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Big Cohen-Macaulay Module

A big Cohen-Macaulay module over a commutative Noetherian local ring R is a module B on which some system of parameters x_1,...,x_d is a ring regular sequence (Hochster 1975). In particular, each x_i acts injectively on B/<x_1,...,x_(i-1)>B, and B/<x_1,...,x_d>B!=0. The module need not be finitely generated.

If every system of parameters is a ring regular sequence on B, the module is called balanced big Cohen-Macaulay. For a nonzero finitely generated module, these two conditions are equivalent and describe a small Cohen-Macaulay module. For example, a Cohen-Macaulay ring R, considered as a module over itself, is balanced big Cohen-Macaulay.

See also

Cohen-Macaulay Ring, Module Depth, Ring Regular Sequence, Small Cohen-Macaulay Module, System of Parameters

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References

Bruns, W. and Herzog, J. Cohen-Macaulay Rings, 2nd ed. Cambridge, England: Cambridge University Press, 1998.Hochster, M. Topics in the Homological Theory of Modules over Commutative Rings. Providence, RI: Amer. Math. Soc., 1975.

Cite this as:

Weisstein, Eric W. "Big Cohen-Macaulay Module." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BigCohen-MacaulayModule.html

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