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Constant-sheaf sections have finite local representatives

PDF p. 5: Finite refinements and integer sections

Statement

For any set $A$ , let $P_{A}$ be the constant presheaf and $a A = a P_{A}$ its cdh sheafification on the specified site. For every $s \in (a A) (X)$ , the sieve of arrows $f : Y \to X$ for which $s ∣_{Y}$ is the image of some $a \in A$ is covering. Hence $s$ has a finite cdh covering family of constant representatives $a_{1}, \dots, a_{m}$ . For $A = Z$ , setting $D = {a_{1}, \dots, a_{m}} \subset Z$ yields a section of the finite constant-set sheaf $a D$ whose image in $a Z$ is $s$ . This is the precise finite-values assertion; it does not posit an unproved interpretation of arbitrary cdh sections as ordinary point functions.

Facts & Assumptions

Given: A constant presheaf and a section of its sheafification.

Goal: Produce finitely many constant representatives, and a finite constant-set factorization.

[F1]

The completed sheafification contract defines $a P = (P^{+})^{+}$ using matching families on covering sieves modulo common covering refinement. Scheme sites and sheafification

[F2]

Every covering sieve contains a finite $B$ -family. A finite-family basis for the generated cdh topology

Proof

1.1F1F2

Represent $s$ by a matching $P_{A}^{+}$ -family on a covering sieve $R$ using [F1]. Each of its entries in turn is represented by a matching $P_{A}$ -family on a covering sieve of its source. On each arrow of this second sieve, the restriction of $s$ is the image of an actual element of $A$ . The sieve $L$ of all arrows having such representatives is closed under precomposition because restrictions of constants remain constant. For each arrow of $R$ , its pullback of $L$ contains the corresponding second covering sieve, so it covers. Transitivity therefore shows $L$ covers $X$ . Apply [F2] to choose a finite family in $L$ and representatives $a_{i}$ for it.

2.1F1step 1.1∎

Sheafification preserves a presheaf injection: if two matching families from $P$ become equal in $Q^{+}$ under an injection $P ↪ Q$ , they agree on a common covering refinement in $Q$ , hence there in $P$ ; thus $P^{+} \to Q^{+}$ is injective, and applying this twice proves the assertion. Consequently $a D \to a A$ is a monomorphism for the subset $D = {a_{i}}$ . The local sections $a_{i}$ in $a D$ have equal restrictions on every pairwise overlap, since their images are the restrictions of $s$ and the map is injective. They glue uniquely to a section of $a D$ by the sheaf condition. Its image is $s$ by uniqueness of gluing in $a A$ . The empty-scheme case uses the empty family and the unique empty-domain section. This proves the finite-values formulation, in particular for integers.

Depends on

Used by · 1

Finite refinement for integer cdh sections

Dependency tree · 3 levels

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