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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 universe and site-size bounds of the supplied comparisons. Let $k$ be an algebraically closed field, let $X$ be an affine finite-type $k$ -scheme of dimension three, put $U = X_{s m o o t h / k}$ , and let $T \subset X$ be a closed subset contained in $U$ . Use the reduced structure on $T$ when a scheme structure is needed. Put $V = U ∖ T$ and $W = X ∖ T$ . 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 support complexes are unshifted cohomological fibers, so $H_{m o t, T}^{a, 3} (X, Z) ≅ H_{6 - a} z_{T}^{3} (U, ∙)$ . These maps respect the support-to-global maps, followed by restriction to $U$ .

Facts & Assumptions

Given: $k, X, U, T, V, W$ and the choice and size assumptions of the statement.

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

[F1]

The actual smooth open pair and its Nisnevich square, 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 foundational hypotheses of the support definition in [F1] and the diagram comparison in [F2] when they are invoked in steps 1.1 and 2.1. The excision and fiber arguments use no additional choice.

Proof

1.1F1F3

The completed EM complex supplier gives derived Nisnevich descent. The geometry lemma supplies the actual two-open cover 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.

2.1F1F2F3step 1.1∎

Taking homotopy fibers is a functor on coherent arrow diagrams. Apply it to the preceding restriction-diagram equivalence. 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.

Depends on

Used by · 2

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