Cohen-Macaulay Rings, Invariant Theory, and the Generic Perfection of Determinantal Loci
1971; Johns Hopkins University Press; Volume: 93; Issue: 4 Linguagem: Inglês
10.2307/2373744
ISSN1080-6377
Autores Tópico(s)Cholinesterase and Neurodegenerative Diseases
Resumo0. Introduction. Let R be a commutative Noetherian ring with identity and let I be a proper ideal of R. A classical result of Krull is that if I can be generated by r elements then the rank or altitude of I (the greatest rank of any minimal prime of I) is at most r. If, moreover, the grade of I (the length of the longest R-sequence contained in I) is r, then I enjoys certain special properties summarized in the term as used by iRees [30, p. 32]: I is perfect if the homological (or projective) dimension of R/I as an R-module is equal to the grade of I. The associated primes of a perfect ideal I all have the same grade as I, that is, perfect ideals are grade unmixed. If R is Cohen-Macaulay, the grade of any ideal is equal to its little rank of height (the least rank of any minimal prime) ; in particular, the notions of grade and rank coincide on -primes, and perfect ideals are rank unmixed. Moreover, if I is perfect in a Cohen-Macaulay ring R, R/I is again (Cohen-Macaulay. Macaulay's famous theorem that in a polynomial ring over a field a rank r ideal which can be generated by r elements is rank unmixed [36, p. 203] is then a consequence of two facts: a polynomial ring over a field is CohenMacaulay, and a grade r ideal generated by r elements is perfect. This is the classical example of a perfect ideal. Good discussions of the subject. are available: see [9], [24, ? 25], [30], [18, Ch. 3], and [36, Appendix 6]. The Noetherian restriction on R is, for certain purposes, unnecessary in the discussion of perfect ideals, if one adopts a suitable definition of grade. This idea is worked out in [1]. Suppose that R is (locally) regular, and I is an ideal of R such that R/I is not the direct product of two rings in a nontrivial way. Then I is perfect if and only if R/I is Cohen-Macaulay. In particular, this is the situation when R is a polynomial ring over a field and I is homogeneous. It is very natural, then, to hunt for perfect ideals. Relatively few classes are known, but several authors [4, 6, 8, 33] have established the perfection
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