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.

Vanishing end groups kill the middle obstruction group

Statement

Let $A, H, D$ be abelian groups and $f : A \to H$ , $g : H \to D$ homomorphisms exact at $H$ , meaning $im f = \ker g$ . If $A$ and $D$ are zero groups, then $H$ is a zero group. Every nonempty set that is a surjective image of the underlying set of $H$ then has exactly one element. No injectivity of $f$ or surjectivity of $g$ is required.

Facts & Assumptions

Given: The groups, exactness, and vanishing assumptions in the Statement.

Goal: Prove the zero-group and singleton assertions.

Proof

1.1givenalgebra

Since $D$ has only its identity element, every $h \in H$ satisfies $g (h) = 0$ , hence $\ker g = H$ . Since $A$ has only its identity and $f$ preserves identity, $im f = {0}$ . Exactness gives $H = \ker g = im f = {0}$ .

2.1step 1.1given∎

A function from the one-element underlying set of $H$ has image consisting of its single value. If it is surjective onto a nonempty set, that set equals this image and is a singleton.

Used by · 1

Unique lifting from vanishing groups yields the Chern classification

Sources

Aravind Asok and Jean Fasel, A cohomological classification of vector bundles on smooth affine threefolds · Theorem 1, p. 2; §6, pp. 22–26