Quasi-unmixed ring

In algebra, specifically in the theory of commutative rings, a quasi-unmixed ring (also called a formally equidimensional ring in EGA^[1]) is a Noetherian ring ${\displaystyle A}$ 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 ${\displaystyle {\widehat {A_{p}}}}$, ${\displaystyle \dim {\widehat {A_{p}}}/q=\dim A_{p}}$ = the Krull dimension of A_p.^[2]

Equivalent conditions

A Noetherian integral domain is quasi-unmixed if and only if it satisfies Nagata's altitude formula.^[3] (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 ${\displaystyle A}$, the following are equivalent:^[4][5]

• ${\displaystyle A}$ is quasi-unmixed.
• For each ideal I generated by a number of elements equal to its height, the integral closure ${\displaystyle {\overline {I}}}$ 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, ${\displaystyle {\overline {I^{n}}}}$ is unmixed.

Formally catenary ring

A Noetherian local ring ${\displaystyle A}$ is said to be formally catenary if for every prime ideal ${\displaystyle {\mathfrak {p}}}$, ${\displaystyle A/{\mathfrak {p}}}$ is quasi-unmixed.^[6] 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.^[7]