LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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Reduction of an exact free complex by a nonzerodivisor stays exact above degree one

Statement

Let $(R, m)$ be a local ring, let $x \in m$ be a nonzerodivisor, and let $F_{∙} : 0 \to F_{e} \to F_{e - 1} \to \dots \to F_{0}$ be a finite complex of finite free $R$ -modules. If $H_{j} (F_{∙}) = 0$ for every $j \geq 1$ , then $H_{j} (F_{∙} / x F_{∙}) = 0$ for every $j \geq 2$ . Equivalently, the reduced complex is exact at all its terms of degree at least two. The degree-one homology can appear from $x$ -torsion in the original cokernel.

Facts & Assumptions

Given: The local ring, nonzerodivisor, finite free complex, and exactness in positive degrees.

[F1]

A short exact sequence of complexes gives a long exact sequence in homology (The long exact sequence in homology).

Proof

technique · use the homology sequence of multiplication by the nonzerodivisor

1.1F1

Because each $F_{j}$ is free and $x$ is a nonzerodivisor on $R$ , multiplication by $x$ is injective on every $F_{j}$ . Thus $0 \to F_{∙} \overset{x}{\to} F_{∙} \to F_{∙} / x F_{∙} \to 0$ is a short exact sequence of complexes.

2.1F1step 1.1∎

By [F1], for $j \geq 2$ the segment $H_{j} (F_{∙}) \overset{x}{\to} H_{j} (F_{∙}) \to H_{j} (F_{∙} / x F_{∙}) \to H_{j - 1} (F_{∙})$ is exact. Both outer groups vanish by hypothesis, so $H_{j} (F_{∙} / x F_{∙}) = 0$ . At $j = 1$ the last group is $H_{0} (F_{∙})$ , which need not be $x$ -torsion-free; no degree-one conclusion is claimed.

Depends on

The long exact sequence in homology

Used by

Exact free complexes have the expected ranks and determinantal regular sequences Lemma

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

The Stacks Project, Algebra, Lemma 10.102.7 (tag 00MZ), reduction of exact free complexes (standard reference, not scraped)