Lecture IX Bloch’s higher Chow groups

§1. Basic properties

Let k be a field. Recall that the simplicial ring k[Δ^{·}] is defined by the equations of the geometric simplices: k[Δ^{n}] = k[t_{0}, …, t_{n}]/(∑ t_{j} = 1). Using the analogy with topology, we can think of the scheme Δ^{n} = Spec(k[Δ^{n}]) as an n-simplex, and talk about its faces Δ^{m} ⊂ Δ^{n}, which correspond to the simplicial inclusion of an m-simplex as a face of an n-simplex.

Suppose that Z is a codimension i subvariety of Δ^{n}, and that Δ^{m} is some face. The intersection cycle Z · Δ^{m} is only defined when Z meets Δ^{m} properly, meaning that every component of the intersection Z ∩ Δ^{m} has codimension ≥ i in Δ^{m}.

Algebraically, Z is defined by a prime ideal P of height i in R = k[Δ^{n}], and Δ^{m} is defined by a second prime ideal of R. To say that Z and Δ^{m} meet properly is to say that the ideal I + P has height i + m in R. If 𝔭_{1}, …, 𝔭_{r} are the minimal associated primes of I + P, they have height i + m in R, and correspond to to codimension i subvarieties Z_{1}, …, Z_{r} in Δ^{m}. We define the multiplicity m_{i} to be the length of the R_{𝔭_{i}}-module (R/I + P)_{𝔭_{i}}. Then the intersection formula is:

(1.0)  Z · Δ^{m} = ∑ m_{j}Z_{j}.

Definition 1.1. Fix a quasi-projective variety X. Let z^{i}(X, n) denote the free abelian group on the set of codimension i subvarieties Z of X × Δ^{n} which meet all faces properly. By construction, the cycle maps z^{i}(X, n) → z^{i}(X, m) are defined for every simplicial map Δ^{m} → Δ^{n}. Thus as n varies these groups form a simplicial abelian group z^{i}(X, ·). We define the higher Chow groups to be:

CH^{i}(X, n) = π_{n} z^{i}(X, ·)

= H_{n} of the associated chain complex.

Examples 1.1.1. (1) Since z^{0}(X, ·) is a constant simplicial group, CH^{0}(X, 0) = H^{0}(X, ℤ) and CH^{0}(X, n) = 0 for n ≠ 0.

(2) CH^{i}(X, n) = 0 if i > n + dim(X). Indeed, no subvariety of X × Δ^{n} can have codimension i > dim(X × Δ^{n}), so we have z^{i}(X, n) = 0.

(3) If n = 0 then CH^{i}(X, 0) is the classical Chow group CH^{i}(X) of codimension i cycles on X modulo rational equivalence.

(4) (Homotopy invariance) If X[t] denotes X × Spec(k[t]) then

CH^{i}(X, n) ≅ CH^{i}(X[t], n).

Functoriality 1.1.2. If f: X → Y is a flat map, pullback of cycles defines a natural map z^{i}(Y, ·) → z^{i}(X, ·), and hence natural maps f^{∗}: CH^{i}(Y, n) → CH^{i}(X, n). That is, each CH^{i}(X, n) is naturally contravariant for flat maps.

If f: X → Y is a proper map, so is each X × Δ^{n} → Y × Δ^{n}. So there are pushforward maps on cycles f_{∗}: z^{i}(X, n) → z^{i+c}(Y, n), where c is the usual codimension shift. Bloch observed that f_{∗} is a simplicial map. Hence there is a natural “transfer” map f_{∗}: CH^{i}(X, n) → CH^{i+c}(Y, n). That is, each CH^{i}(X, n) is naturally covariant for proper maps.

Typeset by AMS-TEX

Proposition 1.2. (Bloch [Bl]) For X smooth, and i = 1,

CH^{1}(X, n) = \{ \begin{matrix}Pic(X), & n = 0 \\ H^{0}(X, O_{X}), & n = 1 \\ 0 & otherwise. \end{matrix}

In addition, if R is a regular local ring with field of fractions F and X = Spec(R), then there is an exact “Gersten resolution” involving the sets X^{r} of height r prime ideals of R:

0 → CH^{i}(R, n) → CH^{i}(F, n) → ∐_{x∈X^{1}} CH^{i−1}(k(x), n − 1) → ⋯

→ ∐_{x∈X^{n}} CH^{i−n}(k(x), 0) → 0.

