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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Regularity Notions And Permutation Invariance Local
Statement
Assume the Axiom of Choice. Let be a finite module over a Noetherian local ring , and let satisfy . Then ordinary, Koszul, and regularity coincide. Moreover, every permutation of an ordinary -regular sequence in is -regular.
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 prerequisites used here are Regularity Notions Coincide Local Finite and Regular Sequences Permutable Local.
Proof
Under the assumed AC, the preceding corollary identifies the three notions in the stated local setting.
Under the same AC premise, the permutation corollary applies to ordinary regularity and equivalence transfers the conclusion. These two corollary applications are the uses of Choice.
Depends on
Used by
- Nonpermutable Regular Sequence Example
Dependency tree · two levels
8 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)