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 a projective resolution of the first variable

Definition

Let $A$ be abelian and let $P$ be supplied projective-resolution data, written homologically with $d_{P} : P_{q + 1} (M) \to P_{q} (M)$ . For $M, N$ in its domain, set $C_{P}^{q} (M, N) = {Hom}_{A} (P_{q} (M), N), d_{P}^{q} (f) = f \circ d_{P}^{q + 1} .$ Then $d_{P}^{q + 1} d_{P}^{q} = 0$ because consecutive differentials of $P (M)$ compose to zero. Define ${Ext}_{P}^{n} (M, N) : = H^{n} (C_{P}^{∙} (M, N)) (n \geq 0) .$ The subscript $P$ records the supplied choice; no equality with the injective construction is being made here.

Ext via a projective resolution of the first variable

Definition

Depends on

Dependency tree · 2 levels

Sources