DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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.

Ext via an injective resolution of the second variable

Definition

Let $A$ be abelian and let $I$ be supplied injective-resolution data. For $M$ and $N$ in its domain, set $C_{I}^{q} (M, N) = {Hom}_{A} (M, I^{q} (N)), d_{I}^{q} (f) = d_{I (N)}^{q} \circ f .$ Thus $C_{I}^{∙} (M, N)$ is a cochain complex. Define ${Ext}_{I}^{n} (M, N) : = H^{n} (C_{I}^{∙} (M, N)) (n \geq 0) .$ The subscript $I$ remains part of the notation: this is a construction relative to the supplied data, not yet an intrinsic Ext bifunctor.

Ext via an injective resolution of the second variable

Definition

Depends on

Dependency tree · 2 levels

Sources