LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-06‡ rests on unproved material (inherited)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Local Koszul Acyclicity Inductive Converse

Statement

Let $(R, m)$ be Noetherian local, $M$ finite, and $x = (x_{1}, \dots, x_{n}) \subseteq m$ be nonempty, with $M / (x) M \neq 0$ . If $K (x; M)$ has no positive homology, then the shorter complex is acyclic and the last element is injective on its preceding quotient.

Facts & Assumptions

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Koszul Mapping Cone Homology Exact Sequence, Regular Sequence On A Module, A local ring is a nonzero commutative ring with a unique maximal ideal, Left and right Noetherian rings, Noetherian modules: every submodule is finitely generated, Generated submodule, cyclic and finitely generated modules, module basis and free module, Assuming the Axiom of Choice, Nakayama's lemma.

Proof

technique · direct

1.1

Put $C = K (x_{1}, \dots, x_{n - 1}; M)$ and $y = x_{n}$ . For every $j > 0$ , the cone exact sequence and $H_{j + 1} (K (x; M)) = 0$ show that multiplication by $y$ on $H_{j} (C)$ is surjective. Each $H_{j} (C)$ is finite because $C$ is a bounded complex of finite modules over the Noetherian ring $R$ .

givenalgebra

2.1

Since $y \in m$ , Nakayama applied to the surjections in step 1.1 gives $H_{j} (C) = 0$ for every $j > 0$ . The segment of the same exact sequence ending in $H_{1} (K (x; M)) = 0$ then shows that multiplication by $y$ on $H_{0} (C) = M / (x_{1}, \dots, x_{n - 1}) M$ is injective. These are the two asserted conclusions.

step 1.1algebra∎

Depends on

Used by

Koszul Acyclicity Characterises Local Regular Sequences Theorem

Dependency tree · two levels

22 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, Koszul complexes and regular sequences (standard reference, not scraped)