PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-06

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Positive projective-resolution Ext vanishes on a projective first variable

Statement

Assume the Axiom of Dependent Choice. Let $A$ be an abelian category with the supplied projective-resolution construction. If $P$ is projective, then for every object $N$ and every $q > 0$ , ${Ext}_{P}^{q} (P, N) = 0.$

Facts & Assumptions

Given: A projective object $P$ and an object $N$ .

[L1]

Projective-resolution Ext is the cohomology of $Hom (P_{∙}, -)$ (Ext via a projective resolution of the first variable).

[L2]

Positive right derived functors computed from arbitrary supplied injective-resolution data vanish on injective objects, assuming Dependent Choice (Positive right derived functors vanish on injective objects).

Proof

technique · direct

1.1

For the additive functor $F = {Hom}_{A} (-, N)$ on $A^{o p}$ , the supplied projective resolution of $P$ in $A$ is an injective resolution in $A^{o p}$ , and $P$ is injective there. Thus the cohomology in [L1] is the corresponding right-derived construction, and [L2] compares it with the length-zero resolution of $P$ .

L1L2givenconstruct

2.1

The Hom cochain complex of that length-zero resolution is concentrated in degree zero, so its cohomology is zero for $q > 0$ . The comparison in step 1.1 therefore gives ${Ext}_{P}^{q} (P, N) = 0$ .

step 1.1algebra∎

Depends on

Used by

FALSE: positive Ext vanishes whenever either variable is injective False statement
Ext dimension shifting in the first variable Theorem
Projective dimension at most n iff higher Ext vanishes Theorem

Dependency tree · two levels

7 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)