Lecture VIII The higher K-theory of fields

Note: References cited as I.x.y, II.x.y and III.x.y refer to chapters I, II and III in the virtual “K-book” located at http://math.rugers.edu/w̃eibel/kbook.html.

§1. 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 1.1. 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 Relation) {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 1.1.1. K_{2}(𝔽_{q}) = 1 for every finite field 𝔽_{q}.

Example 1.1.2. 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)}.

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

The next result is useful for manipulations with symbols.

Lemma 1.1.4 (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.

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

Corollary 1.1.5. 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 1.1.6. 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

Typeset by AMS-TEX

their inverses. We will see in Theorem 1.4 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 1.1.7. Let F be an algebraically closed field. By Lemma 1.1.4, K_{2}F (t) is generated by linear symbols of the form {a, b} and {t − a, b}. It will follow from 1.5.2 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 1.1.2.

Steinberg symbols

Definition 1.2. 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 1.2.1. 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 relation (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 1.2.2 (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 relation is trivial, because x = y = 1 is always a solution when r + s = 1.

Example 1.2.3 (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 name “norm residue” comes from the fact that for each x, the map y ↦ {x, y} is trivial iff 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 1.2.4. If F is a local field, then K_{2}(F) is the direct sum of a uniquely divisible abelian group V and a finite cyclic group, isomorphic under the norm residue symbol to the group μ = μ_{n} of roots of unity in F.

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 1.3. 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.

Ramification 1.3.1. 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 1.3.2. 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).

Theorem 1.4 (Bass-Tate). When F is an algebraically closed field, K_{2}(F) is a uniquely divisible abelian group.

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. Therefore, the theorem is an immediate consequence of our next proposition.

Proposition 1.4.1 (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.

Corollary 1.4.2. 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 1.5. 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 prome 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 1.5.1 (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 1.2.1, this sequence is split by the symbol (r, s)_{∞}, so we have K_{2}(ℚ) ≅ K_{2}(ℤ) ⊕ ∐ 𝔽_{p}^{×}.

Application 1.5.2 (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 1.1.2. Therefore we have a split exact sequence

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

Weil’s Reciprocity Formula 1.5.3. 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 1.1.7, 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 Merkur’ev and Suslin in 1981, and published in their landmark paper [MS]. Recall that an automorphism σ of a field E induces an automorphism of K_{2}(E) sending {x, y} to {σx, σy}.

Theorem 1.6 (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).

Merkur’ev 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 1.6.1 (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 full proof of Hilbert’s Theorem 90 for K_{2}, since it uses higher algebraic K-theory in a crucial way to reduce to the following special case.

Proposition 1.6.2. 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 1.1.4, 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 1.4.1.

Theorem 1.7. If char(F) = p ≠ 0, then the group K_{2}(F) has no p-torsion.

Example 1.7.1. 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 1.8. 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^{×}.

Application 1.8.1. We can use Theorem 1.8 to give another proof of Theorem 1.4, 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 1.7.

Application 1.8.2 (K_{2}ℝ). Theorem 1.8 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 1.2.1 or Example 5.9.1. Recall from Example 1.1.6 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 1.9. 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 1.9.1. 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).

Proposition 1.9.2 (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.

Merkur’ev–Suslin Theorem 1.9.3. K_{2}(F)/nK_{2}(F) is isomorphic to the subgroup _{n}Br(F) of elements of order n in Br(F).

By Matsumoto’s Theorem, this implies that the n-torsion in the Brauer group is generated by cyclic algebras.

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 1.10. 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_{n} of all n^{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_{n}^{⊗2} = μ_{n} ⊗ μ_{n} is also a G-discrete module. Note that the three G-modules ℤ/n, μ_{n} and μ_{n}^{⊗2} have the same underlying abelian group, but are isomorphic G_{F} -modules only when μ_{n} ⊂ 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 1.10.1 (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 n is prime to char(F), the exact sequence of discrete modules

1 → μ_{n} → G_{m} ⟶^{n} G_{m} → 1

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

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

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

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

(1.10.2)  F^{×} ⊗ F^{×} → H_{et}^{1}(F; μ_{n}) ⊗ H_{et}^{1}(F; μ_{n}) ⟶^{∪} H_{et}^{2}(F; μ_{n}^{⊗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 1.10.3 (Galois symbol). The bilinear pairing (1.10.2) induces a Steinberg symbol K_{2}(F) → H_{et}^{2}(F; μ_{n}^{⊗2}) for every n prime to char(F).

