⟶^{i^{∗}}  K_{2}F (x)  ⟶^{1−σ}  K_{2}F (x)
∂′↓  _{∂′}↓  _{∂′}↓
∐_{𝔭_{y}≠(y)}(F [y]/𝔭_{y})^{×} ⟶^{Δ} ∐_{𝔭≠(x)}(F [x]/𝔭)^{×} ⟶^{1−σ} ∐_{𝔭≠(x)}(F [x]/𝔭)^{×}

As before, we may modify v by an element from K_{2}F (y) to arrange that ∂_{𝔭}(v) = 1 for all 𝔭 ≠ (x). For 𝔭 = (x), let a ∈ F = F [x]/(x) be such that ∂_{(x)}(v) = a and set v′ = v{a, x}. Then ∂_{(x)}(v′) = 1 and ∂_{𝔭}(v′) = ∂_{𝔭}(v) = 1 for every other 𝔭. It follows from 6.5.2 that v′ is in K_{2}(F). Therefore (1 − σ)v′ = 1; since v = v′{a, x}^{−1} this implies that u has the asserted form:

u = (1 − σ){a, x}^{−1} = {a, x}^{−1}{a, ζx} = {a, ζ}.

Now we proceed inductively, supposing that n = mp and that the theorem has been proven for m (and p). If u ∈ K_{2}(F) has exponent n then u^{p} has exponent m, so there is an x ∈ F^{×} so that u^{p} = {ζ^{p}, x}. The element u{ζ^{p}, x}^{−1} has exponent p, so it equals {ζ^{m}, y} = {ζ, y^{m}} for some y ∈ F^{×}. Hence u = {ζ, xy^{m}}, as required.

Remark 6.8.1. Suslin also proved the following result in [Su87]. Let F be a field containing a primitive p^{th} root of unity ζ, and let F_{0} ⊂ F be the subfield of constants. If x ∈ F_{0}^{×} and {ζ, x} = 1 in K_{2}(F) then {ζ, x} = 1 in K_{2}(F_{0}). If {ζ, y} = 1 in K_{2}(F) for some y ∈ F^{×} then y = xz^{p} for some x ∈ F_{0}^{×} and z ∈ F^{×}.

Application 6.8.2. We can use Theorem 6.8 to give another proof of Theorem 6.4, that when F is an algebraically closed field, the group K_{2}(F) is uniquely divisible. Fix a prime p. For each a ∈ F^{×} there is an α with α^{p} = a. Hence {a, b} = {α, b}^{p}, so K_{2}(F) is p-divisible. If p ≠ char(F) then there is no p-torsion because {ζ, a} = {ζ, α}^{p} = 1. Finally, if char(F) = p, there is no p-torsion either by Theorem 6.7.

Application 6.8.3 (K_{2}ℝ). Theorem 6.8 states that {−1, −1} is the only element of order 2 in K_{2}ℝ. Indeed, if r is a positive real number then:

\{−1, r\} = \{−1, \sqrt{r}\}^{2} = 1, and \{−1, −r\} = \{−1, −1\}\{−1, r\} = \{−1, −1\}.

Note that {−1, −1} is in the image of K_{2}(ℤ), which is a summand by either Example 6.2.1 or Example 5.9.1. Recall from Example 6.1.6 that the image of K_{2}ℝ in the uniquely divisible group K_{2}ℂ is the eigenspace K_{2}ℂ^{+}, and that the composition K_{2}ℝ → K_{2}ℂ ⟶^{N} K_{2}ℝ is multiplication by 2, so its kernel is K_{2}(ℤ). It follows that

K_{2}ℝ ≅ K_{2}(ℤ) ⊕ K_{2}ℂ^{+}.

K_{2} and the Brauer group

Let F be a field. Recall from II.5.4.3 that the Brauer group Br(F) is generated by the classes of central simple algebras with two relations: [A⊗_{F} B] = [A]·[B] and [M_{n}(F)] = 0. Here is one classical construction of elements in the Brauer group; it is a special case of the construction of crossed product algebras.

Cyclic algebras 6.9. Let ζ be a primitive n^{th} root of unity in F, and α, β ∈ F^{×}. The cyclic algebra A = A_{ζ}(α, β) is defined to be the associative algebra with unit, which is generated by two elements x, y subject to the relations x^{n} = α · 1, y^{n} = β · 1 and yx = ζxy. Thus A has dimension n^{2} over F, a basis being the monomials x^{i}y^{j} with 0 ≤ i, j < n. The identity (x + y)^{n} = (α + β) · 1 is also easy to check.

When n = 2 (so ζ = −1), cyclic algebras are called quaternion algebras. The name comes from the fact that the usual quaternions ℍ are the cyclic algebra A(−1, −1) over ℝ. Quaternion algebras arise in the Hasse invariant of quadratic forms.

It is classical, and not hard to prove, that A is a central simple algebra over F; see [BA, §8.5]. Moreover, the n-fold tensor product A ⊗_{F} A ⊗_{F} ⋯ ⊗_{F} A is a matrix algebra; see [BA, Theorem 8.12]. Thus we can consider [A] ∈ Br(F) as an element of exponent n. We shall write _{n} Br(F) for the subgroup of Br(F) consisting of all elements x with x^{n} = 1, so that [A] ∈ _{n} Br(F)

For example, the following lemma shows that A_{ζ}(1, β) must be a matrix ring because x^{n} = 1. Thus [A_{ζ}(1, β)] = 1 in Br(F).

Lemma 6.9.1. Let A be a central simple algebra of dimension n^{2} over a field F containing a primitive n^{th} root of unity ζ. If A contains an element u ∉ F such that u^{n} = 1, then A ≅ M_{n}(F).

Proof. The subalgebra F [u] of A spanned by u is isomorphic to the commutative algebra F [t]/(t^{n} − 1). Since t^{n} − 1 = ∏(t − ζ^{i}), the Chinese Remainder Theorem yields F [u] ≅ F ×F ×⋯×F. Hence F [u] contains n idempotents e_{i} with e_{i}e_{j} = 0 for i ≠ j. Therefore A splits as the direct sum e_{1}A ⊕ ⋯ ⊕ e_{n}A of right ideals. By the Artin-Wedderburn theorem, if A = M_{d}(D) then A can be the direct sum of at most d right ideals. Hence d = n, and A must be isomorphic to M_{n}(F).

Proposition 6.9.2 (The n^{th} power norm residue symbol). If F contains a primitive n^{th} root of unity, there is a homomorphism K_{2}(F) → Br(F) sending {α, β} to the class of the cyclic algebra A_{ζ}(α, β).

Since the image is a subgroup of exponent n, we shall think of the power norm residue symbol as a map K_{2}(F)/nK_{2}(F) → _{n} Br(F).

This homomorphism is sometimes also called the Galois symbol.

Proof. From Ex. 6.12 we see that in Br(F) we have [A_{ζ}(α, β)] · [A_{ζ}(α, γ)] = [A_{ζ}(α, βγ)]. Thus the map F^{×} × F^{×} → Br(F) sending (α, β) to [A_{ζ}(α, β)] is bilinear. To see that it is a Steinberg symbol we must check that A = A_{ζ}(α, 1 − α) is isomorphic to the matrix algebra M_{n}(F). Since the element x + y of A satisfies (x + y)^{n} = 1, Lemma 6.9.1 implies that A must be isomorphic to M_{n}(F).

Remark 6.9.3. Merkurjev and Suslin proved in [MS] that K_{2}(F)/mK_{2}(F) is isomorphic to the subgroup _{m} Br(F) of elements of order m in Br(F) when μ_{m} ⊂ F. By Matsumoto’s Theorem, this implies that the m-torsion in the Brauer group is generated by cyclic algebras. The general description of K_{2}(F)/m, due to Merkurjev-Suslin, is given in 6.10.4; see VI.3.1.1.

The Galois symbol

We can generalize the power norm residue symbol to fields not containing enough roots of unity by introducing Galois cohomology. Here are the essential facts we shall need; see [WHomo] or [Milne].

Sketch of Galois Cohomology 6.10. Let F_{sep} denote the separable closure of a field F, and let G = G_{F} denote the Galois group Gal(F_{sep}/F). The family of subgroups G_{E} = Gal(F_{sep}/E), as E runs over all finite extensions of F, forms a basis for a topology of G. A G-module M is called discrete if the multiplication G × M → M is continuous.

For example, the abelian group G_{m} = F_{sep}^{×} of units of F_{sep} is a discrete module, as is the subgroup μ_{m} of all m^{th} roots of unity. We can also make the tensor product of two discrete modules into a discrete module, with G acting diagonally. For example, the tensor product μ_{m}^{⊗2} = μ_{m} ⊗ μ_{m} is also a G-discrete module. Note that the three G-modules ℤ/m, μ_{m} and μ_{m}^{⊗2} have the same underlying abelian group, but are isomorphic G_{F} -modules only when μ_{m} ⊂ F.

