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]

References

1. Grothendieck & Dieudonné 1965, 7.1.1
2. Ratliff 1974, Definition 2.9. NB: "depth" there means dimension
3. Ratliff 1974, Remark 2.10.1.
4. Ratliff 1974, Theorem 2.29.
5. Ratliff 1974, Remark 2.30.
6. Grothendieck & Dieudonné 1965, 7.1.9
7. 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.

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.