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

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

Tate has shown in [Tate] that this isomorphism identifies the Galois symbol of Proposition 1.10.3 with the n^{th} power norm residue symbol of Proposition 1.9.2. The Merkur’ev-Suslin Theorem 1.9.3 extends to a more general isomorphism, also proven in [MS]:

Merkurjev-Suslin Theorem 1.11. K_{2}(F)/nK_{2}(F) ≅ H_{et}^{2}(F; μ_{n}^{⊗2}) for all fields F of characteristic prime to n.

EXERCISES

1.1 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 1.1.2 is a special case of this, in which π = t^{−1}. 1.2 (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 1.1.4. 1.3 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}}. 1.4 If F̄ denotes the algebraic closure of a field F, show that K_{2}(ℚ̄) = K_{2}(𝔽̄_{p}) = 1. 1.5 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}.

1.6 Let ((r, s))_{p} denote the Hilbert symbol on ℚ̂_{p} (1.2.2), 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}^{×}. 1.7 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. 1.5.

Hint: From 1.5.1 and Ex. 1.6, the 2-adic symbol of Ex. 1.5 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. 1.8 (Suslin) Suppose that a field F is algebraically closed in a larger field E. Use Lemma 1.1.3 and Remark 1.8.1 to show that K_{2}(F) injects into K_{2}(E). 1.9 Let F be a field, and let Ω_{F} = Ω_{F/ℤ} denote the vector space of absolute Kähler differentials (see Ex. 2.6). The second exterior power of Ω_{F} is written as Ω_{F}^{2} . Show that there is a homomorphism K_{2}(F) → Ω_{F}^{2} sending \{x, y\} to \frac{dx}{x} ∧ \frac{dy}{y}.

1.10 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].)

§2. 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 2.1. 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 1.1 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 2.1.1. (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. 1.1.1. If F has transcendence degree 1 over 𝔽_{q}, 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 1.4, and is relegated to Ex. 2.3.

(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 1.2.1.

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 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 K_{n}^{M}(ℝ)^{r_{1}} ≅ (ℤ/2)^{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}}.

There is a natural graded ring map from K_{∗}^{M}(F) to Quillen’s K_{∗}(F), induced from the identification of K_{1}^{M}(F) with K_{1}(F) = F^{×} and Matsumoto’s relations for K_{2}(F) in 1.1. The following result was proven by Suslin [Su84].

Theorem 2.2. There is a morphism c_{n}: K_{n}(F) → K_{n}^{M}(F) such that the composition K_{n}^{M}(F) → K_{n}^{M}(F) is multiplication by (−1)^{n−1}(n − 1)!.

Suslin actually proved more:

Theorem 2.2.1. If F is an infinite field then H_{n}(GL(F)) = H_{n}(GL_{n}(F)), and c_{n} is just the Hurewicz map, followed by an isomorphism

K_{n}^{M}(F) ≅ H_{n}(GL_{n}(F))/H_{n}(GL_{n−1}(F)).

Tame symbols

Recall from Lemma 1.3 and Ex. 1.1 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.

Theorem 2.3 (Higher tame symbols). 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 1.3, and λ: K_{2}(F) → K_{2}(k) is the map of Example 1.1.2 and Ex. 1.1.

Corollary 2.3.1 (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.

Leading Coefficients 2.3.2. As in Example 1.1.2, 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 1.5.3. 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 2.4 (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.

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 2.4.1. 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}.

Lemma 2.4.2. 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].

The Transfer Map

Let v_{∞} be the valuation on F (t) with parameter t^{−1}. The formulas in Theorem 2.3 defining ∂_{∞} show that it vanishes on K_{∗}^{M}(F). By Theorem 2.4, there are unique homomorphisms N_{𝔭}: K_{n}^{M}(F [t]/𝔭) → K_{n}^{M}(F) so that −∂_{∞} = ∑_{𝔭} N_{𝔭}∂_{𝔭}.

Definition 2.5. 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. 2.5. We will show in 2.6 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 2.5.1. ∑_{v} N_{v}∂_{v}(x) = 0 for all x ∈ K_{n}^{M}F (t).

