DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How 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.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Depth with respect to an ideal

Definition

Let $R$ be a commutative ring, $I \subseteq R$ an ideal, and $M$ a finite $R$ -module. If $I M \neq M$ , define ${depth}_{I} (M) = \sup {r \in N : there is an M -regular sequence of length r in I} .$ If $I M = M$ , set ${depth}_{I} (M) = \infty$ . In particular, the zero module has depth $\infty$ . For a local ring $(R, m)$ , write $depth (M)$ for ${depth}_{m} (M)$ .

Depends on

Regular Sequence On A Module

Used by

Depth as the first nonzero Ext degree Corollary
Depth is additive for a flat local homomorphism Corollary
Depth zero and associated primes Corollary
A normal singular surface: the quadric cone Counterexample
Cohen--Macaulay local modules and rings Definition
serre r k and s k conditions Definition
A Cohen--Macaulay ring with zero divisors Example
A depth computation before and after completion Example
A non-Cohen--Macaulay local quotient Example
Depth of a union of planes Example
The zero-module and surjective-ideal depth conventions Example
Three sharp Depth Lemma inequalities Example
auslander buchsbaum base case free module Lemma
Completion reflects depth Lemma
Depth at a prime is bounded by local support dimension Lemma
Depth from first nonzero Koszul cohomology Lemma
Depth is infinite when the ideal acts surjectively Lemma
Reflection basic invariants form a regular sequence Lemma
The highest positive homology of a depth-bounded finite complex has positive depth Lemma
Depth equals the maximal regular-sequence length Theorem
Depth is bounded by support dimension Theorem
regular local rings are domains and cohen macaulay Theorem

Dependency tree · two levels

3 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)