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.

Quotienting a compatible surjection

PDF p. 13: Divisibility of the normalization Picard group

Statement

Let $r : A \to D$ be a surjective homomorphism of abelian groups, and let $P \subseteq A$ be a subgroup. Then $r$ induces a surjection $\overset{ˉ}{r} : A / P \to D / r (P)$ whose kernel is $(\ker r + P) / P$ .

Facts & Assumptions

Given: $r$ , $A$ , $D$ and $P$ as stated. A quotient $A / P$ consists of cosets with addition induced by that of $A$ .

Goal: Establish the induced map, its surjectivity and its kernel.

Proof

1.1givenconstruct

Define $\overset{ˉ}{r} (a + P) = r (a) + r (P)$ . If $a - a^{'} \in P$ , then $r (a) - r (a^{'}) \in r (P)$ , so this assignment is well-defined and is a homomorphism. Surjectivity follows from that of $r$ .

2.1step 1.1∎

The coset $a + P$ lies in the kernel exactly when $r (a) = r (p)$ for some $p \in P$ , equivalently $a - p \in \ker r$ . Thus the kernel is precisely $(\ker r + P) / P$ .

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Localization of a presented divisor class group