Projection Formula 2.5.2. 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)}.

Taking y = 1 in K_{0}^{M}(E) = ℤ, so N(y) = [E : F] by Ex. 2.5, this yields

Corollary 2.5.3. If the extension E/F has degree d, 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 2.6. 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 2.5:

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 2.6.1 (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′}.

This theorem is a consequence of the following three preliminary results. 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 2.5.3 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 2.6.2. (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).

Corollary 2.6.3. 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})

Proposition 2.6.4 (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).

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. 1.9, 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 2.7. 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 2.7.1. 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 2.7.2. (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 1.7; see [Izh].

Izhboldin’s Theorem 2.8. 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 1.7. As in the proof of theorem 1.7, 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 2.4, 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 2.8.2 below.

Fix u ∈ K_{n}^{M}(F) with pu = 0. The projection formula 2.5.2 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 2.8.2, 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 2.3.2; since λ(σv) = λ(v) we have: u = λ(u) = λ(v) − λ(σv) = 0. This proves Izhboldin’s theorem.

Before proceeding to proposition 2.8.2, we need some facts about the group I(E).

Lemma 2.8.1. (a) The group I(E) is p-divisible.

(b) The transcendental extension E ⊂ E(t) induces an isomorphism I(E) ≅ I(E(t)).

(c) If E/E′ is a purely inseparable field extension then I(E′) → I(E) is onto.

Proposition 2.8.2. P (E) = I(E) for all E containing F (y), linearly disjoint from F (x).

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; it was used in proving the Merkurjev-Suslin theorem. 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 2.4. 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 1.6.2, and we leave the details to Ex. 2.13.

This completes the proof of Izhboldin’s Theorem 2.8.

Corollary 2.8.3 (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).

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 2.9 (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.

Example 2.9.1. 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 2.2(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 2.9.2. 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. In 1996, Voevodsky proved this result for all fields and all n; see [V-MC].

Definition 2.10 (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 2.10.1. 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.

Example 2.10.2. 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 2.10.3. 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 2.9.

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 2.7.2 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.5.12(d).

Theorem 2.10.4 (Kato [K82]). Let F be a field of characteristic 2. Then the map s_{n} of Theorem 2.9 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 Galois symbol

For the next result, we need some facts about Galois cohomology, expanding slightly upon the facts mentioned in 1.10. 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 2.11 (Galois 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 higher Galois symbols.

Remark 2.11.1. Milnor studied the Galois symbol for n = 2 and stated (on p.340 of [M-QF]) that, “I do not know any examples for which the homomorphism h_{F} fails to be bijective.” In 1996, Voevodsky announced a proof of this result for n = 2; see [V-MC]. The following partial had been obtained a decade earlier, by Rost and Merkurjev-Suslin [MS2].

Theorem 2.11.2. If char(F) ≠ 2, the Galois symbol

K_{3}^{M}(F)/2 → H_{et}^{3}(F; ℤ/2) = H_{et}^{3}(F; μ_{2}^{⊗3})

is an isomorphism

EXERCISES

2.1 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 2.3 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). 2.2 Continuing Exercise 2.1, show that the kernel of the map d: K_{n}^{M}(F) → L_{n} of Theorem 2.3 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). 2.3 (Bass-Tate) Generalize Theorem 1.4 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. 2.4 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: To show it is p-divisible, use Moore’s Theorem 1.2.4 to reduce to the case when F has a p^{th} root of unity. Note that for every x ∉ F^{×p} there is a y ∉ F^{×p} so that {x, y} generates K_{2}(F)/p. Given a, b, c with {b, c} ∉ pK_{2}(F), find a′, b′ ∉ F^{×p} so that {b′, c} ≡ 0 and {a′, b′} ≡ {a, b} modulo pK_{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]. 2.5 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 2.5. Use Weil’s Formula (2.5.1) 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. 2.6 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. 1.2 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^{×}. 2.7 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′)

2.8 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 1.3.1.) Use the formulas for ∂_{v} and ∂_{w} in Theorem 2.3 to show that for every x ∈ K_{n}^{M}(F) we have ∂_{w}(x) = e · ∂_{v}(x) in K_{n−1}^{M}(k_{w}) 2.9 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. 2.7 and Lemma 2.6.3 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)

2.10 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).

