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.

Localization of a presented divisor class group

PDF p. 13: Divisibility of the normalization Picard group

Statement

Let $S$ be a set, $B \subseteq S$ , $K$ an abelian group, and $d : K \to Z^{(S)}$ a homomorphism. Let $r$ be restriction to $S ∖ B$ and put $d^{'} = r \circ d$ . Then the natural sequence $Z^{(B)} \to coker (d) \to coker (d^{'}) \to 0$ is exact. Here $coker (d)$ denotes $Z^{(S)} / d (K)$ , and the first map extends by zero and passes to the quotient.

Facts & Assumptions

[F1]

Restriction $r : Z^{(S)} \to Z^{(S ∖ B)}$ is surjective with kernel the extensions by zero of $Z^{(B)}$ (Restriction of finite-support divisors).

[F2]

A surjective homomorphism $r : A \to D$ induces a surjection $A / P \to D / r (P)$ with kernel $(\ker r + P) / P$ for every subgroup $P$ (Quotienting a compatible surjection).

Given: The data in the statement.

Goal: Prove the stated cokernel sequence exact.

Proof

1.1F1F2given

Apply [F2] to $A = Z^{(S)}$ , $D = Z^{(S ∖ B)}$ , and $P = d (K)$ . Since $r (P) = d^{'} (K)$ , the induced map is the displayed map of cokernels and is surjective.

2.1F1F2step 1.1∎

By [F1] and the kernel formula in [F2], its kernel consists exactly of classes represented by divisors supported in $B$ . These are the image of $Z^{(B)}$ , proving exactness.

Depends on

Used by · 1

Picard group of an affine curve through its boundary

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.