• Source: Quasi-unmixed ring

In algebra, specifically in the theory of commutative rings, a quasi-unmixed ring (also called a formally equidimensional ring in EGA) is a Noetherian ring



A


{\displaystyle A}

such that for each prime ideal p, the completion of the localization Ap is equidimensional, i.e. for each minimal prime ideal q in the completion







A

p


^





{\displaystyle {\widehat {A_{p}}}}

,



dim
⁡




A

p


^




/

q
=
dim
⁡

A

p




{\displaystyle \dim {\widehat {A_{p}}}/q=\dim A_{p}}

= the Krull dimension of Ap.




Equivalent conditions


A Noetherian integral domain is quasi-unmixed if and only if it satisfies Nagata's altitude formula. (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


{\displaystyle A}

, the following are equivalent:




A


{\displaystyle A}

is quasi-unmixed.
For each ideal I generated by a number of elements equal to its height, the integral closure





I
¯




{\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,






I

n


¯




{\displaystyle {\overline {I^{n}}}}

is unmixed.




Formally catenary ring


A Noetherian local ring



A


{\displaystyle A}

is said to be formally catenary if for every prime ideal





p




{\displaystyle {\mathfrak {p}}}

,



A

/



p




{\displaystyle A/{\mathfrak {p}}}

is quasi-unmixed. 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.




References



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.

Kata Kunci Pencarian:

• Quasi-unmixed ring
• Cohen–Macaulay ring
• Completion of a ring
• Minimal prime ideal
• Glossary of commutative algebra
• Regular chain
• Dionysus