Quasi-unmixed ring

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.

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]

• 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.

• 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.