The G-invariant subgroup M^{G} of a discrete G-module M is a left exact functor on the category of discrete G_{F} -modules. The Galois cohomology groups H_{et}^{i}(F; M) are defined to be its right derived functors. In particular, H_{et}^{0}(F; M) is just M^{G}.

If E is a finite separable extension of F then G_{E} ⊂ G_{F}. Thus there is a forgetful functor from G_{F} -modules to G_{E}-modules, inducing maps H_{et}^{i}(F; M) → H_{et}^{i}(E; M). In the other direction, the induced module functor from G_{E}-modules to G_{F} -modules gives rise to cohomological transfer maps tr_{E/F}: H_{et}^{i}(E; M) → H_{et}^{i}(F; M); see [WHomo, 6.3.9 and 6.11.11].

Example 6.10.1 (Kummer Theory). The cohomology of the module G_{m} is of fundamental importance. Of course H_{et}^{0}(F, G_{m}) = F^{×}. By Hilbert’s Theorem 90 for units, and a little homological algebra [WHomo, 6.11.16], we also have H_{et}^{1}(F; G_{m}) = 0 and H_{et}^{2}(F; G_{m}) ≅ Br(F).

If m is prime to char(F), the exact sequence of discrete modules

1 → μ_{m} → G_{m} ⟶^{m} G_{m} → 1

is refered to as the Kummer sequence. Writing μ_{m}(F) for the group μ_{m}^{G} of all m^{th} roots of unity in F, the corresponding cohomology sequence is called the Kummer sequence.

1 →μ_{m}(F) → F^{×} ⟶^{m} F^{×} → H_{et}^{1}(F; μ_{m}) → 1
1 →H_{et}^{2}(F; μ_{m}) → Br(F) ⟶^{m} Br(F)

This yields isomorphisms H_{et}^{1}(F; μ_{m}) ≅ F^{×}/F^{×m} and H_{et}^{2}(F; μ_{m}) ≅ _{m} Br(F). If μ_{m} ⊂ F^{×}, this yields a natural isomorphism H_{et}^{2}(F; μ_{m}^{⊗2}) ≅ _{m} Br(F) ⊗ μ_{m}(F).

There are also natural cup products in cohomology, such as the product

(6.10.2)  F^{×} ⊗ F^{×} → H_{et}^{1}(F; μ_{m}) ⊗ H_{et}^{1}(F; μ_{m}) ⟶^{∪} H_{et}^{2}(F; μ_{m}^{⊗2})

which satisfies the following projection formula: if E/F is a finite separable extension, a ∈ F^{×} and b ∈ E^{×}, then tr_{E/F} (a ∪ b) = a ∪ N_{E/F} (b).

Proposition 6.10.3 (Galois symbol). The bilinear pairing (6.10.2) induces a Steinberg symbol K_{2}(F)/mK_{2}(F) → H_{et}^{2}(F; μ_{m}^{⊗2}) for every m prime to char(F).

Proof. It suffices to show that a∪(1−a) vanishes for every a ∈ F −{0, 1}. Fixing a, factor the separable polynomial t^{m} −a = ∏ f_{i} in F [t] with the f_{i} irreducible, and let F_{i} denote the field F (x_{i}) with f_{i}(x_{i}) = 0. Setting t = 1, 1−a = ∏_{i} N_{F_{i}/F} (1−x_{i}). Writing H_{et}^{2} additively, we have

a ∪ (1 − a) = ∑_{i} a ∪ N_{F_{i}/F} (1 − x_{i}) = ∑_{i} tr_{F_{i}/F} (a ∪ (1 − x_{i}))

= m ∑_{i} tr_{F_{i}/F} (x_{i} ∪ (1 − x_{i})).

Since the group H_{et}^{2}(F; μ_{m}^{⊗2}) has exponent m, all these elements vanish, as desired.

Remark 6.10.4. Suppose that F contains a primitive m^{th} root of unity ζ. If we identify ℤ/m with μ_{m} via 1 ↦ ζ, we have a natural isomorphism

m Br(F) ≅ _{m} Br(F) ⊗ ℤ/m ≅ _{m} Br(F) ⊗ μ_{m} ≅ H_{et}^{2}(F; μ_{m}^{⊗2}).

Tate showed in [Tate] that this isomorphism identifies the Galois symbol of Proposition 6.10.3 with the m^{th} power norm residue symbol of Proposition 6.9.2. The Merkurjev-Suslin isomorphism of [MS] cited above in Remark 6.9.3 is a special case of the more general assertion that this symbol induces an isomorphism for all fields F of characteristic prime to m: K_{2}(F)/mK_{2}(F) ≅ H_{et}^{2}(F; μ_{m}^{⊗2}). See chapter VI, 3.1.1.

EXERCISES

6.1 Given a discrete valuation on a field F, with residue field k and parameter π, show that there is a surjection λ: K_{2}(F) → K_{2}(k) given by the formula λ{uπ^{i}, vπ^{j}} = {ū, v̄}. Example 6.1.2 is a special case of this, in which π = t^{−1}. 6.2 (Bass-Tate) If E = F (u) is a field extension of F, and e_{1}, e_{2} ∈ E are monic polynomials in u of some fixed degree d > 0, show that {e_{1}, e_{2}} is a product of symbols {e_{1}, e′_{2}} and {e, e′′_{2}} with e, e′_{2}, e′′_{2} polynomials of degree < d. This generalizes Lemma 6.1.4. 6.3 (Bass-Tate) Let k be a field and set F = k((t)).

(a) Show that K_{2}(F) ≅ K_{2}(k) × k^{×} × K_{2}(k[[t]], t).

(b) Show that the group K_{2}(k[[t]], t) is torsionfree; by Ex. 5.14, it is uniquely divisible if char(k) = 0. Hint: Use 6.7 and the proof of 6.4.1.

6.4 If F is a number field with r_{1} distinct embeddings F ↪ ℝ, show that the r_{1} symbols (,)_{∞} on F define a surjection K_{2}(F) → {±1}^{r_{1}}. 6.5 If F̄ denotes the algebraic closure of a field F, show that K_{2}(ℚ̄) = K_{2}(𝔽̄_{p}) = 1. 6.6 2-adic symbol on ℚ. Any nonzero rational number r can be written uniquely as r = (−1)^{i}2^{j}5^{k}u, where i, k ∈ {0, 1} and u is a quotient of integers congruent to 1 (mod 8). If s = (−1)^{i′}2^{j′}5^{k′}u′, set (r, s)_{2} = (−1)^{ii′+jj′+kk′}. Show that this is a Steinberg symbol on ℚ, with values in {±1}.

6.7 Let ((r, s))_{p} denote the Hilbert symbol on ℚ̂_{p} (6.2.2), and (r, s)_{p} the tame symbol K_{2}(ℚ̂_{p}) → 𝔽_{p}^{×}. Assume that p is odd, so that there is a unique surjection ε: 𝔽_{p}^{×} → {±1}. Show that ((r, s))_{p} = ε (r, s)_{p} for all r, s ∈ ℚ̂_{p}^{×}.

6.8 Quadratic Reciprocity. If r, s ∈ ℚ^{×}, show that (r, s)_{∞}(r, s)_{2} ∏_{p≠2}((r, s))_{p} = +1. Here (r, s)_{2} is the 2-adic symbol of Ex. 6.6.

Hint: From 6.5.1 and Ex. 6.7, the 2-adic symbol of Ex. 6.6 must satisfy some relation of the form

(r, s)_{2} = (r, s)_{∞}^{ε_{∞}} ∏_{p≠2}((r, s))_{p}^{ε_{p}},

where the exponents ε_{p} are either 0 or 1. Since (−1, −1)_{2} = (−1, −1)_{∞} we have ε_{∞} = 1. If p is a prime not congruent to 1 (mod 8), consider {2, p} and {−1, p}. If p is a prime congruent to 1 (mod 8), Gauss proved that there is a prime q < \sqrt{p} such that p is not a quadratic residue modulo q. Then ((p, q))_{q} = −1, even though (p, q)_{∞} = (p, q)_{2} = 1. Since we may suppose inductively that ε_{q} equals 1, this implies that ε_{p} ≠ 0. 6.9 (Suslin) Suppose that a field F is algebraically closed in a larger field E. Use Lemma 6.1.3 and Remark 6.8.1 to show that K_{2}(F) injects into K_{2}(E). 6.10 Let F be a field, and let Ω_{F}^{1} = Ω_{F/ℤ}^{1} denote the vector space of absolute Kähler differentials (see Ex. 2.6). The n^{th} exterior power of Ω_{F}^{1} is written as Ω_{F}^{n}. Show that there is a homomorphism K_{2}(F) → Ω_{F}^{2} sending \{x, y\} to \frac{dx}{x} ∧ \frac{dy}{y}. This map is not onto, because the image is in the kernel of the de Rham differential d : Ω_{F}^{2} → Ω_{F}^{3} . 6.11 If F is a field of transcendence degree κ over the ground field, Ω_{F} is a vector space of dimension κ. Now suppose that κ is an infinite cardinal number, so that Ω_{F}^{n} is also a vector space of dimension κ for all n > 1. Show that the image of the map K_{2}(F) → Ω_{F}^{2} in the previous exercise is an abelian group of rank κ.

