DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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.

The balanced Ext bifunctor

Definition

Assume the Axiom of Dependent Choice. Let $A$ be an abelian category with enough projectives and enough injectives, and fix supplied projective and injective resolution data on all objects of $A$ . For each $n \geq 0$ , define ${Ext}_{A}^{n} (M, N)$ to mean either ${Ext}_{P}^{n} (M, N)$ or ${Ext}_{I}^{n} (M, N)$ , identified by the natural comparison isomorphism already proved. This notation is justified by the comparison theorem, its independence of comparison data, its two-variable naturality, and its change-of-resolution cocycle law; it is not a definition by equality of the two complexes.

Depends on

Used by

Ext can be computed from any injective resolution of the second variable Corollary
Ext can be computed from any projective resolution of the first variable Corollary
Extension classes form a set whenever derived Ext one does Corollary
Positive Ext need not vanish for an injective first variable Counterexample
The derived Ext-one class of an extension Definition
Ext zero as Hom in both constructions Example
Every Ext-one class is represented by an extension Lemma
Ext one of Z modulo n by Z is Z modulo n Lemma
The cohomological universal-coefficient extension map Lemma
Two extensions with the same Ext class are equivalent Lemma
Exact functors compatible with Hom transport Ext under stated adjunction hypotheses Proposition
Semisimple rings have vanishing positive Tor and Ext Proposition
Ext dimension shifting in the first variable Theorem
Projective dimension at most n iff higher Ext vanishes Theorem
The long exact Ext sequence in the first variable Theorem
The long exact Ext sequence in the second variable Theorem
Yoneda Ext one is naturally isomorphic to derived Ext one Theorem

Dependency tree · two levels

15 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)