Generic formal fibers and analytically ramified stable rings

A commutative ring R is stable provided every ideal of R containing a nonzerodivisor is projective as a module over its ring of endomorphisms. The class of stable rings includes the one-dimensional local Cohen-Macaulay rings of multiplicity at most 2, as well as certain rings of higher multiplicity, necessarily analytically ramified. The former are important in the study of modules over Gorenstein rings, while the latter arise in a natural way from generic formal fibers and derivations.We characterize one-dimensional stable local rings in several ways. The characterizations involve the integral closure R of R and the completion of R in a relevant ideal-adic topology. For example, we show: If R is a reduced stable ring, then there are exactly two possibilities for R: (1) R is a Bass ring, that is, R is a reduced Noetherian local ring such that R is finitely generated over R and every ideal of R is generated by two elements; or (2) R is a bad stable domain, that is, R is a one-dimensional stable local domain such that R is not a finitely generated R-module. (Bruce Olberding) 1 Our definition of a stable ring differs slightly from Sally and Vasconcelos [39,40] in that we require only that every regular ideal is stable, not that every ideal is stable. However, for a reduced one-dimensional Cohen-Macaulay ring, the two definitions agree [35, p. 260].

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Prescribed subintegral extensions of local Noetherian domains

Olberding

2014

Journal of Pure and Applied Algebra

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We show how subintegral extensions of certain local Noetherian domains S can be constructed with specified invariants including reduction number, Hilbert function, multiplicity and local cohomology. The construction behaves analytically like Nagata idealization but rather than a ring extension of S, it produces a subring R of S such that R ⊆ S is subintegral.Let i > 0, and choose 0 = c ∈ J i (J + I) ∩ C. By (2.4), the mapping g : R/cR → S/cS ⋆ K/cK induced by f is an isomorphism of rings. Therefore, from (3) we haveThus since g is an isomorphism, there is an isomorphism of A-modules:The second isomorphism is also an isomorphism of R-modules, where the R-module structure is the usual one on the direct sum. Indeed, since S = A + cS, we have R = A + cS ∩ R. Moreover, c ∈ J, and J = JS ∩ R, so R = A + cS ∩ R = A + J, and from this it follows that this isomorphism of A-modules is also an isomorphism of R-modules. This verifies (1). Now a similar argument using (4) and (5) instead of (3) shows that when I ⊆ J there is an isomorphism of R-modules for all i ≥ 0,and this verifies (2). Now to prove statement (1) of the theorem, we make use of the isomorphisms (1) and (2). We consider the case first where we make no assumption about whether I ⊆ J. Since S = R + JS, it follows that the lengths of the S/JS-modules, J i IS/J i+1 IS and J i−1 (I + J)K/J i (I + J)K, agree with their lengths as R/J-modules. Therefore, we have by (1) that for all n ≥ 1, H J,I (n) = length J n I/J n+1 I = length J n IS/J n+1 IS + length J n−1 (J + I)K/J n (J + I)K = H JS,IS (n) + H JS,(I+J)K (n − 1).

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Generic formal fibers and analytically ramified stable rings

Cited by 3 publications

References 27 publications

Finitely Stable Rings

Finitely Stable Rings

One-dimensional stable rings

Prescribed subintegral extensions of local Noetherian domains

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