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

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Positive injective-resolution Ext vanishes on an injective second variable

Statement

Assume the Axiom of Dependent Choice. Let $A$ be an abelian category and let $I$ be a supplied injective-resolution datum on a class $D$ of its objects. If $J \in D$ is injective, then ${Ext}_{I}^{n} (M, J) = 0$ for every $M \in D$ and every integer $n > 0$ .

Facts & Assumptions

Given: The supplied datum $I$ on $D$ , an object $M \in D$ , an injective object $J \in D$ , and an integer $n > 0$ .

Proof

technique · direct

1.1

Apply Positive right derived functors vanish on injective objects to the supplied injective-resolution datum and the additive left exact functor $Hom (M, -)$ . It compares the supplied resolution of $J$ with the length-zero injective resolution and gives vanishing in every positive degree.

givenconstruct

2.1

By Ext via an injective resolution of the second variable, that right-derived group is precisely ${Ext}_{I}^{n} (M, J)$ , so it is zero for $n > 0$ .

step 1.1algebra∎

Depends on

Used by

FALSE: positive Ext vanishes whenever either variable is injective False statement
Ext dimension shifting in the second variable Theorem
Injective 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)