TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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The long exact Ext sequence in the second variable

Statement

Assume the Axiom of Dependent Choice. Let $A$ be abelian with enough projectives and enough injectives, and fix supplied projective and injective resolution data on all its objects. For $0 \to N^{'} \to N \to N^{''} \to 0$ and every $M$ , there is a natural exact sequence $0 \to Hom (M, N^{'}) \to Hom (M, N) \to Hom (M, N^{''}) \overset{δ^{0}}{\to} {Ext}^{1} (M, N^{'}) \to {Ext}^{1} (M, N) \to \dots,$ where $δ^{q} : {Ext}^{q} (M, N^{''}) \to {Ext}^{q + 1} (M, N^{'})$ ; it is natural in the short exact sequence and contravariantly natural in $M$ .

Facts & Assumptions

Given: A short exact sequence $0 \to N^{'} \to N \to N^{''} \to 0$ and an object $M$ .

Proof

technique · direct

1.1givenconstruct

Apply Right derived functors form a cohomological delta functor to the left exact functor $Hom (M, -)$ ; it supplies the displayed long exact sequence for the injective construction.

2.1step 1.1algebra∎

Replace its terms by balanced Ext using The balanced Ext bifunctor. Delta-functor naturality gives naturality in the short exact sequence, while precomposition in $M$ gives the stated contravariant naturality.

Depends on

Used by

Depth as the first nonzero Ext degree Corollary
The derived Ext-one class of an extension Definition
auslander buchsbaum projective dimension one Lemma
Depth drops by one after quotienting by a regular element Lemma
Equivalent extensions have the same Ext class Lemma
Maximal regular sequences have a common Ext length Lemma
Reflexive surface modules and codimension-one lattice extension Lemma
The first nonzero Ext shifts across a regular element Lemma
The two Ext long exact sequences agree under balance Proposition
Ext dimension shifting in the second variable Theorem

Dependency tree · two levels

19 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)