2.11 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). Show that the Stiefel-Whitney invariant w(z) is equal to: 1 + {a_{n}, …, a_{n}, −1, −1, …, −1} if n is odd, and to 1 + {a_{n}, …, a_{n}, −1, −1, …, −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}. 2.12 (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}. 2.13 In this exercise we complete the proof of proposition 2.8.2, and extablish a special case of 2.8.3. 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 1.6.2 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.

§3. Algebraically closed fields

In his lectures, Friedlander has already mentioned Quillen’s computation of the K-theory of finite fields. If we fix p > 0 and take the direct limit over all q = p^{ν}, we get the K-theory of the algebraic closure 𝔽̄_{p}:

Theorem 3.1. (Quillen)

K_{n}(𝔽̄_{p}) = \{ \begin{matrix}ℤ & for n = 0, \\ ℚ/ℤ[\frac{1}{p}] & for n > 0 odd, \\ 0 & otherwise. \end{matrix}

Moreover, the Frobenius automorphism Φ(x) = x^{p} of 𝔽̄_{p} induces multiplication by p^{i} on K_{2i−1}(𝔽̄_{p}), and the subgroup fixed by Φ^{ν} is K_{2i−1}(𝔽_{q}), q = p^{ν}.

Browder [Br] observed that if we took coefficients mod m, where m is relatively prime to p, and let β ∈ K_{2}(𝔽̄_{p}) be the Bott element, then K_{2i}(𝔽̄_{p}; ℤ/m) ≅ ℤ/m on β^{i}, so that we get a polynomial ring.

Corollary 3.2. (Browder) K_{∗}(𝔽̄_{p}; ℤ/m) ≅ ℤ/m[β].

Notation 3.2.1. To indicate the Galois action, we write ℚ/ℤ[\frac{1}{p}](i) for the abelian group μ = ℚ/ℤ[\frac{1}{p}] where Φ acts as p^{i}. If F is any field of characteristic p, we can make G = Gal(F̄/F) act upon ℚ/ℤ[\frac{1}{p}](i) by identifying μ with the roots of unity in F̄ and letting g ∈ G act as g^{i}.

Similarly, if char(F) = 0, we write ℚ/ℤ(i) for the abelian group μ = ℚ/ℤ made into a G-module by identifying μ with the roots of unity in F̄ and letting g ∈ G act as g^{i}.

Rigidity Theorem 3.3 (for presheaves with transfer). Let V be a contravariant functor from smooth schemes over F = F̄. Suppose that V is homotopy invariant, that V (X) is a torsion group for every X, and that V has covariant transfer maps f_{∗}: V (X) → V (Y) for finite flat maps f: X → Y, satisfying the usual properties.

If x, y: Spec(F) → X are two points, and X is connected, then x^{∗} = y^{∗} as maps V (X) → V (F).

Proof. ([Su86, 2.3]) Since x, y lie on a curve in X, and the curve may be chosen so that x and y are smooth points, we may suppose (after removing a finite number of points from the curve) that X is a smooth affine curve, with a finite set S of points at infinity. If X = Spec(F [t]) the result follows from homotopy invariance: V (F) = V (F [t]). In general, there is a pairing

Div(X) ⊗ V (X) ⟶ V (F), x ⊗ v ↦ x^{∗}(v).

Since [x] − [y] belongs to Div^{0}(X), we must show that the restriction to Div^{0}(X) ⊗ V (X) is zero.

If f is a rational function on X, taking the value 1 at all points of S, there is a finite flat map from X − f^{−1}(1) to ℙ^{1} −{1} ≅ Spec(F [t]). The usual properties of transfers imply that if div(f) = ∑ n_{i}x_{i} then ∑ n_{i}x_{i}^{∗} = 0. Let J_{S} denote the quotient of Div^{0}(X) by all such divisors; J_{S} is the Jacobian with modulus S, and is a divisible group. We have shown that the pairing factors through the quotient J_{S} ⊗ V (X), which is zero since V (X) is torsion.

Corollary 3.3.1. If in addition V commutes with filtered colimits, then V (F) = V (L) for every extension F ⊂ L of algebraically closed fields.