Application 1.2.1. Let CH^{i}(n) denote the Zariski sheaf associated to the presheaf U ↦ CH^{i}(U, n). Then the arguments of Quillen [Q341], explained in Lecture IV(6.4), show that there is an Atiyah-Hirzebruch style spectral sequence

E_{2}^{pq} = H^{p}(X, CH^{i}(−q)) ⇒ CH^{i}(X, −p − q).

Theorem 1.3. ([Bl] and [Bl2], or [Le-1]) For any quasi-projective X,

CH^{i}(X, n) ⊗ ℚ ≅ K_{n}^{(i)}(X) ⊗ ℚ,

where K_{n}^{(i)}(X) ⊗ ℚ denotes the eigenspace of ψ^{k} acting on K_{n}(X) ⊗ ℚ as ψ^{k} = k^{i}.

Localization Theorem 1.4. ( [Bl]) If U = X − Z, where Z is closed of pure codimension c in a quasi-projective X, then there is a long exact sequence

⋯ → CH^{i}(U, n + 1) ⟶^{∂} CH^{i−c}(Z, n) ⟶^{i_{∗}} CH^{i}(X, n) ⟶^{j^{∗}} CH^{i}(U, n) ⟶^{∂} ⋯

Here i and j denote the inclusions of Z and U in X, respectively.

Example 1.4.1. If X[t, t^{−1}] denotes X ×Spec(k[t, t^{−1}]), the localization sequences for X[t, t^{−1}] ⊂ X[t] and X[t] ⊂ X × ℙ^{1} yield:

CH^{i}(X[t, t^{−1}], n) ≅ CH^{i}(X, n) ⊕ CH^{i−1}(X, n − 1),
CH^{i}(X × ℙ^{1}, n) ≅ CH^{i}(X, n) ⊕ CH^{i−1}(X, n).

These results show that in order to understand the groups CH^{i}(X, n) we must understand the higher Chow groups CH^{i}(k, n) of a field k. Recall from 1.1.1(1) that if i > n then CH^{i}(k, n) = 0 because z^{i}(k, n) = 0: no subvariety of Δ^{n} can have a codimension greater than n. Thus if i is fixed, the lowest nontrivial Chow group of a field is CH^{i}(k, i).

Theorem 1.5. (Nestorenko-Suslin [NS]) For every field k, and all n,

K_{n}^{M}(k) ≅ CH^{n}(k, n).

Explanation. The group z^{n}(k, n) is the free abelian group on the points x = (x_{1}, …, x_{n}) of Δ^{n} = Spec(k[t_{1}, …, t_{n}]) which do not lie on any face: each x_{j} is nonzero and x_{0} = 1 − (x_{1} + ⋯ + x_{n}) is also nonzero. Of course, the residue field k(x) of x need not be k, but rather a finite extension of k, so “point” has to be taken in the right sense. The isomorphism ν: CH^{n}(k, n) → K_{n}^{M}(k) is defined by sending the class of the point x to the norm of a Steinberg symbol, namely

N_{k(x)/k}\{\frac{x_{1}}{x_{0}} , ⋯, \frac{x_{n}}{x_{0}} \}.

Its inverse map ϕ: K_{n}^{M}(k) → CH^{n}(k, n) sends the Steinberg symbol {a_{1}, …, a_{n}} to the class of the k-rational point

(\frac{a_{1}}{a_{0}} , ⋯, \frac{a_{n}}{a_{0}} )

of Δ^{n}, provided that a_{0} = 1−(a_{1}+⋯+a_{n}) is nonzero; if a_{0} = 0 then this Steinberg symbol maps to zero.

Exercise 1.5.1. Show that ϕ is well-defined, and is inverse to the map ν.

