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Finite-coefficient motivic to cdh comparison

PDF p. 6: Proof of Proposition 2.6, EM23 Theorem 4.10(4)

Statement

Let k be an algebraically closed field of characteristic zero, X an affine threefold over k as in Proposition 2.6, and p a prime. The canonical comparison induces an isomorphism

$H_{m o t}^{5, 3} (X, Z / p) ≅ H_{c d h}^{5, 3} (X, Z / p)$ .

Here the left group is $H^{5} (Z (3)_{m o t} (X) \otimes_{Z}^{L} Z / p)$ , as in Motivic complexes and cohomology. The right group is $H^{5} (Z (3)_{c d h} (X) \otimes_{Z}^{L} Z / p)$ , where $Z (3)_{c d h}$ is Elmanto–Morrow's cdh-local motivic complex on qcqs Q-schemes. Finite coefficients are taken after evaluating each complex at X. The isomorphism is induced by the actual canonical comparison map; this statement does not identify the paper's cdh site with a separately chosen small site.

Facts & Assumptions

Given: k, X and p as stated.

Goal: Prove the displayed canonical group isomorphism.

[F1]

For a qcqs Q-scheme and a nonnegative weight, the canonical motivic-to-cdh map has rational cofiber. Elmanto–Morrow rational comparison cofiber

[F2]

A map in D(Z) with rational cofiber becomes an isomorphism after derived tensoring with Z/n for every positive n, and therefore induces an isomorphism on every cohomology group. Rational cofiber implies finite-coefficient equivalence

[F3]

Derived tensoring with Z/p is represented by the cone of multiplication by p. Derived finite coefficients and the Bockstein sequence

Proof

1.1givenF1

Since k has characteristic zero, the unique homomorphism Q → k exists. Composition X → Spec(k) → Spec(Q) makes X a Q-scheme. An affine scheme is quasi-compact and quasi-separated, so X satisfies [F1].

2.1F1F2step 1.1

Apply [F1] with $j = 3$ to obtain the canonical map Z(3)_mot(X) → $Z (3)_{c d h}$ (X) with rational cofiber. Since p is positive, [F2] makes its derived tensor with Z/p an isomorphism in D(Z).

3.1F2F3step 2.1∎

Taking $H^{5}$ and using the definitions in the statement gives the displayed isomorphism. By [F3] these are derived finite-coefficient groups; the map is induced from the canonical comparison before taking the quotient. This is the characteristic-zero application used on the cited article page.

Depends on

Dependency tree · 3 levels

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Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · Printed p. 6, proof of Proposition 2.6
Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Theorem 4.10(2)–(4), printed pp. 20–21