In particular, if F is a local field then the uniquely divisible summand U of K_{2}(F) in Moore’s Theorem (6.2.4) is uncountable. 6.12 Show that A_{ζ}(α, β) ⊗ A_{ζ}(α, γ) ≅ M_{n}(A), where A = A_{ζ}(α, βγ). Hint: Let x′, y′ generate A_{ζ}(α, β) and x′′, y′′ generate A_{ζ}(α, γ), and show that x′, y = y′y′′ generate A. Then show that u = (x′)^{−1}x′′ + y′′ has u^{n} = 1. (For another proof, see [BA, Ex. 8.5.2].)

§7. Milnor K-theory of fields

Fix a field F, and consider the tensor algebra of the group F^{×},

T (F^{×}) = ℤ ⊕ F^{×} ⊕ (F^{×} ⊗ F^{×}) ⊕ (F^{×} ⊗ F^{×} ⊗ F^{×}) ⊕ · · · .

To keep notation straight, we write l(x) for the element of degree one in T (F^{×}) corresponding to x ∈ F^{×}.

Definition 7.1. The graded ring K_{∗}^{M}(F) is defined to be the quotient of T (F^{×}) by the ideal generated by the homogeneous elements l(x) ⊗ l(1 − x) with x ≠ 0, 1. The Milnor K-group K_{n}^{M}(F) is defined to be the subgroup of elements of degree n. We shall write {x_{1}, …, x_{n}} for the image of l(x_{1}) ⊗ ⋯ ⊗ l(x_{n}) in K_{n}^{M}(F).

That is, K_{n}^{M}(F) is presented as the group generated by symbols {x_{1}, …, x_{n}} subject to two defining relations: {x_{1}, …, x_{n}} is multiplicative in each x_{i}, and equals zero if x_{i} + x_{i+1} = 1 for some i.

The name comes from the fact that the ideas in this section first arose in Milnor’s 1970 paper [M-QF]. Clearly we have K_{0}^{M}(F) = ℤ, and K_{1}^{M} = F^{×} (with the group operation written additively). By Matsumoto’s Theorem 6.1 we also have K_{2}^{M}(F) = K_{2}(F), the elements {x, y} being the usual Steinberg symbols, except that the group operation in K_{2}^{M}(F) is written additively.

Since {x_{i}, x_{i+1}}+{x_{i+1}, x_{i}} = 0 in K_{2}^{M}(F), we see that interchanging two entries in {x_{1}, …, x_{n}} yields the inverse. It follows that these symbols are alternating: for any permutation π with sign (−1)^{π} we have

{x_{π1}, …, x_{πn}} = (−1)^{π}{x_{1}, …, x_{n}}.

Examples 7.2. (a) If 𝔽_{q} is a finite field, then K_{n}^{M}(𝔽_{q}) = 0 for all n ≥ 2, because K_{2}^{M}(𝔽_{q}) = 0 by Cor. 6.1.1. If F has transcendence degree 1 over a finite field (a global field of finite characteristic), Bass and Tate proved in [BT] that K_{n}^{M}(F) = 0 for all n ≥ 3.

(b) If F is algebraically closed then K_{n}^{M}(F) is uniquely divisible. Divisibility is clear because F^{×} is divisible. The proof that there is no p-torsion is the same as the proof for n = 2 given in Theorem 6.4, and is relegated to Ex. 7.3.

(c) When F = ℝ we can define a symbol K_{n}^{M}(ℝ) → {±1} by the following formula: (x_{1}, …, x_{n})_{∞} equals −1 if all the x_{i} are negative, and equals +1 otherwise. When n = 2 this will be the symbol defined in Example 6.2.1.

To construct it, extend ℤ → ℤ/2 to a ring homomorphism T (ℝ^{×}) → (ℤ/2)[t] by sending l(x) to t if x < 0 and to 0 if x > 0. This sends the elements l(x) ⊗ l(1 − x) to zero (as in 6.2.1), so it induces a graded ring homomorphism K_{∗}^{M}(ℝ) → (ℤ/2)[t]. The symbol above is just the degree n part of this map.

By induction on n, it follows that K_{n}^{M}(ℝ) is the direct sum of a cyclic group of order 2 generated by {−1, …, −1}, and a divisible subgroup. In particular, this shows that K_{∗}^{M}(ℝ)/2K_{∗}^{M}(ℝ) is the polynomial ring (ℤ/2)[ϵ] on ϵ = l(−1). Using the norm map we shall see later that the divisible subgroup of K_{n}^{M}(ℝ) is in fact uniquely divisible. This gives a complete description of each K_{n}^{M}(ℝ) as an abelian group.

(d) When F is a number field, let r_{1} be the number of embeddings of F into ℝ. Then we have a map from K_{n}^{M}(F) to the torsion subgroup (ℤ/2)^{r_{1}} of K_{n}^{M}(ℝ)^{r_{1}}.

Bass and Tate proved in [BT] that this map is an isomorphism for all n ≥ 3: K_{n}^{M}(F) ≅ (ℤ/2)^{r_{1}}.

Tame symbols

Recall from Lemma 6.3 and Ex. 6.1 that every discrete valuation v on F induces a Steinberg symbol K_{2}(F) ⟶^{∂_{v}} k_{v}^{×} and a map K_{2}(F) ⟶^{λ} K_{2}(k_{v}). These symbols extend to all of Milnor K-theory; the ∂_{v} are called higher tame symbols, and the λ are called specialization maps.

Theorem 7.3 (Higher tame symbols and Specialization). For every discrete valuation v on F, there are two surjections

K_{n}^{M}(F) ⟶^{∂_{v}} K_{n−1}^{M}(k_{v}) and K_{n}^{M}(F) ⟶^{λ} K_{n}^{M}(k_{v})

satisfying the following conditions. Let R = {r ∈ F : v(r) ≥ 0} be the valuation ring, and π a parameter for v. If u_{i} ∈ R^{×}, and ū_{i} denotes the image of u_{i} in k_{v} = R/(π) then

λ{u_{1}π^{i_{1}}, …, u_{n}π^{i_{n}}} = {ū_{1}, …, ū_{n}}, ∂_{v}{π, u_{2}, …, u_{n}} = {ū_{2}, …, ū_{n}}.

In particular, ∂_{v}: K_{2}^{M}(F) → k_{v}^{×} is the tame symbol of Lemma 6.3, and λ: K_{2}(F) → K_{2}(k) is the map of Example 6.1.2 and Ex. 6.1.

Proof. (Serre) Let L denote the graded K_{∗}^{M}(k_{v})-algebra generated by an indeterminate Π in L_{1}, with the relation {Π, Π} = {−1, Π}. We claim that the group homomorphism

d: F^{×} → L_{1} = l(k_{v}^{×}) ⊕ ℤ · Π, d(uπ^{i}) = l(ū) + iΠ

satisfies the relation: for r ≠ 0, 1, d(r)d(1 − r) = 0 in L_{2}. If so, the presentation of K_{∗}^{M}(F) shows that d extends to a graded ring homomorphism d: K_{∗}^{M}(F) → L. Since L_{n} is the direct sum of K_{n}^{M}(k_{v}) and K_{n−1}^{M}(k_{v}), we get two maps: λ: K_{n}^{M}(F) → K_{n}^{M}(k_{v}) and ∂_{v}: K_{n}^{M}(F) → K_{n−1}^{M}(k_{v}). The verification of the relations is routine, and left to the reader.

If 1 ≠ r ∈ R^{×}, then either 1 − r ∈ R^{×} and d(r)d(1 − r) = {r̄, 1 − r̄} = 0, or else v(1 − r) = i > 0 and d(r) = l(1) + 0 · Π = 0 so d(r)d(1 − r) = 0 · d(1 − r) = 0. If v(r) > 0 then 1 − r ∈ R^{×} and the previous argument implies that d(1 − r)d(r) = 0. If r ∉ R then 1/r ∈ R, and we see from (5.10.3) and the above that d(r)d(1 − r) = d(1/r)d(−1/r). Therefore it suffices to show that d(r)d(−r) = 0 for every r ∈ R. If r = π this is the given relation upon L, and if r ∈ R^{×} then d(r)d(−r) = {r, −r} = 0 by (5.10.3). Since the product in L is anticommutative, the general case r = uπ^{i} follows from this.

Corollary 7.3.1 (Rigidity). Suppose that F is complete with respect to the valuation v, with residue field k = k_{v}. For every integer q prime to char(k), the maps λ ⊕ ∂_{v}: K_{n}^{M}(F)/q → K_{n}^{M}(k)/q ⊕ K_{n−1}^{M}(k)/q are isomorphisms for every n.