Proof. If A is a f.g. smooth subalgebra of L, then V (F) → V (A) is injective because it is split by V (A) → V (A/𝔪) = V (F) for any maximal ideal 𝔪 of A. Similarly, V (B) → V (A ⊗_{F} B) is injective for every F -algebra B. Taking the colimit over all A shows that both V (F) → V (L) and V (B) → V (L ⊗_{F} B) are injective.

Now define maps x: A ⊗_{F} L → (A/𝔪) ⊗_{F} L ≅ L ⊗_{F} L and y: A ⊗_{F} L → L ⊗_{F} L by y(a ⊗ x) = ax. Setting X = Spec(A ⊗_{F} L), theorem 3.3 yields x^{∗} = y^{∗}. Thus V (F) and V (A) have the same image in V (L), because the map V (A) → V (L) induced by A ⊂ L factors through V (F) ≅ V (L).

Application 3.4. If F ⊂ L is an extension of algebraically closed fields, then K_{n}(F)/m ≅ K_{n}(L)/m for all m, and the torsion subgroups of K_{n}(F) and K_{n}(L) are isomorphic.

Indeed, the functors K_{n}(F)/m, K_{n}(F; ℤ/m) and “torsion subgroup of K_{n}(F)” all satisfy the conditions of 3.3 and 3.3.1.

Corollary 3.4.1. If F is any algebraically closed field of characteristic p, then each K_{2i}(F) is uniquely divisible (except for i = 0), while for some uniquely divisible group W:

K_{2i−1}(F) ≅ ℚ/ℤ[\frac{1}{p}](i) ⊕ W.

Finally, for every m prime to p we have a ring isomorphism:

K_{∗}(F; ℤ/m) ≅ (ℤ/m)[β].

For fields of characteristic zero, the entering wedge is this.

Theorem 3.5. (Suslin [S-LF]) The natural maps

BGL(ℝ)^{+} → BO  and  BGL(ℂ)^{+} → BU

induce isomorphisms on both homology and homotopy with finite coefficients.

Corollary 3.6. Modulo uniquely divisible groups, the K-theory groups of ℝ and ℂ are given by the following table.

n (mod 8)  1  2  3  4  5  6  7  8^{K_{n}(ℝ)  ℤ/2  ℤ/2  ℚ/ℤ  0  0  0  ℚ/ℤ  0}^{↓  incl.  0  2  0  0  0  =  0} K^{n}(ℂ)  ℚ/ℤ  0  ℚ/ℤ  0  ℚ/ℤ  0  ℚ/ℤ  0

Corollary 3.7. If F is any algebraically closed field of characteristic zero, then each K_{2i}(F) is uniquely divisible (except for i = 0), while for some uniquely divisible group W:

K_{2i−1}(F) ≅ ℚ/ℤ(i) ⊕ W.

Finally, for every m > 0 we have a ring isomorphism

K_{∗}(F; ℤ/m) ≅ (ℤ/m)[β].

§4. K_{3} of fields

For any field F, let D(F) denote the free abelian group on the classes [x], where x runs through the elements of F − {0, 1}. The formula r([x]) = x ∧ (1 − x) induces a map r: D(F) → Λ^{2}(F^{×}), where Λ^{2}(F^{×}) = (F^{×} ⊗ F^{×})/(x ⊗ b + b ⊗ a).

Exercise 4.1.0. Show that if x ≠ y then r(CR_{xy}) = 0, where CR_{xy} is the crossratio

CR_{xy} = [x] − [y] + [\frac{y}{x}] − [\frac{1 − x^{−1}}{1 − y^{−1}} ] + [\frac{1 − x}{1 − y}]

We let T (F) denote the quotient of D(F) by the subgroup generated by the cross-ratios CR_{xy}.

Definition 4.1. Bloch’s group B(F) is the kernel of the induced map r: T (F) → Λ^{2}(F^{×}). Thus we have an exact sequence:

0 → B(F) → T (F) ⟶^{r} Λ^{2}(F^{×}) → K_{2}(F) → 0.

Let μ_{F} denote the group of roots of unity in F. As an abelian group it is isomorphic to the group μ_{F} (2) = Tor(μ_{F} , μ_{F}), but the Galois group of F̄/F acts differently upon these two groups.