The following remarkable theorem was proven recently by Bloch and Lichtenbaum [BL]. It shows that we have really been talking about K-theory all along.

Theorem 1.6. (Bloch-Lichtenbaum) For every field k there is a third quadrant spectral sequence (p, q ≤ 0):

E_{2}^{pq} = CH^{−q}(k, −p − q) ⇒ K_{−p−q}(k).

The edge maps along the y-axis of this spectral sequence are the natural maps from the Milnor K-groups K_{n}^{M}(F) ≅ CH^{n}(F, n) to K_{n}(F). The first nontrivial part of the spectral sequence, which implies that CH^{2}(F, 3) ≅ K_{3}(F)_{ind}, is the exact sequence:

K_{4}(F) → CH^{2}(F, 4) ⟶^{d} K_{3}^{M}(F) → K_{3}(F) → CH^{2}(F, 3) → 0.

Remark 1.6.1. (Soulé) Adams operations ψ^{k} act on this spectral sequence, as multiplication by k^{i} on CH^{i}(F, n). Hence for each r > 1 there is an integer N = gcd{k(k^{r−1} − 1)} so that Nd_{r}^{pq} = 0 for every differential d_{r}^{pq}. For example, if r = 2 then N = 2. Therefore if we tensor with ℚ the spectral sequence degenerates. This gives the formula of theorem 1.3: CH^{i}(X, n)⊗ℚ ≅ K_{n}^{(i)}(X)⊗ℚ.

In the next lecture, we will see that the higher Chow groups may be interpreted in terms of motivic cohomology, with CH^{i}(X, n) ≅ H^{2i−n}(X, ℤ(i)) when char(k) = 0. With this indexing, the Bloch-Lichtenbaum spectral sequence becomes

(1.6.2)  E_{2}^{pq} = H^{p−q}(k, ℤ(−q)) ⇒ K_{−p−q}(k),  if char(k) = 0.

Beilinson has conjectured that the motivic groups H^{n}(k, ℤ(i)) should vanish if n < 0. If so, this spectral sequence would lie in the octant 0 ≥ p ≥ q.

For example, if we fix i = 2, then z^{2}(k, ∗) may be viewed as a cochain complex which is zero in non-negative degrees, as well as in degree −1. If we shift 4 steps to the right we get a cochain complex of abelian groups Γ = z^{2}(k, ∗)[4] whose positive cohomology is:

(1.6.3)  H^{n}(Γ) = \{ \begin{matrix}CH^{2}(k, 4) & if n = 0, \\ CH^{2}(k, 3) ≅ K_{3}(k)_{ind} & if n = 1, \\ CH^{2}(k, 2) ≅ K_{2}(k) & if n = 2, \\ 0 & if n ≥ 3. \end{matrix}

Conjecturally, CH^{2}(k, 4) should also vanish; this is unknown at present. Bloch and Lichtenbaum have proven that the truncation τ_{≥1}Γ, which eliminates H^{n} for n ≤ 0, is isomorphic to the “weight two motivic complex” introduced by Lichtenbaum [L87] in 1987.

§2. Chow groups with finite coefficients

We now consider higher Chow groups with finite coefficients ℤ/m. The idea is that these should be the hyper-Tor groups of the Bloch complex. Since the chain complex asociated to the simplicial group z^{i}(X, ·) is free in every degree, its total tensor product with ℤ/m is obtained by tensoring with ℤ/m.

Definition 2.1.

CH_{n}^{i}(X; ℤ/m) = π_{n}(z^{i}(X, ·) ⊗ ℤ/m)

We remark that the more awkward notation CH^{i}(X, n; ℤ/m) is used in the literature for CH_{n}^{i}(X; ℤ/m).

The following result, while not immediate from the construction of the Bloch-Lichtenbaum spectral sequence 1.6, is obtained from the exact couple used in the Bloch-Lichtenbaum construction by straightforward homotopy theory. A proof is given by Rognes and Weibel in Appendix B of [RW].

Theorem 2.2. There is also a third quadrant spectral sequence for every field k and every m:

E_{2}^{pq} = CH_{−p−q}^{−q}(k; ℤ/m) ⇒ K_{−p−q}(k; ℤ/m).

