PropositionStatement: 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.

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

1.1

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 $G X$ .

givenconstruct

2.1

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.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

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)