CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 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.

Extension classes form a set whenever derived Ext one does

Statement

Assume the Axiom of Dependent Choice and the hypotheses of The balanced Ext bifunctor. If ${Ext}^{1} (M, N)$ is a set, then extensions of $M$ by $N$ have a set of equivalence classes: explicitly, there is a set $S$ and an assignment $c$ from extensions to $S$ , every element of $S$ occurs, and $c (e) = c (e^{'})$ exactly when $e$ and $e^{'}$ are equivalent. This is a set realization of the classes; it does not assert that each literal class of isomorphic extension diagrams is itself a set.

Facts & Assumptions

Given: Dependent Choice, the balanced-Ext resolution hypotheses, and the set $S = {Ext}^{1} (M, N)$ .

[F1]

Equivalent extensions have identical connecting classes: Equivalent extensions have the same Ext class.

[F2]

Every derived Ext-one element is represented by an extension: Every Ext-one class is represented by an extension.

[F3]

Extensions with the same connecting class are equivalent: Two extensions with the same Ext class are equivalent.

Proof

technique · direct

1.1

For an extension $e : 0 \to N \to E \to M \to 0$ , assign its connecting class $c (e) = δ_{e} (1_{M}) \in S$ . By [F1] and [F3], $c (e) = c (e^{'})$ if and only if $e$ and $e^{'}$ are equivalent. These lemmas concern individual extensions and do not assume a set of all extension classes.

givenF1F3construct

2.1

The supplied projective resolution gives the presentation $0 \to \ker (P_{0} (M) \to M) \to P_{0} (M) \to M \to 0$ required by [F2]. Hence every $s \in S$ equals $c (e)$ for some extension $e$ . Together with step 1.1 this makes the already given set $S$ a set realization of precisely the extension equivalence classes. No quotient of a proper class or simultaneous choice of representatives is used.

givenF2step 1.1algebra∎

Depends on

Used by

FALSE: extension classes automatically form a set False statement

Dependency tree · two levels

13 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)
The Stacks Project, Extensions and Ext groups (standard reference, not scraped)