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On the Gorenstein Property of Rees and Form Rings of Power of Ideals

1994; American Mathematical Society; Volume: 342; Issue: 2 Linguagem: Inglês

10.2307/2154645

1088-6850

M. Herrmann, J. Ribbe, S. Zarzuela,

Rings, Modules, and Algebras

In this paper we determine the exponents n for which the Rees ring R(In) and the form ring grA (In) are Gorenstein rings, where I is a strongly Cohen-Macaulay ideal of linear type (including complete and almost complete intersections) or an m-primary ideal in a local ring A with maximal ideal m.Given an ideal I in a local ring (A, m) it is well known that the Cohen-Macaulayness of the Rees algebra R(I) implies the Cohen-Macaulayness of all Rees algebras R(In).The same is true for the form rings grA (I) and grA (In); see [3, (2.7.8) and (8.8.5)].In this paper we show that, in contrast to the Cohen-Macaulay property, the Gorenstein property of R(In) and grA(In) only holds for special exponents n.If in particular grA(I) is Gorenstein and R(I) is Cohen-Macaulay it turns out that these special exponents are closely related to the a-invariant of the form ring grA(I).Mainly under this aspect we prove some results concerning the Gorenstein property of Rees and form rings of powers of (i) strongly Cohen-Macaulay ideals of linear type (including almost complete intersections) in Gorenstein rings, (ii) m-primary ideals in Cohen-Macaulay rings, and (iii) equimultiple prime ideals p, $ m in a generalized Cohen-Macaulay ring.Our investigations are essentially based on the explicit computation of the a-invariants of form (and Rees) rings (see ?2).For the above classes of idealsusing a structure theorem for the canonical module of R(I) in [7]-we can determine in ?3 the exponents n > 1 for which R(In) and grA (In) are Gorenstein rings (see in particular Theorem (3.5)).A more geometrical interpretation of the results in ?3 is the observation that in these situations the Gorenstein property of Proj(R(I)) can be deduced from the Gorenstein property of a certain Veronesean subring R(In) of R(I).It might be an interesting question when the Gorenstein property of a blow up Proj(R(I)) is inherited to an appropriate Rees ring R(J) with Proj(R(J)) = Proj(R(I)) over SpecA.In ?4,Part I we characterize-for Cohen-Macaulay (or Gorenstein) rings A-the Gorenstein property of R(md-i), i = 1, 2, 3, by conditions on the