TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Baer sum makes extension classes an abelian group

Statement

For fixed $M, N$ in an abelian category, assume that the equivalence classes of extensions of $M$ by $N$ form a set. With Baer sum, that set is an abelian group. Its zero is the split extension and the inverse of a class is its pushout along $- 1_{N}$ .

Facts & Assumptions

Given: The set of extension classes of $M$ by $N$ .

Proof

technique · direct

1.1

The Baer sum is independent of representatives makes the operation well-defined. The associativity and symmetry maps of finite biproducts transport the two iterated diagonal-pullback/codiagonal-pushout constructions into equivalent extensions; the required additive identities are those in On a biproduct, the injections and projections satisfy the identity-sum relation.

givenconstruct

2.1

The split extension is neutral because its diagonal pullback and codiagonal pushout recover the original extension. Pushing out an extension along $- 1_{N}$ gives the inverse: the codiagonal of $1_{N}$ and $- 1_{N}$ is zero, hence the resulting Baer sum is split.

step 1.1algebra∎

Depends on

Used by

Baer sum of two extensions of cyclic groups Example
The split extension as the zero Baer class Example
FALSE: Baer sum only takes the direct sum of middle objects False statement
Extension classes are contravariant in the quotient and covariant in the subobject Proposition
Yoneda Ext one is naturally isomorphic to derived Ext one Theorem

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapters 3–4 (standard reference, not scraped)