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.

Projective and injective constructions of Ext agree for supplied resolutions

Statement

For supplied resolutions of $M, N$ , the canonical maps from $Hom (P_{∙}, N)$ and $Hom (M, I^{∙})$ to $Tot Hom (P_{∙}, I^{∙})$ are quasi-isomorphisms. Consequently ${Ext}_{P}^{q} (M, N) ≅ {Ext}_{I}^{q} (M, N)$ for all $q \geq 0$ .

Facts & Assumptions

Given: A projective resolution $P_{∙} \to M$ and an injective resolution $N \to I^{∙}$ .

Proof

technique · direct

1.1

The augmented columns are exact after applying $Hom (P_{p}, -)$ , and the augmented rows are exact after applying $Hom (-, I^{q})$ . Both augmentations commute with the other differential.

given

2.1

Finite-diagonal acyclic assembly applied first to columns and then to rows makes both edge-to-total maps quasi-isomorphisms. Taking cohomology yields the displayed isomorphism; the maps themselves are retained for the later naturality proof.

step 1.1∎

Depends on

Used by

The balanced Ext bifunctor Definition
FALSE: balance of Ext requires spectral-sequence pages False statement
FALSE: projective and injective Ext are equal by definition False statement
The Ext balance isomorphism is independent of resolution comparison data Lemma
The Ext balance isomorphism is natural in both variables Proposition
The two Ext long exact sequences agree under balance Proposition

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, Chapter 2 (standard reference, not scraped)