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Finite rank strata detect Elmanto-Morrow motivic cohomology

PDF p. 8: Computations after the splitting principle

Statement

Assume AC and the article's field convention: $k$ is algebraically closed unless otherwise specified. Let $X$ be any qcqs $k$ -scheme and $X = ∐_{α \in A} X_{α}$ the finite common rank partition of finitely many finite locally free sheaves. For the actual Elmanto-Morrow motivic complex of every weight $j \geq 0$ , restriction gives $Z (j)_{m o t} (X) ≃ \prod_{α} Z (j)_{m o t} (X_{α}) .$ Thus for every $i \in Z$ and $R = Z$ or $Z / n$ with $n > 0$ , the joint restriction map $H_{m o t}^{i, j} (X, R) \overset{≅}{\to} \prod_{α} H_{m o t}^{i, j} (X_{α}, R)$ is an isomorphism and in particular injective. No smoothness, finite-type, noetherian, dimension or regularity hypothesis is required. Weights $j < 0$ are not defined here unless an additional zero-weight convention has explicitly been adopted.

Facts & Assumptions

Given: AC, the field, scheme, family and integer/coefficient bounds above.

Goal: Prove the exact motivic detection property required by a varying-rank flag tower.

[F1]

The common rank strata and their finite union are qcqs and clopen. Finite clopen common rank strata, The Axiom of Choice

[F2]

The specified complex is the restriction from the prime field of a homotopy-coherent $D (Z)$ -valued Nisnevich sheaf on qcqs schemes. Elmanto–Morrow motivic complex input

[F3]

Finite clopen descent for such a sheaf is the product equivalence, including its value zero on the empty scheme. Derived Nisnevich descent on a finite clopen partition

[F4]

Integral motivic groups are $H^{i}$ of the complex; finite coefficients apply derived tensor with $Z / n$ first. These operations commute with the finite product. Motivic complexes and cohomology, Finite derived products and motivic coefficient operations

Proof

1.1F1F2given

Every $k$ -scheme under consideration is a scheme over the prime field $F \subset k$ by composition of structure maps, and its underlying qcqs property is unchanged. The clopen inclusions and their fiber products in [F1] remain the same Nisnevich cover after forgetting to $F$ . Thus [F2] supplies descent for precisely this cover, not a different smooth or cycle-theoretic cohomology.

2.1F2F3step 1.1

Apply [F3] to the restricted weight- $j$ sheaf, with $j \geq 0$ . It gives the displayed equivalence of actual motivic complexes, whose components are the restriction maps. The empty-base case is included by its empty-cover clause.

3.1F4step 2.1∎

Applying the two operations in [F4] and their finite-product compatibility to step 2.1 gives the displayed group isomorphisms in every integer degree $i$ . Hence the joint restriction is injective. Only the hypotheses of the existing qcqs motivic sheaf and the finite clopen partition have been used, proving all claims.

Depends on

Used by · 1

Detection on a simultaneous flag tower

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