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Complete Local Ring

A complete local ring is a local ring (R,m) for which the canonical ring homomorphism to its completion is an isomorphism,

R=lim_(<--)nR/m^n.

The inverse limit consists of compatible residue classes modulo successive powers of the maximal ideal. Completeness refers to the topology defined by these powers, not to an arbitrary topology on R (Bruns and Herzog 1998, Stacks Project Authors 2026).

For example, the power series ring k[[x_1,...,x_d]] of formal power series over a field k is complete with respect to its maximal ideal (x_1,...,x_d). The small Cohen-Macaulay module conjecture asks an existence question over complete local rings that are integral domains.

See also

Discrete Valuation Ring, Formal Power Series, Inverse Limit, Local Ring, Maximal Ideal, Power Series Ring, 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. "The Cohen Structure Theorem." Definition 10.160.1 in The Stacks Project, Tag 0324, 2026. https://stacks.math.columbia.edu/tag/0324.

Cite this as:

Weisstein, Eric W. "Complete Local Ring." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CompleteLocalRing.html

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