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.

Group cohomology as a derived functor

Definition

Assume the Axiom of Dependent Choice and fix supplied injective resolution data $I$ on all left $G$ -modules. Define $H_{I}^{n} (G; M) : = R_{I}^{n} ((-)^{G}) (M) .$ By invariants-as-Hom, this is the injective-resolution construction ${Ext}_{I, Z [G]}^{n} (Z, M)$ . The cited change-of-resolution theorem gives natural isomorphisms for two supplied data; after making that identification, write the resolution-independent notation $H^{n} (G; M)$ .

Depends on

Used by

Restriction and corestriction in group cohomology Definition
Cohomology of a finite cyclic group Example
Group cohomology is the derived functor of coinvariants False statement
This page defines H¹ by crossed homomorphisms False statement
This page defines H² by group extensions False statement
Degree-zero group (co)homology Proposition
Cohomological dimension is detected by vanishing Theorem
Long exact sequence in group cohomology Theorem
Shapiro lemma for group cohomology Theorem
The bar cochain complex computes group cohomology Theorem
Universal coefficients in degree two Theorem

Dependency tree · two levels

20 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

Weibel, Definition 6.1.2 (standard reference, not scraped)