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Quasi-unmixed ring

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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 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 , = the Krull dimension of Ap.[2]

Equivalent conditions

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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 , the following are equivalent:[4][5]

  • 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

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A Noetherian local ring is said to be formally catenary if for every prime ideal , 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

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  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

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  • 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.