DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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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.

Depends on

Supplied projective resolution data

Used by

Ext can be computed from any projective resolution of the first variable Corollary
The Hom double complex of projective and injective resolutions Definition
Ext from a two-term projective resolution Example
FALSE: Ext is defined before choosing or supplying resolutions False statement
FALSE: projective and injective Ext are equal by definition False statement
Positive projective-resolution Ext vanishes on a projective first variable Proposition
Projective-resolution Ext has the stated bifunctor variance Proposition
The degree-zero projective construction of Ext is Hom Proposition
Higher Yoneda Ext agrees with derived Ext Theorem
Projective and injective constructions of Ext agree for supplied resolutions Theorem
Projective dimension at most n iff higher Ext vanishes Theorem

Dependency tree · two levels

3 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)