Beilinson’s ‘office hours’ on Bloch’s lecture of 11.3.03.

The γ-filtration; an unorthodox approach.

We use some basic facts about Karoubi-Villamayor K-groups. For a ring R and i ≥ 1 one has K_{n}^{KV} (R) := π_{n−1}GL(R_{·}) = π_{n}BGL(R_{·}). Here R_{·} is the usual simplicial ring, R_{n} := R[t_{0}, .., t_{n}]/Σt_{i} = 1, GL := ∪GL_{n}, so GL(R_{·}) is a simplicial group, and we consider the homotopy groups of its realization.

Since elementary matrices lie in the connected component of 1 in GL(R_{·}), we see that K_{1}^{KV} (R) is commutative. Therefore the obvious embedding BGL(R) ↪ BGL(R_{·}) extends (canonically) to a morphism BGL(R)^{+} → BGL(R_{·}). Passing to the homotopy groups, we get a canonical morphism K_{n}(R) → K_{n}^{KV} (R).

From now on R is a commutative regular ring. Then K_{n}(R) →^{∼} K_{n}^{KV} (R). The key fact needed for the proof is that the assumption on R implies that K_{n}(R) →^{∼} K_{n}(R[t]). (Drinfeld’s question: is it true if R is not supposed to be commutative?)

Below Δ_{R}^{n} := SpecR_{n}, and Σ_{R}^{n−1} ⊂ Δ_{R}^{n} is the union of hyperplanes t_{i} = 0; let I_{n} ⊂ R_{n} be its ideal. We also denote by Δ^{n} the standard simplex (which is a simplicial set), and Σ^{n−1} ⊂ Δ^{n} is its boundary (a simplicial sphere).

Proposition. For n ≥ 1 one has K_{n}(R) →^{∼} Coker(K_{1}(R) → K_{1}(Σ_{R}^{n−1})).

Proof. Use the fact that GL(R_{·}) (being a simplicial group) is a Kahn’s simplicial set: K_{n}(R) = π_{n−1}(R_{·}) = Coker(Maps(Δ^{n}, GL(R_{·})), Maps(Σ^{n−1}, GL(R_{·}))) = Coker(GL(Δ_{R}^{n}) → GL(Σ_{R}^{n−1})). Recall that K_{1}(R) = K_{1}(R_{n}) = GL(R_{n})/E(R_{n}) and K_{1}(Σ_{R}^{n−1}) = GL(Σ_{R}^{n−1})/E(Σ_{R}^{n−1}) where E is the subgroup generated by the elementary matrices. Taking quotients mod E, we get a morphism Coker(GL(Δ_{R}^{n}) → GL(Σ_{R}^{n−1})) → Coker(K_{1}(R) → K_{1}(Σ_{R}^{n−1})). This is an isomorphism since the map E(R_{n}) → E(Σ_{R}^{n−1}) is surjective (this immediately follows from the surjectivity of the map R_{n} → Σ_{R}^{n−1}). □

Now let Z_{R}^{n−1} ⊂ Δ_{R}^{n} be the union of all closed subschemes that do not intersect Σ_{R}^{n−1}; equivalently, this is the union of hypersurfaces (f = 0) where f is a function such that f(Σ_{R}^{n−1}) = 1. Our Z_{R}^{n−1} is an ind-scheme; we define the Chow groups CH_{a}(Z^{n−1}) as the inductive limit of CH_{a}(Y) with respect to all subschemes Y ⊂ Z^{n−1}. Define K′_{0}(Z^{n−1}) in a similar way; it carries a filtration K′_{0}(Z^{n−1})_{·} by the dimension of support, and one has the usual identifications gr_{a}K′_{0}(Z^{n−1})_{ℚ} →^{∼} CH_{a}(Z^{n−1})_{ℚ}.

