Quasi-unmixed ring

In algebra, specifically in the theory of commutative rings, a quasi-unmixed ring (also called a formally equidimensional ring in EGA{{harvnb|Grothendieck|Dieudonné|1965|loc=7.1.1}}) 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 $\widehat{A_p}$ , $\dim \widehat{A_p}/q = \dim A_p$ = the Krull dimension of A_p.{{harvnb|Ratliff|1974|loc=Definition 2.9. NB: "depth" there means dimension}}

Equivalent conditions

A Noetherian integral domain is quasi-unmixed if and only if it satisfies Nagata's altitude formula.{{harvnb|Ratliff|1974|loc=Remark 2.10.1.}} (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:{{harvnb|Ratliff|1974|loc=Theorem 2.29.}}{{harvnb|Ratliff|1974|loc=Remark 2.30.}}

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

Formally catenary ring

A Noetherian local ring $A$ is said to be formally catenary if for every prime ideal $\mathfrak{p}$ , $A/\mathfrak{p}$ is quasi-unmixed.{{harvnb|Grothendieck|Dieudonné|1965|loc=7.1.9}} 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

{{EGA | book=IV-2}}
Appendix of Stephen McAdam, Asymptotic Prime Divisors. Lecture notes in Mathematics.
{{cite journal |last=Ratliff |first=Louis |year=1974 |title=Locally quasi-unmixed Noetherian rings and ideals of the principal class |journal=Pacific Journal of Mathematics |doi=10.2140/pjm.1974.52.185 |volume=52 |issue=1 |pages=185–205|doi-access=free }}

Quasi-unmixed ring

Equivalent conditions

Formally catenary ring

References

Further reading