Proof. Since the valuation ring R is complete, Hensel’s Lemma implies that the group 1+πR is q-divisible. It follows that l(1+πR)·K_{n−1}^{M}(F) is also q-divisible. But by Ex. 7.2 this is the kernel of the map d: K_{n}^{M}(F) → L_{n} ≅ K_{n}^{M}(k_{v})⊕K_{n−1}^{M}(k_{v}).

Leading Coefficients 7.3.2. As in Example 6.1.2, K_{n}^{M}(F) is a direct summand of K_{n}^{M}F (t). To see this, we consider the valuation v_{∞}(f) = −deg(f) on F (t) of Example 6.5.3. Since t^{−1} is a parameter, each polynomial f = ut^{−i} has lead(f) = ū. The map λ: K_{n}^{M}F (t) → K_{n}^{M}(F), given by λ{f_{1}, …, f_{n}} = {lead(f_{1}), …, lead(f_{n})}, is clearly inverse to the natural map K_{n}^{M}(F) → K_{n}^{M}F (t).

Except for v_{∞}, every discrete valuation v on F (t) which is trivial on F is the 𝔭-adic valuation v_{𝔭} associated to a prime ideal 𝔭 of F [t]. In this case k_{v} is the field F [t]/𝔭, and we write ∂_{𝔭} for ∂_{v}.

Theorem 7.4 (Milnor). There is a split exact sequence for each n, natural in the field F, and split by the map λ:

0 → K_{n}^{M}(F) → K_{n}^{M}F (t) \underrightarrow{∂=∐ ∂_{𝔭}} ∐_{𝔭} K_{n−1}^{M}(F [t]/𝔭) → 0.

Proof. Let L_{d} denote the subgroup of K_{n}^{M}F (t) generated by those symbols {f_{1}, …, f_{r}} such that all the polynomials f_{i} have degree ≤ d. By Example 7.3.2, L_{0} is a summand isomorphic to K_{n}^{M}(F). Since K_{n}^{M}F (t) is the union of the subgroups L_{d}, the theorem will follow from Lemma 7.4.2 below, using induction on d.

Let π be an irreducible polynomial of degree d and set k = k_{π} = F [t]/(π). Then each element ā of k is represented by a unique polynomial a ∈ F [t] of degree < d.

Lemma 7.4.1. There is a unique homomorphism h = h_{π}: K_{n−1}^{M}(k) → L_{d}/L_{d−1} carrying {ā_{2}, …, ā_{n}} to the class of {π, a_{2}, …, a_{n}} modulo L_{d−1}.

Proof. The formula gives a well-defined set map h from k^{×} × ⋯ × k^{×} to L_{d}/L_{d−1}. To see that it is linear in ā_{2}, suppose that ā_{2} = ā′_{2}ā′′_{2}. If a_{2} ≠ a′_{2}a′′_{2} then there is a nonzero polynomial f of degree < d with a_{2} = a′_{2}a′′_{2} + fπ. Since fπ/a_{2} = 1 − a′_{2}a′′_{2}/a_{2} we have {fπ/a_{2}, a′_{2}a′′_{2}/a_{2}} = 0. Multiplying by {a_{3}, …, a_{n}} gives

{π, a′_{2}a′′_{2}/a_{2}, a_{3}, …, a_{n}} ≡ 0 modulo L_{d−1}.

Similarly, h is linear in a_{3}, …, a_{n}. To see that the multilinear map h factors through K_{n−1}^{M}(k), we observe that if ā_{i} + ā_{i+1} = 1 in k then a_{i} + a_{i+1} = 1 in F.

Lemma 7.4.2. The homomorphisms ∂_{(π)} and h_{π} induce an isomorphism between L_{d}/L_{d−1} and the direct sum ⊕_{π}K_{n−1}^{M}(k_{π}) as π ranges over all monic irreducible polynomials of degree d in F [t].

Proof. Since π cannot divide any polynomial of degree < d, the maps ∂_{(π)} vanish on L_{d−1} and induce maps ¯∂_{(π)}: L_{d}/L_{d−1} → K_{n−1}^{M}(k_{π}). By inspection, the composition of ⊕h_{π} with the direct sum of the ¯∂_{(π)} is the identity on ⊕_{π}K_{n−1}^{M}(k_{π}). Thus it suffices to show that ⊕h_{π} maps onto L_{d}/L_{d−1}. By Ex. 6.2, L_{d} is generated by L_{d−1} and symbols {π, a_{2}, …, a_{n}} where π has degree d and the a_{i} have degree < d. But each such symbol is h_{π} of an element of K_{n−1}^{M}(k_{π}), so ⊕h_{π} is onto.

The Transfer Map

Let v_{∞} be the valuation on F (t) with parameter t^{−1}. The formulas in Theorem 7.3 defining ∂_{∞} show that it vanishes on K_{∗}^{M}(F). By Theorem 7.4, there are unique homomorphisms N_{𝔭}: K_{n}^{M}(F [t]/𝔭) → K_{n}^{M}(F) so that −∂_{∞} = ∑_{𝔭} N_{𝔭}∂_{𝔭}.

Definition 7.5. Let E be a finite field extension of F generated by an element a. Then the transfer map, or norm map N = N_{a/F}: K_{∗}^{M}(E) → K_{∗}^{M}(F), is the unique map N_{𝔭} defined above, associated to the kernel 𝔭 of the map F [t] → E sending t to a.

We can calculate the norm of an element x ∈ K_{n}^{M}(E) as N_{𝔭}(x) = −∂_{v_{∞}}(y), where y ∈ K_{n+1}^{M}F (t) is such that ∂_{𝔭}(y) = x and ∂_{𝔭′}(y) = 0 for all 𝔭′ ≠ 𝔭.

If n = 0, the transfer map N: ℤ → ℤ is multiplication by the degree [E : F] of the field extension, while if n = 1 the map N: E^{×} → F^{×} is the usual norm map; see Ex. 7.5. We will show in 7.6 below that N is independent of the choice of a ∈ E for all n. First we make two elementary observations.

If we let N_{∞} denote the identity map on K_{n}^{M}(F), and sum over the set of all discrete valuations on F (t) which are trivial on F, the definition of the N_{v} yields:

Weil Reciprocity Formula 7.5.1. ∑_{v} N_{v}∂_{v}(x) = 0 for all x ∈ K_{n}^{M}F (t).

Projection Formula 7.5.2. Let E = F (a). Then for x ∈ K_{∗}^{M}(F) and y ∈ K_{∗}^{M}(E) the map N = N_{a/F} satisfies N{x, y} = {x, N(y)}.

Proof. The inclusions of F in F (t) and F [t]/𝔭 allow us to view K_{∗}^{M}F (t) and K_{∗}^{M}(F [t]/𝔭) as graded modules over the ring K_{∗}^{M}(F). It follows from Theorem 7.4 that each ∂_{𝔭} is a graded module homomorphism of degree −1. This remark also applies to v_{∞} and ∂_{∞}, because F (t) = F (t^{−1}). Therefore each N_{𝔭} is a graded module homomorphism of degree 0.

Taking y = 1 in K_{0}^{M}(E) = ℤ, so N(y) = [E : F] by Ex. 7.5, this yields

Corollary 7.5.3. If the extension E/F has degree d, then the composition K_{∗}^{M}(F) → K_{∗}^{M}(E) ⟶^{N} K_{∗}^{M}(F) is multiplication by d. In particular, the kernel of K_{∗}^{M}(F) → K_{∗}^{M}(E) is annihilated by d.

Definition 7.6. Let E = F (a_{1}, …, a_{r}) be a finite field extension of F. The transfer map N_{E/F}: K_{∗}^{M}(E) → K_{∗}^{M}(F) is defined to be the composition of the transfer maps defined in 7.5:

K_{n}^{M}(E) ⟶^{N_{ar}} K_{n}^{M}(F (a_{1}, …, a_{r−1})) ⟶^{N_{a_{r−1}}} ⋯ K_{n}^{M}(F (a_{1})) ⟶^{N_{a_{1}}} K_{n}^{M}(F).

The transfer map is well-defined by the following result of K. Kato.

Theorem 7.6.1 (Kato). The transfer map N_{E/F} is independent of the choice of elements a_{1}, …, a_{r} such that E = F (a_{1}, …, a_{r}). In particular, if F ⊂ F′ ⊂ E then N_{E/F} = N_{F′/F} N_{E/F′}.

The key trick used in the proof of this theorem is to fix a prime p and pass from F to the union F′ of all finite extensions of F of degree prime to p. By Corollary 7.5.3 the kernel of K_{n}^{M}(F) → K_{n}^{M}(F′) has no p-torsion, and the degree of every finite extension of F′ is a power of p.

Lemma 7.6.2. (Kato) If E is a normal extension of F, and [E : F] is a prime number p, then the map N_{E/F} = N_{a/F}: K_{∗}^{M}(E) → K_{∗}^{M}(F) does not depend upon the choice of a such that E = F (a).

