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. EGA IV. Part 2, Definition 7.1.1. harvnb error: no target: CITEREFEGA_IV._Part_2 (help)
2. Ratliff 1974, Definition 2.9. NB: "depth" there means dimension harvnb error: no target: CITEREFRatliff1974 (help)
3. Ratliff 1974, Remark 2.10.1. harvnb error: no target: CITEREFRatliff1974 (help)
4. Ratliff 1974, Theorem 2.29. harvnb error: no target: CITEREFRatliff1974 (help)
5. Ratliff 1974, Remark 2.30. harvnb error: no target: CITEREFRatliff1974 (help)
6. EGA IV. Part 2, Definition 7.1.9. harvnb error: no target: CITEREFEGA_IV._Part_2 (help)
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.
• L.J Ratliff Jr., Locally quasi-unmixed Noetherian rings and ideals of the principal class Pacific J. Math., 52 (1974), pp. 185–205

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.