We now turn to the connection with étale cohomology. If j + n = 2i, there is a canonical map from CH_{j}^{i}(X; ℤ/m) to H_{et}^{n}(X; μ_{m}^{⊗i}), called the (generalized) norm residue homomorphism. Conjecturally, it is an isomorphism for n ≤ i. If char(k) = 0 there is some evidence for this guess, based upon the fact that the groups CH_{j}^{i}(X; ℤ/m) are isomorphic to the “motivic cohomology” groups H^{n}(X, ℤ/m(i)), which will be defined in the next lecture.

Theorem 2.3. (Suslin [S-2]) Suppose X is smooth and quasi-projective over an algebraically closed field k of characteristic zero. Then for all m > 0, n ≥ 0 and i ≥ dim(X) we have:

CH_{2i−n}^{i}(X; ℤ/m) ≅ H^{n}(X, ℤ/m(i)) ≅ H_{et}^{n}(X, μ_{m}^{⊗i}).

When X is singular, we should think of the higher Chow groups as a “Borel-Moore” homology theory, since over ℂ it agrees with the homology theory first constructed in [BM]. To see this, suppose that X is equidimensional of dimension d and i ≥ d. Then the Localization Theorem 1.4 implies that CH_{2i−n}^{i}(X; ℤ/m) ≅ H^{n}(X, ℤ/m(i)) is dual to H_{c}^{2d−n}(X, ℤ/m(d − i)), which is an étale cohomology group with compact supports.

As mentioned in Friedlander’s Lecture VI, Voevodsky has recently proven (in [V-MC]) that if m is a power of two then the higher Chow groups of a field k coincide with its étale cohomology:

Theorem 2.4. (Voevodsky [V-MC]) If char(k) = 0 and m = 2^{ν} then for all n ≤ i:

CH_{2i−n}^{i}(k; ℤ/m) ≅ H^{n}(k, ℤ/m(i)) ≅ H_{et}^{n}(k, μ_{m}^{⊗i}).

In particular, when n = i we have K_{n}^{M}(k)/m ≅ H_{et}^{n}(k, μ_{m}^{⊗n}).

The restriction on n is because the left-hand side is zero for n > i (i > 2i − n) by example 1.1.1(1). In contrast, the étale cohomology groups H_{et}^{n}(k, μ_{m}^{⊗i}) on the right-hand side can be nonzero for all n ≥ 0.

One way to describe this result is to interpret both sides as the cohomology of a cochain complex of Zariski sheaves, and to truncate the complex whose Zariski cohomology gives étale cohomology. Thus if π: X_{et} → X_{zar} then the étale cohomology of X is the Zariski cohomology of the cochain complex Rπ_{∗}(μ_{m}^{⊗i}), and the motivic cohomology of X is the Zariski cohomology of a cochain complex called ℤ/m(i). Moreover, there is a natural map in the derived category ℤ/m(i) → τ_{≤i}Rπ_{∗}(μ_{m}^{⊗i}). The Beilinson-Lichtenbaum conjecture states that this map should be a quasiisomorphism (of complexes of Zariski sheaves). A key ingredient in Voevodsky’s proof of theorem 2.4 is the reduction in [SV-3] of the Beilinson-Lichtenbaum conjecture to the following conjecture, first stated by Milnor for ℓ = 2 in [M-QF] and later by Bloch and Kato for odd ℓ.

Conjecture 2.5 (Milnor and Bloch-Kato). For every prime ℓ except char(k), the Galois symbol (Lecture VIII(2.11)) should be an isomorphism.

Is K_{n}^{M}(k)/ℓ ≅ H_{et}^{n}(k, μ_{ℓ}^{⊗n}) ?

Of course, this conjecture is only open for ℓ odd, since Voevodsky’s Theorem 2.4 establishes the case ℓ = 2.