Proof. If also E = F (b), then from Corollary 7.5.3 and Ex. 7.7 with F′ = E we see that δ(x) = N_{a/F} (x) − N_{b/F} (x) is annihilated by p. If δ(x) ≠ 0 for some x ∈ K_{n}^{M}(E) then, again by Corollary 7.5.3, δ(x) must be nonzero in K_{n}^{M}(F′), where F′ is the union of all finite extensions of F of degree prime to p. Again by Ex. 7.7, we see that we may replace F by F′ and x by its image in K_{n}^{M}(EF′). Since the degree of every finite extension of F′ is a power of p, the assertion for F′ follows from Ex. 7.6, since the Projection Formula 7.5.2 yields N_{a/F′}{y, x_{2}, …, x_{n}} = {N(y), x_{2}, …, x_{n}}.

Corollary 7.6.3. If in addition F is a complete discrete valuation field with residue field k_{v}, and the residue field of E is k_{w}, the following diagram commutes.

K_{n}^{M}(E) ⟶^{∂_{w}} K_{n−1}^{M}(k_{w})
N↓  ↓^{N}
K_{n}^{M}(F) ⟶^{∂_{v}} K_{n−1}^{M}(k_{v})

Proof. Ex. 7.6 implies that for each u ∈ K_{n}^{M}(E) there is a finite field extension F′ of F such that [F′ : F] is prime to p and the image of u in K_{n}(EF′) is generated by elements of the form u′ = {y, x_{2}, …, x_{n}} (y ∈ EF′, x_{i} ∈ F′). By Ex. 7.7 and Ex. 7.8 it suffices to prove that N_{k_{w}/k_{v}}∂_{w}(u) = ∂_{v}(N_{EF′/F′}u) for every element u of this form. But this is an easy computation.

Proposition 7.6.4 (Kato). Let E and F′ = F (a) be extensions of F with E/F normal of prime degree p. If E′ = E(a) denotes the composite field, the following diagram commutes.

K_{∗}^{M}(E′) ⟶^{N_{a/E}} K_{∗}^{M}(E)
N↓  ↓^{N}
K_{∗}^{M}(F′) ⟶^{N_{a/F}} K_{∗}^{M}(F).

Proof. The vertical norm maps are well-defined by Lemma 7.6.2. Let π ∈ F [t] and π′ ∈ E[t] be the minimal polynomials of a over F and E, respectively. Given x ∈ K_{n}^{M}(E′), we have N_{a/E}(x) = −∂_{∞}(y), where y ∈ K_{n+1}^{M}E(t) satisfies ∂_{π′}(y) = x and ∂_{w}(y) = 0 if w ≠ w_{π′}. If v is a valuation on F (t), Ex. 7.9 gives:

∂_{v}(N_{E(t)/F (t)}y) = ∑_{w|v} N_{E(w)/F (v)}(∂_{w}y) = \{ \begin{matrix}N_{E′/F′}(x) & if v = v_{π} \\ N_{E/F} (∂_{∞}y) & if v = v_{∞} \\ 0 & else \end{matrix}

in K_{∗}^{M}(F′). Two applications of Definition 7.5 give the desired calculation:

N_{a/F} (N_{E′/F′}x) = −∂_{∞}(N_{E(t)/F (t)}y) = −N_{E/F} (∂_{∞}y) = N_{E/F} (N_{a/F} x).

Proof of Theorem 7.6.1. As in the proof of Lemma 7.6.2, we see from Corollary 7.5.3 and Ex. 7.7 with F′ = E that the indeterminacy is annihilated by [E : F]. Using the key trick of passing to a larger field, we may assume that the degree of every finite extension of F is a power of a fixed prime p.

Let us call a tower of intermediate fields F = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{r} = E maximal if [F_{i} : F_{i−1}] = p for all i. By Lemma 7.6.2, the transfer maps N: K_{∗}^{M}(F_{i}) → K_{∗}^{M}(F_{i−1}) are independent of the choice of a such that F_{i} = F_{i−1}(a). If F ⊂ F_{1} ⊂ E and F ⊂ F′ ⊂ E are maximal towers, Proposition 7.6.4 states that N_{F′/F} N_{E/F′} = N_{F_{1}/F} N_{E/F_{1}}, because if F′ ≠ F_{1} then E = F′F_{1}. It follows by induction on [E : F] that if F = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{r} = E is a maximal tower then the composition of the norm maps

K_{n}^{M}(E) ⟶^{N} K_{n}^{M}(F_{r−1}) ⟶^{N} ⋯ K_{n}^{M}(F_{1}) ⟶^{N} K_{n}^{M}(F)

is independent of the choice of maximal tower.

Comparing any tower to a maximal tower, we see that it suffices to prove that if F ⊂ F_{1} ⊂ F′ is a maximal tower and F′ = F (a) then N_{a/F} = N_{F_{1}/F} N_{F′/F_{1}}. But this is just Proposition 7.6.4 with E = F_{1} and E′ = F′.

The dlog symbol and ν(n)_{F}

For any field F, we write Ω_{F}^{n} for the nth exterior power of the vector space Ω_{F} = Ω_{F/ℤ} of Kähler differentials (Ex. 2.6). The direct sum over n forms a gradedcommutative ring Ω_{F}^{∗}, and the map dlog: F^{×} → Ω_{F} sending a to \frac{da}{a} extends to a graded ring map from the tensor algebra T (F^{×}) to Ω_{F}^{∗}. By Ex. 6.10, l(a)⊗ l(1−a) maps to zero, so it factors through the quotient ring K_{∗}^{M}(F) of T (F^{×}). We record this observation for later reference.

Lemma 7.7. If F is any field, there is a graded ring homomorphism

dlog: K_{∗}^{M}(F) → Ω_{F}^{∗} , dlog\{a_{1}, …, a_{n}\} = \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} .

Now let F be a field of characteristic p ≠ 0, so that d(a^{p}) = p da = 0. In fact, if {x_{i}} is a p-basis of F over F^{p} then the symbols dx_{i} form a basis of the F -vector space Ω_{F}. Note that the set dΩ_{F}^{n−1} of all symbols da_{1} ∧⋯∧da_{n} forms an F^{p}-vector subspace of Ω_{F}^{n}.

Definition 7.7.1. If char(F) = p ≠ 0, let ν(n)_{F} denote the kernel of the Artin-Schrier operator ℘: Ω_{F}^{n} → Ω_{F}^{n} /dΩ_{F}^{n−1}, which is defined by

℘(x \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} ) = (x^{p} − x) \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} .

(In the literature, ℘ + 1 is the inverse of the “Cartier” operator.)

Clearly ℘(dlog{a_{1}, …, a_{n}}) = 0, so the image of the dlog map lies in ν(n)_{F}. The following theorem, which implies that these symbols span ν(n)_{F}, was proven by Kato [K82] for p = 2, and for general p by Bloch, Kato and Gabber [BK, 2.1].

Theorem 7.7.2. (Bloch-Kato-Gabber) Let F be a field of characteristic p ≠ 0. Then the dlog map induces an isomorphism K_{n}^{M}(F)/pK_{n}^{M}(F) ≅ ν(n)_{F} for every n ≥ 0.

Using this result, Bloch and Kato also proved that the p-torsion subgroup of K_{n}^{M}(F) is divisible [BK, 2.8]. Using this divisibility, Izhboldin found the following generalization of theorem 6.7; see [Izh].

Izhboldin’s Theorem 7.8. If char(F) = p, the group K_{n}^{M}(F) has no p-torsion.

Proof. We proceed by induction on n, the case n = 2 being theorem 6.7. As in the proof of theorem 6.7, let x be an indeterminate and y = x^{p} − x; the field extension F (x)/F (y) is an Artin-Schrier extension, and its Galois group is generated by an automorphism σ satisfying σ(x) = x + 1. By theorem 7.4, we can regard K_{n}^{M}(F) as a subgroup of both K_{n}^{M}F (x) and K_{n}^{M}F (y).

For all field extensions E of F (y) linearly disjoint from F (x), i.e., with no root of t^{p} − t − y, write E(x) for the field E ⊗_{F (y)} F (x). Let I(E) denote the set of all ptorsion elements in K_{n}^{M}E(x) of the form v−σ(v), v ∈ K_{n}^{M}E(x), and let P (E) denote the p-torsion subgroup of the kernel of the norm map N_{x/E}: K_{n}^{M}E(x) → K_{n}^{M}(E). Since Nσ(v) = N(v), I(E) ⊆ P (E). Both I(E) and P (E) vary naturally with E, and are equal by proposition 7.8.2 below.

Fix u ∈ K_{n}^{M}(F) with pu = 0. The projection formula 7.5.2 shows that the norm map K_{n}^{M}F (x) → K_{n}^{M}F (y) sends u to pu = 0. Hence u ∈ P (F (y)). By proposition 7.8.2, u ∈ I(F (y)), i.e., there is a v ∈ K_{n}^{M}F (x) so that u = v − σ(v) in K_{n}^{M}F (x). Now apply the leading coefficient symbol λ of 7.3.2; since λ(σv) = λ(v) we have: u = λ(u) = λ(v) − λ(σv) = 0. This proves Izhboldin’s theorem.

