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Smooth motivic comparison

PDF p. 4: Property (4); PDF p. 13: Smooth groups in Proposition 6.1

Statement

Assume the Axiom of Choice. Let k be a field, X a smooth finite-type k-scheme, j ≥ 0 and i an integer. The Elmanto–Morrow complex Z(j)_mot(X) is equivalent to $z^{j} (X, ∙) [- 2 j]$ , where the homological cycle complex is placed in cochain degree −n in homological degree n. Hence $H_{m o t}^{i, j} (X, Z) ≅ C H^{j} (X, 2 j - i)$ , and these groups agree with classical Voevodsky motivic cohomology. The motivic filtration of K(X) agrees, by a natural multiplicative filtered equivalence, with the filtration arising from Voevodsky's slice filtration. In particular $H_{m o t}^{2 r, r} (X, Z) ≅ C H^{r} (X)$ for r ≥ 0, and $H_{m o t}^{7, 3} (X, Z) = 0$ ; the latter includes the case of a smooth threefold.

Here $C H^{j} (X, n)$ is degree-n homology of Bloch's cycle complex, with negative degrees zero, and $C H^{r} (X)$ is codimension-r cycles modulo principal-divisor rational equivalence. The filtration is the decreasing Elmanto–Morrow tower on nonconnective K-theory of perfect complexes; its image filtration consists of the images of stage-to-zero maps on homotopy groups. The classical comparison and slice filtration are the constructions recalled in the imported smooth-comparison input.

Facts & Assumptions

Given: The Axiom of Choice, a field k, a smooth finite-type k-scheme X, j ≥ 0 and i in Z.

Goal: Establish the smooth complex and filtration comparisons and the stated cohomological consequences.

[F1]

The imported Elmanto–Morrow comparison supplies the derived equivalence, classical identification, and natural multiplicative filtered equivalence under these hypotheses. Elmanto–Morrow smooth comparison input

[F2]

Motivic cohomology is cohomology of the Elmanto–Morrow complex. Motivic complexes and cohomology

[F3]

Equivalent derived complexes have isomorphic cohomology; the cohomology of the shifted cycle complex is higher Chow homology in degree 2j−i. Cohomology of equivalent derived complexes, Cycle-complex degree translation, Bloch cycle complexes and higher Chow groups

[F4]

Degree-zero cycle homology is the ordinary Chow group, and negative cycle homology vanishes. Degree zero and negative homology of the cycle complex

[F5]

The assumed Choice premise, The Axiom of Choice, discharges the AC hypothesis of the selected perfect-complex realization when Motivic filtration and image groups is invoked. For nonnegative stages, the image filtration uses transitions to stage zero, and a filtered equivalence identifying stage zero with $K (X)$ identifies these images by Filtered equivalences preserve image filtrations. The integer-indexed convention of Motivic filtration and image groups extends the nonnegative tower constantly below zero, with stage $K (X)$ and identity augmentation. Apply the same constant extension to the classical tower when comparing integer indices. At every negative stage, both augmentation images are therefore the whole group $K_{i} (X)$ . No transition from a negative stage to zero is used, and the image comparison makes no further choice.

Proof

1.1givenF1

Apply the imported smooth comparison [F1] to X and j. This gives the displayed derived equivalence and the identification with the classical motivic complex; it also gives the multiplicative filtered equivalence.

2.1F2F3step 1.1

Apply cohomology to the derived equivalence and use the explicit chain-to-cochain and shift translation in [F3]. Together with the definition in [F2], this gives $H_{m o t}^{i, j} (X, Z) ≅ C H^{j} (X, 2 j - i)$ . The classical identification in step 1.1 identifies the same groups with classical Voevodsky motivic cohomology.

3.1F4step 2.1algebra

Put i = 2r and $j = r$ in step 2.1. The higher Chow index is zero, so [F4] identifies the result with $C H^{r} (X)$ . Put $i = 7$ and $j = 3$ : the index is −1, so the group is zero by [F4], independently of dimension.

4.1F5step 1.1step 2.1step 3.1∎

Invoke the local motivic-filtration interface under the assumed Choice premise as specified in [F5]. The filtered equivalence of step 1.1 gives the asserted comparison of nonnegative towers. For each nonnegative stage, its compatibility with the transition to zero gives the commuting square of [F5], so the images in $K_{i} (X)$ agree. Under the common constant extension to negative indices, extend the equivalence there by its stage-zero identification; both negative-stage augmentations are identities on $K (X)$ and thus have image $K_{i} (X)$ . This also compares the integer-indexed image filtrations without a negative-stage transition to zero. Steps 1.1 through 3.1 prove the cohomological assertions.

Depends on

Used by · 8

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Sources

Vector bundles on singular surfaces and threefolds · PDF p. 4, property (4); p. 13, Proposition 6.1 discussion