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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Cohen--Macaulayness and a regular parameter quotient
Statement
Assume the Axiom of Choice (The Axiom of Choice).
Under the hypotheses of
lem-regular-quotient-preserves-depth-dimension-gap, is Cohen--Macaulay
if and only if is Cohen--Macaulay.
Facts & Assumptions
Given: The Axiom of Choice; the depth--dimension gap is defined for both nonzero modules.
Proof
Under the assumed Choice, the gap lemma says the two modules have the same value.
Each module is Cohen--Macaulay exactly when that value is zero. Hence one is Cohen--Macaulay exactly when the other is.
Depends on
Used by
- Induction along a Cohen--Macaulay parameter sequence Lemma
- Normality and fibre cohomology of a rational surface point blowup Lemma
- Positive conormal degree for a fibre divisor on a normal surface Lemma
- Reduced Cartier multiples on a normal local surface Lemma
- Regular quotients and Cohen--Macaulayness Theorem
Dependency tree · two levels
9 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)