Corollary 2.6. If char(k) = 0 and m = 2^{ν}, the third quadrant spectral sequence of theorem 2.2 is:

E_{2}^{pq} = \{ \begin{matrix}H_{et}^{p−q}(k, μ_{m}^{⊗−q}), & q ≤ p ≤ 0 \\ 0, & otherwise \end{matrix} \} ⇒ K_{−p−q}(k; ℤ/m).

Using Voevodsky’s theorem 2.4, Rognes and Weibel [RW] have calculated the 2-primary Sylow subgroups of the K-theory of a number field F, and of its ring of integers O_{F}. The answer uses certain numbers w_{i} = w_{i}(F), which are defined as the order of the finite cyclic groups H_{et}^{0}(F, ℚ/ℤ(i)). Here are two special cases of their calculation. The notation “odd” represents an unknown finite group of odd order.

Theorem 2.7. Let F be a totally imaginary number field, and set r_{2} = \frac{1}{2}[F : ℚ], X = Spec(O_{F} [\frac{1}{2}]). Then for n > 1,

K_{n}(O_{F}) = \{ \begin{matrix}ℤ^{r_{2}} ⊕ ℤ/w_{i} ⊕ (odd) & n = 2i − 1 \\ H_{et}^{2}(X, ℤ_{2}(i + 1)) ⊕ (odd) & n = 2i. \end{matrix}

Theorem 2.8. Let F be a totally real number field, and set r_{1} = [F : ℚ]. Then if the Galois group Gal(F/ℚ) is an abelian group, the zeta function ζ_{F} satisfies: ζ_{F} (1 − i) is a rational number, and its 2-part ζ_{F} (1 − i)_{(2)} is related to the order of the 2-primary subgroup K_{n}(O_{F}){2} by the formula

ζ_{F} (1 − i)_{(2)} = 2^{r_{1}} · \frac{K_{2i−2}(O_{F})\{2\}}{K_{2i−1}(O_{F})\{2\}}.

This result was conjectured by Lichtenbaum in [L73] for all totally real number fields, except for the additional factor of 2^{r_{1}}. A similar formula with 2 replaced by any other prime ℓ was also conjectured by Lichtenbaum in [L73], and is still open.

§3. Finite characteristic

Although Voevodsky’s result 2.4 requires characteristic zero, the following tricks allow us to deduce similar results for any field k of characteristic p > 0. Recall that for each such k there is a complete discrete valuation ring R with residue field k, whose field of fractions has characteristic zero.

Valuation Trick 3.1. Choose a field k of characteristic p > 0, and let m be prime to p. We construct a henselian valuation domain V with value group ℤ[\frac{1}{m}] and residue field k, whose field of fractions L has characteristic zero. Choose a henselian discrete valuation ring R with residue field k, whose field of fractions L_{0} has characteristic zero. If π is a parameter for R (i.e., ν(π) = 1) set R′ = R[t]/(t^{m} − π). By Eisenstein’s criterion, t^{m} − π is irreducible, so R′ is a domain. In fact R′ is a henselian discrete valuation ring with residue field k, and ν′(π) = \frac{1}{m}. Now take

V = \underrightarrow{lim} (R ⊂ R′ ⊂ R′′ ⊂ ⋯).

Note that a natural choice of the initial R and π yields a natural construction of V. For example, we could start with the Cohen p-ring (the Witt ring if k is perfect) and take π = p.

Lemma 3.2. Let F_{∗}(X) denote one of the functors K_{∗}(X; ℤ/m) or H_{et}^{∗}(X, μ_{m}^{⊗i}), where m is prime to p = char(k). Then there is a natural isomorphism

F_{∗}(k) ⟵^{≅} F_{∗}(V) ⟶^{≅} F_{∗}(L).

Proof. Both these functors satisfy a Rigidity Theorem that has

F_{∗}(R) ⟶^{≅} F_{∗}(R′) ⟶^{≅} ⋯ ⟶^{≅} F_{∗}(V) ⟶^{≅} F_{∗}(k).

And we also have maps of exact sequences

F_{∗}(k) ⟶ F_{∗}(R) ⟶ F_{∗}(L_{0}) ⟶ F_{∗−1}(k)
↓^{m=0}  ↓≅  ↓  ↓^{m=0}
F_{∗}(k) ⟶ F_{∗}(R′) ⟶ F_{∗}(L_{1}) ⟶ F_{∗−1}(k).

