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 $A$ 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.^[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 $A$ , the following are equivalent:^[4]^[5]

$A$ 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.^[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]

References

^ Grothendieck & Dieudonné 1965, 7.1.1
^ Ratliff 1974, Definition 2.9. NB: "depth" there means dimension
^ Ratliff 1974, Remark 2.10.1.
^ Ratliff 1974, Theorem 2.29.
^ Ratliff 1974, Remark 2.30.
^ Grothendieck & Dieudonné 1965, 7.1.9
^ L. J. Ratliff, Jr., Characterizations of catenary rings, Amer. J. Math. 93 (1971)

Grothendieck, Alexandre; Dieudonné, Jean (1965). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie". Publications Mathématiques de l'IHÉS. 24. doi:10.1007/bf02684322. MR 0199181.
Appendix of Stephen McAdam, Asymptotic Prime Divisors. Lecture notes in Mathematics.
Ratliff, Louis (1974). "Locally quasi-unmixed Noetherian rings and ideals of the principal class". Pacific Journal of Mathematics. 52 (1): 185–205. doi:10.2140/pjm.1974.52.185.

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Equivalent conditions

Formally catenary ring

References

Further reading