An invariant of a module (or local ring) given by the length of a maximal regular sequence contained in a given ideal or the maximal length of nonzerodivisor sequences; it measures nondegeneracy of relations.

Principle

Depth counts how many successive elements act as nonzerodivisors on a module; a regular sequence of length d indicates the module resists certain degeneracies up to d steps.

Demonstration

For a local Noetherian ring R and a finitely generated module M, if there exists a sequence x1,...,xr of elements of the maximal ideal that are nonzerodivisors on successive quotients, then depth(M) ≥ r, and equality holds when the sequence is maximal.

Misapplication

Interpreting depth as codimension or Krull dimension without qualification is erroneous; depth and dimension are related by inequalities but are distinct invariants and can differ markedly in pathological cases.

Consequence

Depth controls homological properties: it appears in the Auslander–Buchsbaum formula and determines Cohen–Macaulayness when depth equals Krull dimension, implying favorable vanishing of local cohomology.

Reversal

Reversing the notion focuses on the presence of zerodivisors: instead of counting nonzerodivisors, one would measure how early zero divisors appear, yielding complementary obstruction information.

Boundary

Defined primarily for modules over commutative Noetherian rings and local rings; in non-Noetherian or noncommutative settings depth may be undefined or require modified definitions.

Semantic Tension

Depth competes with dimension and projective dimension: while dimension measures size of prime chains, depth measures resistance to zerodivisors; their interaction is central and sometimes paradoxical.

Synthesis

Depth is the homological measure of how many nonzerodivisor steps a module admits; combined with Krull dimension it indicates regularity properties and categorizes modules such as Cohen–Macaulay ones.