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Excision and the exact fixed-support cycle fiber

PDF p. 13: Proposition 6.1 support argument

Statement

Assume the Axiom of Choice and the fixed site and universe bounds of the support and comparison constructions. Let $k$ be an algebraically closed field of arbitrary characteristic, let $X$ be an affine finite-type $k$ -scheme of Krull dimension three, and let $T \subset X$ be a closed subset contained in the smooth locus $U = X_{s m o o t h}$ of $X \to Spec k$ . Equip $T$ with its reduced induced structure, and put $V = U ∖ T$ and $W = X ∖ T$ . Write $Z (3)_{m o t, T} (X) : = fib (Z (3)_{m o t} (X) \to Z (3)_{m o t} (W))$ for the unshifted cohomological support fiber. There is an actual equivalence of support fibers $Z (3)_{m o t, T} (X) ≃ fib (Z (3)_{m o t} (U) \to Z (3)_{m o t} (V)) ≃ z_{T}^{3} (U, ∙) [- 6] .$ Here $z_{T}^{3} (U, ∙)$ is reindexed from homological degree $n$ to cochain degree $- n$ before shifting. Support complexes are unshifted cohomological fibers, so, for every integer $a$ , $H_{m o t, T}^{a, 3} (X, Z) ≅ H_{6 - a} z_{T}^{3} (U, ∙)$ . These maps respect the support-to-global maps, followed by restriction to $U$ .

Facts & Assumptions

Given: The hypotheses in the statement, with the Axiom of Choice and fixed universe bounds.

Goal: Construct the stated actual maps and prove their consequence.

[F1]

The actual smooth open pair and its choice scope, Motivic and filtered K theory with support, Elmanto–Morrow motivic complex input and Open excision for a descent theory.

[F2]

A coherent comparison of the particular open restriction and Cycle complexes with fixed support.

[F3]

The assumed Axiom of Choice, The Axiom of Choice, discharges the inherited Choice hypotheses of the geometry, support and comparison suppliers; their constructions use the fixed size bounds.

Proof

1.1F1F3given

Under the Choice and size assumptions in [F3], the EM complex supplier gives derived Nisnevich descent. The geometry lemma supplies the actual two-open cover $X = U \cup W$ with $U \cap W = V$ and its restriction square, so that square is homotopy cartesian. Apply the completed open-excision supplier to obtain the first support equivalence, with its support-to-global compatibility. All these objects are qcqs, so the arbitrary-open extension convention is unnecessary here.

2.1F1F2step 1.1∎

Taking homotopy fibers is a functor on coherent arrow diagrams. Apply it to the restriction-diagram equivalence of [F2] for this $V ↪ U$ . The completed fixed-support cycle definition gives the quasi-isomorphism from $z_{T}^{3} (U, ∙)$ to the homotopy fiber of the actual unshifted homological cycle restriction. Reindex to cochains and shift by $- 6$ , which commutes with fibers. This gives the second equivalence and $H^{a} = H_{6 - a}$ , with the indicated projection maps.

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