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Pointwise kernels of abelian sheaves

PDF p. 4: Derived Nisnevich and cdh sections

Statement

For a morphism $f : F \to G$ of abelian sheaves on a small site, the presheaf $K (U) = \ker (F (U) \to G (U))$ is a sheaf and is the categorical kernel of $f$ among abelian sheaves. A morphism of abelian sheaves is a natural transformation whose components are group homomorphisms.

Facts & Assumptions

Given: A small site and a morphism $f : F \to G$ of abelian sheaves.

Goal: Establish the sheaf and kernel properties of the pointwise kernel $K$ .

[F1]

Covering-sieve matching families of an abelian sheaf have unique amalgamations. Scheme sites and sheafification Matching families of abelian presheaves

Proof

technique · Glue in the source sheaf and use uniqueness in the target sheaf

1.1F1given

A compatible family in $K$ glues uniquely to $s \in F (U)$ by [F1]. Its image $f (s)$ restricts to zero on the covering sieve, so uniqueness of amalgamation in $G$ gives $f (s) = 0$ . Thus $s \in K (U)$ and $K$ is a sheaf.

2.1step 1.1construct∎

The inclusions $K (U) \subseteq F (U)$ form a natural transformation killed by $f$ . Any sheaf morphism into $F$ killed by $f$ factors uniquely through these pointwise kernels, and the factor maps are natural. Thus $K$ has the categorical kernel universal property.

Depends on

Used by · 1

Evaluation preserves kernels

Dependency tree · 3 levels

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