Formally Catenary Ring

In algebra, specifically in the theory of commutative rings, a quasi-unmixed ring (also called a formally equidimensional ring in EGA) is a Noetherian ring $A$ such that for each prime ideal ''p'', the completion of the localization ''A_p'' is equidimensional, i.e. for each minimal prime ideal ''q'' in the completion $\widehat$, $\dim \widehat/q = \dim A_p$ = the Krull dimension of ''A_p''.

Equivalent conditions

A Noetherian integral domain is quasi-unmixed if and only if it satisfies Nagata's altitude formula. (See also: #formally catenary ring below.)

Precisely, a quasi-unmixed ring is a ring in which the unmixed theorem, which characterizes a Cohen–Macaulay ring, holds for integral closure of an ideal; specifically, for a Noetherian ring $A$, the following are equivalent:
*$A$ is quasi-unmixed.
*For each ideal ''I'' generated by a number of elements equal to its height, the integral closure $\overline$ is unmixed in height (each prime divisor has the same height as the others).
*For each ideal ''I'' generated by a number of elements equal to its height and for each integer ''n'' > 0, $\overline$ is unmixed.

Formally catenary ring

A Noetherian local ring $A$ is said to be formally catenary if for every prime ideal $\mathfrak$, $A/\mathfrak$ is quasi-unmixed. As it turns out, this notion is redundant: Ratliff has shown that a Noetherian local ring is formally catenary if and only if it is universally catenary.L. J. Ratliff, Jr., Characterizations of catenary rings, Amer. J. Math. 93 (1971)

References

*
*Appendix of Stephen McAdam, Asymptotic Prime Divisors. Lecture notes in Mathematics.
*

Further reading

*Herrmann, M., S. Ikeda, and U. Orbanz: Equimultiplicity and Blowing Up. An Algebraic Study with an Appendix by B. Moonen. Springer Verlag, Berlin Heidelberg New-York, 1988.

Ring theory
Commutative algebra

{{commutative-algebra-stub