Before proceeding to proposition 7.8.2, we need some facts about the group I(E) defined in the proof of 7.8. We first claim that the transcendental extension E ⊂ E(t) induces an isomorphism I(E) ≅ I(E(t)). Indeed, since E(x, t) is purely transcendental over E(x), theorem 7.4 and induction on n imply that K_{n}^{M}E(x) → K_{n}^{M}E(x, t) is an isomorphism on p-torsion subgroups, and the claim follows because the leading coefficient symbol 7.3.2 commutes with σ.

We next claim that if E/E′ is a purely inseparable field extension then I(E′) → I(E) is onto. For this we may assume that E^{p} ⊆ E′ ⊂ E. The composition of the Frobenius map E → E^{p} with this inclusion induces the endomorphism of K_{n}^{M}(E) sending {a_{1}, …, a_{n}} to {a_{1}^{p}, …, a_{n}^{p}} = p^{n}{a_{1}, …, a_{n}}. Hence this claim follows from the following result.

Lemma 7.8.1. The group I(E) is p-divisible.

Proof. Pick v ∈ K_{n}^{M}E(x) so that u = v − σ(v) is in I(E). Now we invoke the Bloch-Kato result, mentioned above, that the p-torsion subgroup of K_{n}^{M}(L) is divisible for every field L of characteristic p. By theorem 7.7.2, this implies that u vanishes in K_{n}^{M}E(x)/p ≅ ν(n)_{E(x)}. By Ex. 7.12 and theorem 7.7.2, the class of v mod p comes from an element w ∈ K_{n}^{M}(E), i.e., v − w = pv′ for some v′ ∈ K_{n}^{M}E(x). Then u = v − σ(v) = pv′ − pσ(v′), it follows that u′ = v′ − σ(v′) is an element of I(E) with u = pu′.

Proposition 7.8.2. For all E containing F (y), linearly disjoint from F (x), P (E) = I(E).

Proof. We shall show that the obstruction V (E) = P (E)/I(E) vanishes. This group has exponent p, because if u ∈ P (E) then

pu = pu − N_{EL/E}u = (p − 1 − σ − ⋯ − σ^{p−1})u
= ((1 − σ) + (1 − σ^{2}) + ⋯ + (1 − σ^{p−1}))u

is in (1 − σ)K_{n}^{M}E(x) and hence in I(E). It follows that V (E) injects into I(E′) whenever E′/E is an extension of degree prime to p.

Now we use the “Brauer-Severi” trick; this trick will be used again in chapter V, 1.6, in connection with Severi-Brauer varieties. For each b ∈ E we let E_{b} denote the field E(t_{1}, …, t_{p−1}, β) with t_{1}, …, t_{p−1} purely transcendental over E and β^{p} − β − y + ∑ b^{i}t_{i}^{p} = 0. It is known that b is in the image of the norm map E_{b}(x)^{×} → E_{b}^{×}; see [J37]. Since E · (E_{b})^{p} is purely transcendental over E (on β, t_{2}^{p}, ..., t_{p−1}^{p}), it follows that I(E) → I(E_{b}) is onto. Since E_{b}(x) is purely transcendental over E(x) (why?), I(E(x)) = I(E_{b}(x)) and K_{n}^{M}E(x) embeds in K_{n}^{M}E_{b}(x) by theorem 7.4. Hence K_{n}^{M}(E(x))/I(E) embeds in K_{n}^{M}E_{b}(x)/I(E_{b}). Since V (E) ⊂ K_{n}^{M}E(x)/I(E) by definition, we see that V (E) also embeds into V (E_{b}).

Now if we take the composite of all the fields E_{b}, b ∈ E, and then form its maximal algebraic extension E′ of degree prime to p, it follows that V (E) embeds into V (E′). Repeating this construction a countable number of times yields an extension field E′′ of E such that V (E) embeds into V (E′′) and every element of E′′ is a norm from E′′(x). Hence it suffices to prove that V (E′′) = 0. The proof in this special case is completely parallel to the proof of proposition 6.6.2, and we leave the details to Ex. 7.13.

This completes the proof of Izhboldin’s Theorem 7.8.

Corollary 7.8.3 (Hilbert’s Theorem 90 for K_{∗}^{M}). Let j: F ⊂ L be a degree p field extension, with char(F) = p, and let σ be a generator of G = Gal(L/F). Then K_{n}^{M}(F) ≅ K_{n}^{M}(L)^{G}, and the following sequence is exact for all n > 0:

0 → K_{n}^{M}(F) ⟶^{j^{∗}} K_{n}^{M}(L) ⟶^{1−σ} K_{n}^{M}(L) ⟶^{N} K_{n}^{M}(F).

Proof. Since K_{n}^{M}(F) has no p-torsion, Corollary 7.5.3 implies that j^{∗} is an injection. To prove exactness at the next spot, suppose that v ∈ K_{n}^{M}(L) has σ(v) = v. By Ex. 7.12 and theorem 7.7.2, the class of v mod p comes from an element w ∈ K_{n}^{M}(E), i.e., v − j^{∗}(w) = pv′ for some v′ ∈ K_{n}^{M}E(x). Hence pσ(v′) = σ(pv′) = pv′. Since K_{n}^{M}(L) has no p-torsion, σ(v′) = v′. But then pv′ equals j^{∗}N(v′) = ∑ σ^{i}(v′), and hence v = j^{∗}(w)+j^{∗}(Nv′). In particular, this proves that K_{n}^{M}(F) ≅ K_{n}^{M}(L)^{G}.

To prove exactness at the final spot, note that G acts on K_{n}^{M}(L), and that ker(N)/im(1 − σ) is isomorphic to the cohomology group H^{1}(G, K_{n}^{M}(L)); see [WHomo, 6.2.2]. Now consider the exact sequence of Gal(L/F)-modules

0 → K_{n}^{M}(L) ⟶^{p} K_{n}^{M}(L) ⟶^{7.7.2} ν(n)_{L} → 0.

Using Ex. 7.12, the long exact sequence for group cohomology begins

0 → K_{n}^{M}(F) ⟶^{p} K_{n}^{M}(F) → ν(n)_{F} → H^{1}(G, K_{n}^{M}(L)) ⟶^{p} H^{1}(G, K_{n}^{M}(L)).

But K_{n}^{M}(F) maps onto ν(n)_{F} by theorem 7.7.2, and the group H^{1}(G, A) has exponent p for all G-modules A [WHomo, 6.5.8]. It follows that H^{1}(G, K_{n}^{M}(L)) = 0, so ker(N) = im(1 − σ), as desired.

Remark 7.8.4. Hilbert’s Theorem 90 for K_{n}^{M}, which extends Theorem 6.6, states that for any Galois extension F ⊂ E of degree p, with σ generating Gal(E/F), the following sequence is exact:

K_{n}^{M}(E) ⟶^{1−σ} K_{n}^{M}(E) ⟶^{N_{E/F}} K_{n}^{M}(F).

This is a consequence of the Norm Residue Theorem (Chapter VI, 3.1) and is due to Voevodsky; we refer the reader to [HW, 3.2] for a proof.

Relation to the Witt ring

Let F be a field of characteristic ≠ 2. Recall from §5.6 of chapter II that the Witt ring W (F) is the quotient of the Grothendieck group K_{0}SBil(F) of symmetric inner product spaces over F by the subgroup {nH} generated by the hyperbolic form ⟨1⟩ ⊕ ⟨−1⟩. The dimension of the underlying vector space induces an augmentation K_{0}SBil(F) → ℤ, sending {nH} isomorphically onto 2ℤ, so it induces an augmentation ε: W (F) → ℤ/2.

We shall be interested in the augmentation ideals I = ker(ε) of W (F) and Î of K_{0}SBil(F). Since H ∩ Î = 0, we have Î ≅ I. Now I is generated by the classes ⟨a⟩ − 1, a ∈ F − {0, 1}. The powers I^{n} of I form a decreasing chain of ideals W (F) ⊃ I ⊃ I^{2} ⊃ · · ·.

For convenience, we shall write K_{n}^{M}(F)/2 for K_{n}^{M}(F)/2K_{n}^{M}(F).

Theorem 7.9 (Milnor). There is a unique surjective homomorphism

s_{n}: K_{n}^{M}(F)/2 → I^{n}/I^{n+1}

sending each product {a_{1}, …, a_{n}} in K_{n}^{M}(F) to the product ∏_{i=1}^{n}(⟨a_{i}⟩ − 1) modulo I^{n+1}. The homomorphisms s_{1} and s_{2} are isomorphisms.

