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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Depth is bounded by support dimension
Statement
Assume the Axiom of Choice (The Axiom of Choice). For every nonzero finite module over a Noetherian local ring , The nonzero hypothesis is essential for this formulation: under the adopted convention , whereas the empty support has no nonnegative Krull dimension.
Facts & Assumptions
Given: The Axiom of Choice; is nonzero and finite over the Noetherian local ring .
Depth is the supremum of lengths of regular sequences in the maximal ideal, including the empty sequence (Depth with respect to an ideal).
Under the assumed AC, the depth of a nonzero finite local module is at most its support dimension (A finite local module has depth at most its dimension).
Proof
By [L1], depth is a supremum of a set containing zero, so it is nonnegative. The AC-qualified supplier [L2] gives the upper bound.
These give the displayed double inequality. The last sentence follows directly from the separately declared zero-module depth convention.
Depends on
Used by
- One regular system of parameters implies Cohen--Macaulayness Corollary
- Polynomial extension preserves Cohen--Macaulayness Corollary
- Cohen–Macaulayness over a finite regular local base Example
- Reflection basic invariants form a regular sequence Lemma
- Finite étale covers of a smooth proper family over a complete DVR are determined by the closed fibre Theorem
- regular local rings are domains and cohen macaulay 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)