Alphabeta MathHow 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.
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Local Koszul Acyclicity Iff Regular Sequence
Statement
Assume the Axiom of Choice. For a finite module over a Noetherian local ring, with and , is -regular if and only if its positive Koszul homology vanishes.
Facts & Assumptions
Given: The Axiom of Choice (The Axiom of Choice) and the rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisite used here is Koszul Acyclicity Characterises Local Regular Sequences.
Proof
The forward direction is regular-sequence Koszul acyclicity.
Under the assumed AC, the reverse direction of the local criterion applies; together with the terminal quotient hypothesis it gives regularity. This is the exact use of Choice.
Depends on
Used by
Dependency tree · two levels
5 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)