Let R̃_{n} be the localization of R_{n} with respect to all functions that equal 1 on Σ_{R}^{n−1}, Δ̃_{R}^{n} := SpecR̃_{n}. So Σ_{R}^{n−1} is a closed subscheme of Δ̃_{n}^{R}; let Ĩ_{n} ⊂ R̃_{n} be its ideal, so 1 + Ĩ_{n} ⊂ R̃_{n}^{×} ⊂ K_{1}(R̃_{n}). We have a usual “boundary” map ∂ : K_{1}(R̃_{n}) → K′_{0}(Z_{R}^{n−1}), so for f ∈ R_{n} which is invertible in R̃_{n}, i.e., on Σ_{R}^{n−1}, one has ∂(f) = [R_{n}/fR_{n}] ∈ K′_{0}(Z_{R}^{n−1}).

Theorem. One has K_{n}(R) = K′_{0}(Z_{R}^{n−1})/∂(1 + Ĩ_{n}).

Proof. (a) We have K_{n}(R) = Coker(K_{1}(R_{n}) → K_{1}(Σ_{R}^{n−1})) →^{∼} K_{0}(Δ_{R}^{n}, Σ_{R}^{n−1}). The latter arrow is the boundary map in the exact sequence for relative K-groups; it is an isomorphism since the restriction map i : K_{0}(R_{n}) → K_{0}(Σ_{R}^{n−1}) is injective (any R-point x on the smooth part of Σ_{R}^{n−1} yields the pull-back map K_{0}(Σ_{R}^{n−1}) → K_{0}(R) →^{∼} K_{0}(R_{n}), which is a left inverse to i).

(b) Consider the exact category of finitely generated R_{n}-modules P such that R̃_{n} ⊗ P is a projective R̃_{n}-module. It contains the exact category of R_{n}-modules supported on Z_{R}^{n−1}, the quotient category equals the category of finitely generated projective R̃_{n}-modules. We get the corresponding exact sequence of K-groups. We also have the exact sequence of relative K-groups with respect to the closed subscheme Σ_{R}^{n−1} which looks as K_{1}(Δ̃_{R}^{n}, Σ_{R}^{n−1}) → K′_{0}(Z_{R}^{n−1}) → K_{0}(Δ_{R}^{n}, Σ_{R}^{n−1}) → K_{0}(Δ̃_{R}^{n}, Σ_{R}^{n−1}). Since Ĩ_{n} lies in the Jakobson ideal of R̃, the last K-group vanishes, and the first equals 1 + Ĩ_{n}, hence K_{0}(Δ_{R}^{n}, Σ_{R}^{n−1}) = Coker(∂ : 1 + Ĩ_{n} → K′_{0}(Z_{R}^{n−1})), and we are done by (a). □

Spencer also claims that the γ-filtration on K_{n}(R) coincides with the decreasing filtration coming from the filtration by the codimension of support on K′_{0}(Z^{n−1}). I will consider the increasing filtration by the dimension of support instead denoting it by K′_{0}(Z^{n−1})_{a}.

Suppose n ≥ 2 and R = k is a field. Then Σ_{k}^{n−1} is connected, and 1 + Ĩ_{n} is the group of all codimension 1 cycles on Δ_{k}^{n} that do not intersect Σ_{k}^{n−1}. So the composition 1 + Ĩ_{n} → K′_{0}(Z^{n−1}) → K′_{0}(Z^{n−1})/K′_{0}(Z^{n−1})_{n−2} is an isomorphism, and we can rewrite the theorem as an identification

(1)  K_{n}(k) →^{∼} K′_{0}(Z^{n−1})_{n−2}.

Passing to gr, and using the motivic cohomology notation, we get for a ≤ n − 2 a canonical identification

(2)  H_{M}^{n−2a}(k, ℚ(n − a)) →^{∼} CH_{a}(Z^{n−1})_{ℚ}.

For a > n − 2 the l.h.s. in (2) vanishes.

Therefore the vanishing conjecture about motivic cohomology groups amounts to the vanishing of the Chow groups CH_{a}(Z^{n−1})_{ℚ} for a ∈ [n−2, n/2]. (Notice that the non-emptiness of the latter interval means that n ≥ 4). Can one prove this?