Quasi-unmixed ring explained
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
,
\dim\widehat{Ap}/q=\dimAp
= 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
, the following are equivalent:
is quasi-unmixed.
- For each ideal I generated by a number of elements equal to its height, the integral closure
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,
is unmixed.
Formally catenary ring
A Noetherian local ring
is said to be
formally catenary if for every prime ideal
,
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.
[1] References
- 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 . 10.2140/pjm.1974.52.185 . 52 . 1 . 185–205. free .
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.
Notes and References
- L. J. Ratliff, Jr., Characterizations of catenary rings, Amer. J. Math. 93 (1971)