Proof. Because (⟨a⟩ − 1) + (⟨b⟩ − 1) ≡ ⟨ab⟩ − 1 modulo I^{2} (II.5.6.5), the map l(a_{1})×⋯×l(a_{n}) ↦ ∏(⟨a_{i}⟩−1) is a multilinear map from F^{×} to I^{n}/I^{n+1}. Moreover, if a_{i} + a_{i+1} = 1 for any i, we know from Ex. II.5.12 that (⟨a_{i}⟩ − 1)(⟨a_{i+1}⟩ − 1) = 0. By the presentation of K_{∗}^{M}(F), this gives rise to a group homomorphism from K_{n}^{M}(F) to I^{n}/I^{n+1}. It annihilates 2K_{∗}^{M}(F) because ⟨a^{2}⟩ = 1:

2s_{n}{a_{1}, …, a_{n}} = s_{n}{a_{1}^{2}, a_{2} … } = (⟨a_{1}^{2}⟩ − 1) ∏_{i=2}^{n}(⟨a_{i}⟩ − 1) = 0.

It is surjective because I is generated by the (⟨a⟩ − 1). When n = 1 the map is the isomorphism F^{×}/F^{×2} ≅ I/I^{2} of chapter II. We will see that s_{2} is an isomorphism in Corollary 7.10.3 below, using the Hasse invariant w_{2}.

Example 7.9.1. For the real numbers ℝ, we have W (ℝ) = ℤ and I = 2ℤ on s_{1}(−1) = ⟨−1⟩ − 1 = 2⟨−1⟩. On the other hand, we saw in Example 7.2(c) that K_{n}^{M}(ℝ)/2 ≅ ℤ/2 on {−1, …, −1}. In this case each s_{n} is the isomorphism ℤ/2 ≅ 2^{n}ℤ/2^{n+1}ℤ.

At the other extreme, if F is algebraically closed then W (F) = ℤ/2. Since K_{n}^{M}(F) is divisible, K_{n}^{M}(F)/2 = 0 for all n ≥ 1. Here s_{n} is the isomorphism 0 = 0.

Remark 7.9.2. In 1970, Milnor asked if the surjection s_{n}: K_{n}^{M}(F)/2 → I^{n}/I^{n+1} is an isomorphism for all n and F, char(F) ≠ 2 (on p. 332 of [M-QF]). Milnor proved this was so for local and global fields. This result was proven for all fields and all n by Orlov, Vishik and Voevodsky in [OVV].

Definition 7.10 (Stiefel-Whitney invariant). The (total) Stiefel-Whitney invariant w(M) of the symmetric inner product space M = ⟨a_{1}⟩ ⊕ ⋯ ⊕ ⟨a_{n}⟩ is the element of ∏_{i=0}^{∞} K_{i}^{M}(F)/2 defined by the formula

w(M) = ∏_{i=1}^{n}(1 + l(a_{i})) = 1 + l(a_{1} ⋯ a_{n}) + ⋯ + {a_{1}, …, a_{n}}

The lemma below shows that w(M) is independent of the representation of M as a direct sum of 1-dimensional forms. We write w(M) = 1 + w_{1}(M) + w_{2}(M) + ⋯, where the i^{th} Stiefel-Whitney invariant w_{i}(M) ∈ K_{i}^{M}(F)/2 equals the i^{th} elementary symmetric function of l(a_{1}), …, l(a_{n}). For example, w_{1}(M) = a_{1} ⋯ a_{n} ∈ F^{×}/F^{×2} is just the classical “discriminant” of M defined in II.5.6.3, while the second elementary symmetric function w_{2}(M) = ∑_{i<j}{a_{i}, a_{j}} lies in K_{2}(F)/2 and is called the Hasse invariant of M; see [M-SBF].

For M = ⟨a⟩ ⊕ ⟨b⟩ we have w_{1}(M) = ab and w_{2}(M) = {a, b}, with w_{i}(M) = 0 for i ≥ 3. In particular, the hyperbolic plane H has w_{i}(H) = 0 for all i ≥ 2.

Lemma 7.10.1. w(M) is a well-defined unit in the ring ∏_{i=1}^{∞} K_{i}^{M}(F)/2. It satisfies the Whitney sum formula

w(M ⊕ N) = w(M)w(N),

so w extends to a function on K_{0}SBil(F). Hence each Stiefel-Whitney invariant w_{i} extends to a function K_{0}SBil(F) ⟶^{w_{i}} K_{i}^{M}(F)/2.

Proof. To show that w(M) is well defined, it suffices to consider the rank two case. Suppose that ⟨a⟩ ⊕ ⟨b⟩ ≅ ⟨α⟩ ⊕ ⟨β⟩. Then the equation ax^{2} + by^{2} = α must have a solution x, y in F. The case y = 0 (or x = 0) is straightforward, since ⟨α⟩ = ⟨ax^{2}⟩ = ⟨a⟩, so we may assume that x and y are nonzero. Since the discriminant w_{1} is an invariant, we have ab = αβu^{2} for some u ∈ F, and all we must show is that {a, b} = {α, β} in K_{2}(F)/2. The equation 1 = ax^{2}/α + by^{2}/α yields the equation

0 = {ax^{2}/α, by^{2}/α} ≡ {a, b} + {α, α} − {a, α} − {b, α} ≡ {a, b} − {α, ab/α}

in K_{2}(F)/2K_{2}(F). Substituting ab = αβu^{2}, this implies that {a, b} ≡ {α, β} modulo 2K_{2}(F), as desired.

Example 7.10.2. Since I ≅ Î, we may consider the w_{i} as functions on I ⊆ W (F). However, care must be taken as w_{i}(M) need not equal w_{2}(M ⊕ H). For example, w_{2}(M ⊕ H) = w_{2}(M) + {w_{1}(M), −1}. In particular, w_{2}(H ⊕ H) = {−1, −1} can be nontrivial. The Hasse-Witt invariant of an element x ∈ I ⊂ W (F) is defined to be h(x) = w_{2}(V, B), where (V, B) is an inner product space representing x so that dim(V) ≡ 0 mod 8.

Corollary 7.10.3. The Hasse invariant w_{2}: Î → K_{2}(F)/2 induces an isomorphism from Î^{2}/Î^{3} ≅ I^{2}/I^{3} to K_{2}^{M}(F)/2, inverse to the map s_{2} of Theorem 7.9.

Proof. By Ex. 7.11, w_{2} vanishes on the ideal Î^{3} ≅ I^{3}, and hence defines a function from Î^{2}/Î^{3} to K_{2}(F)/2. Since the total Stiefel-Whitney invariant of s_{2}{a, b} = (⟨a⟩ − 1)(⟨b⟩ − 1) is 1 + {a, b}, this function provides an inverse to the function s_{2} of Theorem 7.9.

If char(F) = 2, there is an elegant formula for the filtration quotients of the Witt ring W (F) and the W (F)-module W Q(F) (see II.5) due to K. Kato [K82]. Recall from 7.7.2 that K_{n}^{M}(F)/2 ≅ ν(n)_{F}, where ν(n)_{F} is the kernel of the operator ℘. The case n = 0 of Kato’s result was described in Ex. II.5.13(d).

Theorem 7.10.4 (Kato [K82]). Let F be a field of characteristic 2. Then the map s_{n} of Theorem 7.9 induces an isomorphism K_{n}^{M}(F)/2 ≅ ν(n)_{F} ≅ I^{n}/I^{n+1}, and there is a short exact sequence

0 → I^{n}/I^{n+1} → Ω_{F}^{n} ⟶^{℘} Ω_{F}^{n} /dΩ_{F}^{n−1} → I^{n} W Q(F)/I^{n+1} W Q(F) → 0.

The Norm Residue symbol

For the next result, we need some facts about Galois cohomology, expanding slightly upon the facts mentioned in 6.10. Assuming that n is prime to char(F), there are natural cohomology cup products H_{et}^{i}(F; M)⊗H_{et}^{j}(F; N) ⟶^{∪} H_{et}^{i+j}(F; M⊗ N) which are associative in M and N. This makes the direct sum H_{et}^{∗}(F; M^{⊗∗}) = ⊕_{i=0}^{∞}H_{et}^{i}(F; M^{⊗i}) into a graded-commutative ring for every ℤ/n-module M over the Galois group Gal(F_{sep}/F). (By convention, M^{⊗0} is ℤ/n.) In particular, both H_{et}^{∗}(F; ℤ/n) and H_{et}^{∗}(F; μ_{n}^{⊗∗}) are rings, and are isomorphic only when F contains a primitive n^{th} root of unity.

Theorem 7.11 (Norm Residue Symbols). (Bass-Tate) Fix a field F and an integer n prime to char(F).

(1) If F contains a primitive n^{th} root of unity, the Kummer isomorphism from F^{×}/F^{×n} to H_{et}^{1}(F; ℤ/n) extends uniquely to a graded ring homomorphism

h_{F}: K_{∗}^{M}(F)/n → H^{∗}(F; ℤ/n).

(2) More generally, the Kummer isomorphism from F^{×}/F^{×n} to H^{1}(F; μ_{n}) extends uniquely to a graded ring homomorphism

h_{F}: K_{∗}^{M}(F)/n → H_{et}^{∗}(F; μ_{n}^{⊗∗}) = ⊕_{i=0}^{∞}H_{et}^{i}(F; μ_{n}^{⊗i}).

