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 [math]\displaystyle{ A }[/math] 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 [math]\displaystyle{ \widehat{A_p} }[/math], [math]\displaystyle{ \dim \widehat{A_p}/q = \dim A_p }[/math] = the Krull dimension of Ap.[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 [math]\displaystyle{ A }[/math], the following are equivalent:[4][5]
- [math]\displaystyle{ A }[/math] is quasi-unmixed.
- For each ideal I generated by a number of elements equal to its height, the integral closure [math]\displaystyle{ \overline{I} }[/math] 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, [math]\displaystyle{ \overline{I^n} }[/math] is unmixed.
Formally catenary ring
A Noetherian local ring [math]\displaystyle{ A }[/math] is said to be formally catenary if for every prime ideal [math]\displaystyle{ \mathfrak{p} }[/math], [math]\displaystyle{ A/\mathfrak{p} }[/math] 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. http://www.numdam.org/articles/PMIHES_1965__24__5_0.
- 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.
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