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 homology as a derived functor

Definition

Assume the Axiom of Dependent Choice and fix supplied projective resolution data $P$ on all left $G$ -modules. Define $H_{n}^{P} (G; M) : = L_{n}^{P} ((-)_{G}) (M) = H_{n} ⁣ (Z \otimes_{Z [G]} P_{∙} (M)),$ where the first factor is the right trivial module. This is the left-resolution construction of ${Tor}_{n}^{Z [G]} (Z, M)$ . The cited change-of-resolution theorem gives natural isomorphisms for two supplied data; after making that identification, write $H_{n} (G; M)$ .

Depends on

Used by

Schur multiplier Definition
Superperfect group Definition
Group cohomology is the derived functor of coinvariants False statement
Low-degree sequence of a free presentation Lemma
Positive-degree homology of a finite group is order-torsion Lemma
Degree-zero group (co)homology Proposition
Long exact sequence in group homology Theorem
Shapiro lemma for group homology Theorem

Dependency tree · two levels

14 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)