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Every system of parameters is regular in a Cohen--Macaulay module
Statement
Assume the Axiom of Choice (The Axiom of Choice).
Every system of parameters of a nonzero finite Cohen--Macaulay module over a Noetherian local ring is a regular sequence on that module.
Facts & Assumptions
Given: the Axiom of Choice and a system of parameters for .
Under Choice, the parameter-sequence induction lemma makes the first parameter regular and the remaining tuple a parameter system on the Cohen--Macaulay quotient (Induction along a Cohen--Macaulay parameter sequence).
Proof
Induct on . For the empty sequence is regular because . For , [L1] makes regular and identifies as a parameter system on the Cohen--Macaulay quotient .
The induction hypothesis makes the remaining tuple regular on that quotient. Its terminal quotient is the nonzero parameter quotient (Nakayama), so the whole tuple satisfies the adopted definition of an -regular sequence.
Depends on
Used by
- Polynomial extension preserves Cohen--Macaulayness Corollary
- Cohen–Macaulayness over a finite regular local base Example
- Completed local degrees of finite normal surface covers Lemma
- Dualizing modules and trace pairing for normal projective surface modifications Lemma
- Reflection basic invariants form a regular sequence Lemma
- Parameters and regular sequences in Cohen--Macaulay modules Theorem
Dependency tree · two levels
6 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
- Depth and Cohen--Macaulay modules source treatment (standard reference, not scraped)