PropositionStatement: 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.

Exact functors compatible with Hom transport Ext under stated adjunction hypotheses

Statement

Let $F : A \to B$ be exact and have a right adjoint $G$ . If $F$ sends projectives to projectives, then the adjunction isomorphisms induce ${Ext}_{B}^{n} (F M, X) ≅ {Ext}_{A}^{n} (M, G X)$ for every $n \geq 0$ , provided the displayed projective resolutions exist. The dual assertion holds for an exact $G$ that sends injectives to injectives.

Facts & Assumptions

Given: The stated exactness, adjunction, preservation, and resolution hypotheses.

Proof

technique · direct

1.1givenconstruct

Apply $F$ to a projective resolution of $M$ . Exactness preserves its augmentation exactness and the preservation hypothesis makes it a projective resolution of $F M$ . The adjunction Under local smallness, transposition gives the natural hom-set bijection, and conversely identifies its Hom cochain complex into $X$ with the original Hom cochain complex into $G X$ .

2.1step 1.1algebra∎

Taking cohomology and using The balanced Ext bifunctor gives the claimed isomorphism. The dual argument applies the stated injective preservation to the adjoint construction; no assertion is made without these hypotheses.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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)