The outside vertical maps are multiplication by m, and hence are zero. Now use the fact that F_{∗} commutes with direct limits of rings.

Taking m = 2^{ν}, this lemma immediately translates Corollary 2.6 for L into a result for k:

Theorem 3.3. There is also a spectral sequence for every field k of characteristic p > 2 and every m = 2^{ν}:

E_{2}^{pq} = \{ \begin{matrix}H_{et}^{p−q}(k, μ_{m}^{⊗−q}), & q ≤ p ≤ 0 \\ 0, & otherwise \end{matrix} \} ⇒ K_{−p−q}(k; ℤ/m).

Voevodsky has also observed that this kind of trick deduces Milnor’s Conjecture 2.5 in finite characteristic from the result (Voevodsky’s theorem 2.4) in characteristic zero.

Theorem 3.4. (Voevodsky [V-MC]) For every field k of characteristic p > 2, every m = 2^{ν} and for every n ≥ 0:

K_{n}^{M}(k)/m ≅ H_{et}^{n}(k, μ_{m}^{⊗i}).

Of course, this is easy if k is finite, so we may assume that k is an infinite field. To apply the Valuation Trick 3.1 we need to extend the definition of Milnor

K-theory to the rings R and V. If A is a local ring with an infinite residue field k, we define the graded ring K_{∗}^{M}(A) to be the quotient of the tensor algebra of the group A^{×} of units in A, modulo the ideal generated by elements a ⊗ (1 − a), where both a and (1 − a) are units of A. The following result is implicit in [M-QF, 2.6].

Lemma 3.5. If k is infinite, and V and L are as above, then there is a natural isomorphism

K_{∗}^{M}(k)/m ⟵^{≅} K_{∗}^{M}(V)/m ⟶^{≅} K_{∗}^{M}(L)/m.

Proof. Because R and V are henselian we know, say by [SY, 3.4], that

K_{∗}^{M}(R)/m ≅ K_{∗}^{M}(R′)/m ≅ ⋯ ≅ K_{∗}^{M}(V)/m ≅ K_{∗}^{M}(k)/m.

In addition, because R is a discrete valuation ring, we have exact sequences (by [SY, 5.2]):

K_{∗}^{M}(R) → K_{∗}^{M}(L_{0}) ⟶^{∂} K_{∗−1}^{M}(k) → 0.

Tensoring with ℤ/m preserves exactness. If we multiply x ∈ K_{n}^{M}(R) by the parameter π ∈ L_{0}^{×} = K_{1}^{M}(L_{0}) and apply ∂ we have ∂{π, x} = x̄, where x̄ denotes the image of x in K_{n}^{M}(k). Going modulo m, we see that the map K_{∗}^{M}(R)/m → K_{∗}^{M}(L_{0})/m is a split injection.

Given the above facts, the proof of Lemma 3.2 goes through for the functor F_{∗}(A) = K_{∗}^{M}(A)/m.

We can now use the Valuation Trick 3.1 to deduce Theorem 3.4 from Voevodsky’s theorem 2.4. Indeed, one only needs to chase the commutative diagram:

K_{n}^{M}(k)/m ⟵^{≅} K_{n}^{M}(V)/m ⟶^{≅} K_{n}^{M}(L)/m
↓  ↓  ↓≅
H_{et}^{n}(k, μ_{m}^{⊗i}) ⟵^{≅} H_{et}^{n}(V, μ_{m}^{⊗i}) ⟶^{≅} H_{et}^{n}(L, μ_{m}^{⊗i}).

Concluding Remark 3.6. At present we do not know if the map CH_{2i−n}^{i}(k; ℤ/m) → H_{et}^{n}(k, μ_{m}^{⊗i}) is an isomorphism in characteristic p > 2, as it is in characteristic zero by Voevodsky’s theorem 2.4 (when m = 2^{ν}). The results in this section make it appear plausible, and it would follow if we knew that resolution of singularities held in characteristic p.