The individual maps K_{i}^{M}(F) → H_{et}^{i}(F; μ_{n}^{⊗i}) are called the norm residue symbols, and also higher Galois symbols.

Proof. The first assertion is just a special case of the second assertion. As in (6.10.2), the Kummer isomorphism induces a map from the tensor algebra T (F^{×}) to H_{et}^{∗}(F; μ_{n}^{⊗∗}), which in degree i is the iterated cup product

F^{×} ⊗ ⋯ F^{×} = (F^{×})^{⊗n} ≅ (H_{et}^{1}(F; μ_{n}))^{⊗i} ⟶^{∪} H_{et}^{i}(F; μ_{n}^{⊗i}).

By Proposition 6.10.3, the Steinberg identity is satisfied in H_{et}^{2}(F, μ_{n}^{⊗2}). Hence the presentation of K_{∗}^{M}(F) yields a ring homomorphism from K_{∗}^{M}(F) to H_{et}^{∗}(F; μ_{n}^{⊗∗}).

Remark 7.11.1. In his seminal paper [M-QF], Milnor studied the norm residue symbol for K_{n}^{M}(F)/2 and stated (on p.340) that, “I do not know any examples for which the homomorphism h_{F} fails to be bijective.” Voevodsky proved that h_{F} is an isomorphism for n = 2^{ν} in his 2003 paper [V-MC]. The proof that h_{F} is an isomorphism for all n prime to char(F) was proven a few years later; see VI.3.1.1.

EXERCISES

7.1 Let v be a discrete valuation on a field F. Show that the maps λ: K_{n}^{M}(F) → K_{n}^{M}(k_{v}) and ∂_{v}: K_{n}^{M}(F) → K_{n−1}^{M}(k_{v}) of Theorem 7.3 are independent of the choice of parameter π, and that they vanish on l(u) · K_{n−1}^{M}(F) whenever u ∈ (1 + πR). Show that the map λ also vanishes on l(π) · K_{n−1}^{M}(F).

7.2 Continuing Exercise 7.1, show that the kernel of the map d: K_{n}^{M}(F) → L_{n} of Theorem 7.3 is exactly l(1 + πR) · K_{n−1}^{M}(F). Conclude that the kernel of the map λ is exactly l(1 + πR) · K_{n−1}^{M}(F) + l(π) · K_{n−1}^{M}(F).

7.3 (Bass-Tate) Generalize Theorem 6.4 to show that for all n ≥ 2:

(a) If F is an algebraically closed field, then K_{n}^{M}(F) is uniquely divisible.

(b) If F is a perfect field of characteristic p then K_{n}^{M}(F) is uniquely p-divisible.

7.4 Let F be a local field with valuation v and finite residue field k. Show that K_{n}^{M}(F) is divisible for all n ≥ 3. Hint: By Moore’s Theorem 6.2.4, K_{n}^{M}(F) is ℓ-divisible unless F has a ℓ^{th} root of unity. Moreover, for every x ∉ F^{×ℓ} there is a y ∉ F^{×ℓ} so that {x, y} generates K_{2}(F)/ℓ. Given a, b, c with {b, c} ∉ ℓK_{2}(F), find a′, b′ ∉ F^{×ℓ} so that {b′, c} ≡ 0 and {a′, b′} ≡ {a, b} modulo ℓK_{2}(F), and observe that {a, b, c} ≡ {a′, b′, c} ≡ 0.

In fact, I. Sivitskii has shown that K_{n}^{M}(F) is uniquely divisible for n ≥ 3 when F is a local field. See [Siv]. We will give a proof of this in VI.5.1.

7.5 Let E = F (a) be a finite extension of F, and consider the transfer map N = N_{a/F}: K_{n}^{M}(E) → K_{n}^{M}(F) in definition 7.5. Use Weil’s Formula (7.5.1) to show that when n = 0 the transfer map N: ℤ → ℤ is multiplication by [E : F], and that when n = 1 the transfer map N: E^{×} → F^{×} is the usual norm map.

7.6 Suppose that the degree of every finite extension of a field F is a power of some fixed prime p. If E is an extension of degree p and n > 0, use Ex. 6.2 to show that K_{n}^{M}(E) is generated by elements of the form {y, x_{2}, …, x_{n}}, where y ∈ E^{×} and the x_{i} are in F^{×}.

7.7 Ramification and the transfer. Let F′ and E = F (a) be finite field extensions of F, and suppose that the irreducible polynomial π ∈ F [t] of a has a decomposition π = ∏ π^{e_{i}} in F′[t]. Let E_{i} denote F′(a_{i}), where each a_{i} has minimal polynomial π_{i}. Show that the following diagram commutes.

K_{n}^{M}(E) ⟶^{e_{1},...,e_{r}} ⊕K_{n}^{M}(E_{i})
N_{a}/F ↓  ↓^{∑ N_{a_{i}/F′}}
K_{n}^{M}(F) ⟶ K_{n}^{M}(F′)

7.8 Ramification and ∂_{v}. Suppose that E is a finite extension of F, and that w is a valuation on E over the valuation v on F, with ramification index e. (See 6.3.1.) Use the formulas for ∂_{v} and ∂_{w} in Theorem 7.3 to show that for every x ∈ K_{n}^{M}(F) we have ∂_{w}(x) = e · ∂_{v}(x) in K_{n−1}^{M}(k_{w})

7.9 If E/F is a normal extension of prime degree p, and v is a valuation on F (t) trivial on F, show that ∂_{v}N_{E(t)/F (t)} = ∑_{w} N_{E(w)/F (v)}∂_{w}, where the sum is over all the valuations w of E(t) over v. Hint: If F (t)_{v} and E(t)_{w} denote the completions of F (t) and E(t) at v and w, respectively, use Ex. 7.7 and Lemma 7.6.3 to show that the following diagram commutes.

K_{n+1}^{M}E(t) ⟶ ⨁_{w} K_{n+1}^{M}E(t)_{w} ⟶^{∂} ⨁_{w} K_{n}^{M}E(w)
N_{E(t)/F (t)}↓  ↓^{∑_{w} N_{E(t)w/F (t)v}}  ↓^{∑_{w} N_{e(w)/F (v)}}
K_{n}^{M}F (t) ⟶ K_{n}^{M}F (t)_{w}  ⟶^{∂} K_{n}^{M}F (v)

7.10 If v is a valuation on F, and x ∈ K_{i}^{M}(F), y ∈ K_{j}^{M}(F), show that

∂_{v}(xy) = λ(x)∂_{v}(y) + (−1)^{j}∂_{v}(x)ρ(y)

where ρ: K_{∗}^{M}(F) → K_{∗}^{M}(k_{v}) is a ring homomorphism characterized by the formula ρ(l(uπ^{i})) = l((−1)^{i}ū).

7.11 Let t = 2^{n−1} and set z = ∏_{i=1}^{n}(⟨a_{i}⟩ − 1); this is a generator of the ideal Î^{n} in K_{0}SBil(F). Writing s for {a_{1}, …, a_{n}, −1, −1, …, −1} ∈ K_{t}^{M}(F)/2, show that the Stiefel-Whitney invariant w(z) is equal to: 1 + s if n is odd, and to (1 + s)^{−1} if n is even. This shows that the invariants w_{i} vanish on the ideal Î^{n} if i < t = 2^{n−1}, and that w_{t} induces a homomorphism from I^{n}/I^{n+1} ≅ Î^{n}/Î^{n+1} to K_{t}^{M}(F)/2.

For example, this implies that w_{1} vanishes on Î^{2}, while w_{2} and w_{3} vanish on Î^{3}. 7.12 (Izhboldin) Let L/F be a field extension of degree p = char(F), with Galois group G. Show that Ω_{F}^{n} is isomorphic to (Ω_{L}^{n})^{G}, and that Ω_{F}^{n} /dΩ_{F}^{n−1} is isomorphic to (Ω_{L}^{n}/dΩ_{L}^{n−1})^{G}. Conclude that ν(n)_{F} ≅ ν(n)_{L}^{G}. 7.13 In this exercise we complete the proof of proposition 7.8.2, and establish a special case of 7.8.3. Suppose that E(x) is a degree p field extension of E, char(E) = p, and that σ is a generator of Gal(E(x)/E). Suppose in addition that the norm map E(x)^{×} → E^{×} is onto, and that E has no extensions of degree < p. Modify the proof of proposition 6.6.2 to show that the following sequence is exact:

K_{n}^{M}E(x) ⟶^{1−σ} K_{n}^{M}E(x) ⟶^{N} K_{n}^{M}E → 0.

7.14 Suppose that F is a field of infinite transcendence degree κ over the ground field. Show that the image of the dlog symbol of 7.7 lies in the kernel of Ω_{F}^{n} ⟶^{d} Ω_{F}^{n+1}. Using Ex. 6.11, show that K_{n}^{M}(F) has cardinality κ for all n > 0.

If F is a local field, this and Ex. 7.4 implies that K_{n}^{M}(F) is an uncountable, uniquely divisible abelian group.