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Derived Nisnevich descent on a finite clopen partition

PDF p. 8: Computations after the splitting principle

Statement

Let $F$ be a homotopy-coherent sheaf with values in the derived infinity-category $D (Z)$ on a category of schemes with the Nisnevich topology. Suppose the empty scheme and a finite clopen partition $X = ∐_{α \in A} X_{α}$ , its inclusions and their fiber products lie in that category. Then $F (\emptyset) ≃ 0$ , and restriction induces a natural equivalence $F (X) \overset{≃}{\to} \prod_{α \in A} F (X_{α}) .$ Here sheaf means that evaluating on a cover is the homotopy limit of its Cech descent diagram; the empty covering family is included. The finite empty product in $D (Z)$ is its zero object. No assertion about injectivity for general overlapping open covers is made.

Facts & Assumptions

Given: $F$ , its derived sheaf condition and the partition in the Statement.

Goal: Compute descent on this particular finite cover.

[F1]

Open immersions are etale and preserve residue fields, so an open cover is a Nisnevich cover. The empty family covers the empty scheme. Nisnevich covers and distinguished squares

[F2]

By the assumed sheaf condition, $F (X)$ is the totalization (homotopy limit) of the cosimplicial diagram whose degree $n$ term is the product of $F (X_{α_{0}} \times_{X} \dots \times_{X} X_{α_{n}})$ over all $(n + 1)$ -tuples of indices. Descent for the empty family is the limit of the empty diagram, the terminal object.

Proof

1.1F1F2

Apply the empty-cover case of [F1]–[F2]. It identifies $F (\emptyset)$ with the terminal object of $D (Z)$ , which is zero. This also proves the conclusion for an empty partition.

2.1F2step 1.1construct

For a nonempty partition, the intersection indexed by $(α_{0}, \dots, α_{n})$ is $X_{α}$ if all indices equal $α$ , and empty otherwise. Step 1.1 deletes all mixed-index factors, so in every degree the Cech object is canonically $\prod_{α} F (X_{α})$ . Every face and degeneracy map is the identity on these surviving factors, since all the diagonal fiber products of a clopen inclusion are that same open subscheme. Thus this is the constant cosimplicial object on that product.

3.1F1F2step 2.1∎

The totalization of a constant cosimplicial object with its identity structure maps is the object itself: the simplex indexing category has contractible nerve, with its terminal object $[0]$ , so the homotopy limit of this constant diagram has the canonical equivalence to its value. Apply [F2] and step 2.1. The augmentation maps to the surviving factors are precisely restriction to $X_{α}$ , giving the displayed natural equivalence.

Depends on

Nisnevich covers and distinguished squares

Used by · 5

Dependency tree · 3 levels

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