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.

Degree-zero representatives modulo a boundary class

PDF p. 13: Divisibility of the normalization Picard group

Statement

Let $q : A \to D$ and $\deg : A \to Z$ be homomorphisms of abelian groups. If $q$ is surjective and there is $e \in \ker q$ with $\deg (e) = 1$ , then $q$ maps $\ker (\deg)$ onto $D$ .

Facts & Assumptions

Given: The homomorphisms and element in the statement.

Goal: Find a degree-zero preimage of every element of $D$ .

Proof

1.1givenchooseconstruct

Given $y \in D$ , choose $a \in A$ with $q (a) = y$ and put $a_{0} = a - \deg (a) e$ . Then $\deg (a_{0}) = \deg (a) - \deg (a) \deg (e) = 0$ .

2.1givenstep 1.1∎

Since $q (e) = 0$ , we also have $q (a_{0}) = q (a) = y$ . Thus every $y$ has a preimage in $\ker (\deg)$ .

Used by · 1

Picard group of an affine curve through its boundary