TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

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

1.1given

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.

2.1step 1.1∎

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.

Depends on

Used by

The balanced Ext bifunctor Definition
Five-term sequence of a composite functor Example
Grothendieck with an exact outer functor Example
LHS for a split group extension Example
FALSE: balance of Ext requires spectral-sequence pages False statement
FALSE: projective and injective Ext are equal by definition False statement
Grothendieck needs only left exactness False statement
Applying F gives a termwise G-acyclic complex Lemma
The Ext balance isomorphism is independent of resolution comparison data Lemma
Ext balance isomorphisms satisfy change-of-resolution cocycle laws Proposition
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)