It is conjectured in [Su86] that the map K_{3}^{M}(F) → K_{3}(F) is an injection for every field. This was recently proven for fields of characteristic zero by Bruno Kahn, and is also known for fields of characteristic two. Suslin proved in [Su84] that the kernel of this map has exponent 2.

Write K_{3}(F)_{ind} for the cokernel of the map K_{3}^{M}(F) → K_{3}(F).

Theorem 4.2. (Suslin [Su86]) If F is an infinite field, the quotient K_{3}(F)_{ind} naturally fits into an exact sequence

0 → μ̃_{F} (2) → K_{3}(F)_{ind} → B(F) → 0,

where μ̃_{F} (2) is: the group μ_{F} (2) if char(F) = 2, and the unique nontrivial extension of μ_{F} (2) by ℤ/2 otherwise.

Example 4.2.1. If F = 𝔽_{q} is a finite field with q ≥ 5, the result also holds. The group K_{3}^{M}(𝔽_{q}) is zero, and Quillen proved that K_{3}(𝔽_{q}) ≅ ℤ/(q^{2} − 1). Direct calculations show that T (F) is cyclic of order q + 1. If q is even then Λ^{2}(F^{×}) = 0, so B(F) = T (F) ≅ ℤ/(q + 1), and μ̃_{F} (2) is cyclic of order q − 1. On the other hand, if q is odd then Λ^{2}(F^{×}) = ℤ/2, B(F) is cyclic of order (q + 1)/2, and μ̃_{F} (2) is cyclic of order 2(q − 1). See [L89, §3].

The following result was proven independently by Marc Levine [Le-3] and by Merkurjev-Suslin [MS3].

Theorem 4.3. If char(F) = p > 0, the group K_{3}(F) has no p-torsion. If m is prime to char(F), then

(a) The m-torsion subgroup of K_{3}(F)_{ind} is isomorphic to H^{0}(F; μ_{m}^{⊗2}).

(b) There is an exact sequence

K_{3}^{M}(F)/m → K_{3}(F; ℤ/m) → H^{1}(F; μ_{m}^{⊗2}) → 0.

Corollary 4.3.1. If F̄ is the algebraic closure of F and G = Gal(F̄/F) then K_{3}(F)_{ind} injects into K_{3}(F̄)_{ind}, and K_{3}(F)_{ind} ≅ K_{3}(F̄)_{ind}^{G}.

In particular, the torsion subgroup of K_{3}(F)_{ind} is: H^{0}(F; ℚ/ℤ(2)) if char(F) = 0, and H^{0}(F; ℚ/ℤ[\frac{1}{p}](2)) if char(F) = p > 0.

Conjecture 4.3.2. ([MS3, 11.7]) If F_{0} denotes the algebraic closure of the ground field in F, is K_{3}(F_{0})_{ind} = K_{3}(F)_{ind}?

As an application, suppose that F is a number field with r_{1} real embeddings and r_{2} complex embeddings (so [F : ℚ] = r_{1} + 2r_{2}). Note that the group H^{0}(F; μ_{m}^{⊗2}) is a finite cyclic group; we write w_{2} for the order of this group. Combining Borel’s theorem that the rank of K_{3}(F) is r_{2} with the Bass-Tate theorem that K_{3}^{M}(F) ≅ (ℤ/2)^{r_{1}}, and the calculation of K_{3}(ℝ) → K_{3}(ℂ), we obtain:

Corollary 4.4. ( [MS3]) Let F be a number field.

(a) If F is totally imaginary (r_{1} = 0) then

K_{3}(F) ≅ ℤ^{r_{2}} ⊕ ℤ/w_{2}.

(b) If F has a real embedding (r_{1} ≠ 0) then

K_{3}(F) ≅ ℤ^{r_{2}} ⊕ ℤ/(2w_{2}) ⊕ (ℤ/2)^{r_{1}−1}.

Examples 4.4.4. (a) If F = ℚ then μ_{F} = ℤ/2 but H^{0}(ℚ; μ_{m}^{⊗2}) ≅ ℤ/24, and so K_{3}(ℚ) ≅ ℤ/48.

(b) If F = ℚ(i), i = \sqrt{−1}, then μ_{F} = ℤ/4 but H^{0}(ℚ(i); μ_{m}^{⊗2}) ≅ ℤ/24. Thus K_{3}(ℚ(i)) ≅ ℤ ⊕ ℤ/24.