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

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Ext dimension shifting in the first variable

Statement

Assume the Axiom of Dependent Choice. Let $A$ be abelian with enough projectives and enough injectives, fix supplied projective and injective resolution data on all its objects, and let $0 ⟶ Ω M \overset{i}{\to} P_{0} \overset{ε}{\to} M ⟶ 0$ be the first stage of a projective resolution. For every object $N$ there is an exact sequence $0 \to Hom (M, N) \to Hom (P_{0}, N) \to Hom (Ω M, N) \to {Ext}^{1} (M, N) \to 0,$ and for every $q \geq 1$ there is a natural isomorphism ${Ext}^{q} (Ω M, N) ≅ {Ext}^{q + 1} (M, N) .$

Facts & Assumptions

Given: The displayed first stage and the remaining projective resolution $\dots \to P_{2} \to P_{1} \to P_{0} \to M \to 0$ .

[L1]

A short exact sequence in the first variable gives the long exact Ext sequence (The long exact Ext sequence in the first variable).

Proof

technique · direct

1.1L1givenconstruct

Apply [L1] to $0 \to Ω M \to P_{0} \to M \to 0$ . Its relevant terms are ${Ext}^{q} (P_{0}, N) \to {Ext}^{q} (Ω M, N) \to {Ext}^{q + 1} (M, N) \to {Ext}^{q + 1} (P_{0}, N)$ .

2.1step 1.1algebra∎

Since $P_{0}$ is projective, the outer groups vanish for $q \geq 1$ by Positive projective-resolution Ext vanishes on a projective first variable, giving the displayed natural isomorphism. The degree-zero end of the same long exact sequence is exactly the displayed five-term sequence.

Depends on

Used by

An Ext dimension shift Example
Every Ext-one class is represented by an extension Lemma
Projective dimension at most n iff higher Ext vanishes Theorem

Dependency tree · two levels

21 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)