LemmaStatement: AI-generatedProof: AI-generated

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Negative cohomology of a nonnegative evaluated complex

PDF p. 4: Derived Nisnevich and cdh sections

Statement

Let $I^{∙}$ be a cochain complex of abelian sheaves on a small site with $I^{n} = 0$ for $n < 0$ . For every object $U$ and every $n < 0$ , the cohomology group $H^{n} (I^{∙} (U))$ is zero. Here $H^{n}$ is the kernel of the degree- $n$ differential modulo the image of the preceding differential.

Facts & Assumptions

Given: A site object $U$ and a complex $I^{∙}$ vanishing in negative degrees.

Goal: Prove the negative-degree cohomology vanishes after evaluation.

[F1]

Evaluation sends complexes of abelian sheaves to complexes of abelian groups. Evaluation preserves chain homotopies

Proof

technique · Inspect the terms in negative degrees

1.1F1given

For each $n < 0$ , evaluation gives $I^{n} (U) = 0$ . Hence the kernel of the outgoing differential in degree $n$ is zero.

2.1step 1.1∎

The quotient defining $H^{n} (I^{∙} (U))$ has zero numerator and is therefore zero.

Depends on

Evaluation preserves chain homotopies

Used by · 1

Derived global sections on a scheme site

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.