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Module Depth

The module depth of a nonzero finitely generated module M over a Noetherian local ring (R,m) is the largest length of a ring regular sequence for M contained in its maximal ideal m. Such a sequence x_1, ..., x_r has the property that multiplication by x_i is injective on M/(x_1,...,x_(i-1))M and the final quotient module is nonzero (Bruns and Herzog 1998, pp. 3 and 10).

The depth is at most the Krull dimension dimR (Bruns and Herzog 1998, p. 12). For M=R, it is the depth of the ring. A Cohen-Macaulay ring that is a local ring has depth equal to its Krull dimension (Stacks Project Authors 2026).

See also

Cohen-Macaulay Ring, Krull Dimension, Local Ring, Module, Ring Regular Sequence, Serre Condition S2, Small Cohen-Macaulay Module

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References

Bruns, W. and Herzog, J. Cohen-Macaulay Rings, 2nd ed. Cambridge, England: Cambridge University Press, 1998.The Stacks Project Authors. "Cohen-Macaulay Rings." §10.104 in The Stacks Project, Tag 00N7, 2026. https://stacks.math.columbia.edu/tag/00N7.

Cite this as:

Weisstein, Eric W. "Module Depth." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ModuleDepth.html

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