LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Restriction of finite-support divisors

PDF p. 13: Divisibility of the normalization Picard group

Statement

Let $S$ be a set and $B \subseteq S$ . Write $Z^{(S)}$ for the abelian group of integer-valued functions on $S$ with finite support. Restriction to $S ∖ B$ gives an exact sequence $Z^{(B)} \to Z^{(S)} \to Z^{(S ∖ B)} \to 0$ , where the first map extends functions by zero.

Facts & Assumptions

Given: A set $S$ and a subset $B$ .

Goal: Prove the displayed sequence exact.

Proof

1.1givenconstruct

Extension by zero and restriction preserve finite support and addition. Every finitely supported function on $S ∖ B$ extends by zero to $S$ , so restriction is surjective.

2.1step 1.1∎

A finitely supported function on $S$ restricts to zero on $S ∖ B$ exactly when its support is contained in $B$ ; it is then the extension by zero of its restriction to $B$ . This proves exactness.

Used by · 1

Localization of a presented divisor class group