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Evaluation preserves kernels

PDF p. 4: Derived Nisnevich and cdh sections

Statement

For an object $U$ of a small site, evaluation $F \mapsto F (U)$ is additive and preserves kernels of morphisms of abelian sheaves. Consequently, once the category of abelian sheaves is known to be abelian, evaluation is left exact: it preserves exact sequences of the form $0 \to F^{'} \to F \to F^{''}$ .

Facts & Assumptions

Given: An object $U$ of a small site.

Goal: Prove the stated properties of evaluation.

[F1]

Kernels of abelian sheaf morphisms are computed objectwise. Pointwise kernels of abelian sheaves

Proof

technique · Evaluate the kernel universal property

1.1F1given

Addition of morphisms of abelian sheaves is defined componentwise, so evaluation preserves sums and the zero morphism. By [F1], evaluating a kernel gives the kernel of the evaluated homomorphism.

2.1step 1.1assume-hyp∎

In an abelian category, exactness of $0 \to F^{'} \to F \to F^{''}$ means that $F^{'} \to F$ identifies $F^{'}$ with the kernel of $F \to F^{''}$ . Step 1.1 therefore gives exactness after evaluation in abelian groups.

Depends on

Pointwise kernels of abelian sheaves

Used by · 2

Dependency tree · 3 levels

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