An Alexandrov topology for maximal Cohen-Macaulay modules

Using the theory of cohomology annihilators, we define a family of topologies on the set of isomorphism classes of maximal Cohen-Macaulay modules over a Gorenstein ring. We study compactness of these topologies.     

Deformations of arithmetically Cohen-Macaulay subvarieties of pn

We show that every arithmetically Cohen-Macaulay two-codimensional subscheme ofP ⁿ can be deformed to a reduced union of two-codimensional linear subvarieties. This problem (classical for curves with the name of Zeuthen problem) was solved for curves by F.Gaeta.     

Bezout's theorem and Cohen-Macaulay modules

We define very proper intersections of modules and projective subschemes. It turns out that equidimensional locally Cohen-Macaulay modules intersect very properly if and only if they intersect properly. We prove a Bezout theorem for modules which meet very properly. Furthermore, we show for equidimensional subschemes X and Y: If they intersect properly in an arithmetically Cohen-Macaulay subscheme of positive dimension then X and Y are arithmetically Cohen-Macaulay. The module version of this result implies splitting criteria for reflexive sheaves. Received August 26, 1999 / Published online March 12, 2001     

Hilbert-Samuel functions of modules over Cohen-Macaulay rings

For a finitely generated, non-free module over a CM local ring , it is proved that for the length of is given by a polynomial of degree . The vanishing of is studied, with a view towards answering the question: If there exists a finitely generated -module with such that the projective dimension or the injective dimension of is finite, then is regular? Upper bounds are provided for beyond which the question has an affirmative answer.

     

On the Cohen-Macaulay modules of graded subrings

We give several characterizations for the linearity property for a maximal Cohen-Macaulay module over a local or graded ring, as well as proofs of existence in some new cases. In particular, we prove that the existence of such modules is preserved when taking Segre products, as well as when passing to Veronese subrings in low dimensions. The former result even yields new results on the existence of finitely generated maximal Cohen-Macaulay modules over non-Cohen-Macaulay rings.

     

On Cohen-Macaulay modules over non-commutative surface singularities

We generalize the results of Kahn about a correspondence between Cohen-Macaulay modules and vector bundles to non-commutative surface singularities. As an application, we give examples of non-commutative surface singularities which are not Cohen-Macaulay finite, but are Cohen-Macaulay tame.     

A characterization of Cohen-Macaulay modules and local cohomology

Let R be a commutative Noetherian ring, and let N be a non-zero finitely generated locally quasi-unmixed R-module. In this paper, the main result asserts that N is Cohen-Macaulay if and only if, for any N-proper ideal I of R generated by height_N I elements, the set of asymptotic primes of I with respect to N is equal to the set of presistent primes of I with respect to N. In addition, some applications about local cohomology are included. Received: 3 July 2005     

The Hilbert function of a maximal Cohen-Macaulay module

We study Hilbert functions of maximal CM modules over CM local rings. When A is a hypersurface ring with dimension d>0, we show that the Hilbert function of M with respect to is non-decreasing. If A=Q/(f) for some regular local ring Q, we determine a lower bound for e₀(M) and e₁(M) and analyze the case when equality holds. When A is Gorenstein a relation between the second Hilbert coefficient of M, A and S^A(M)= (Syz^A₁(M*))* is found when G(M) is CM and depthG(A)≥d−1. We give bounds for the first Hilbert coefficients of the canonical module of a CM local ring and analyze when equality holds. We also give good bounds on Hilbert coefficients of M when M is maximal CM and G(M) is CM.     

Generalized Cohen-Macaulay modules over rings with approximation property

Let (A, m) be an excellent Henselian ring with isolated singularity and letR be its completion. Then every indecomposable maximal Buchsbaum (resp. generalized Cohen-Macaulay)R-module is isomorphic with the completion of an indecomposable maximal Buchsbaum (resp. generalized Cohen-Macaulay)A-module. Hence one gets examples of non-complete, non-regular rings having finite Buchsbaum representation type.     

Indecomposable modules of large rank over Cohen-Macaulay local rings

A commutative Noetherian local ring is called Dedekind-like provided is one-dimensional and reduced, the integral closure is generated by at most 2 elements as an -module, and is the Jacobson radical of . If is an indecomposable finitely generated module over a Dedekind-like ring , and if is a minimal prime ideal of , it follows from a classification theorem due to L. Klingler and L. Levy that must be free of rank 0, 1 or 2.

Now suppose is a one-dimensional Cohen-Macaulay local ring that is not Dedekind-like, and let be the minimal prime ideals of . The main theorem in the paper asserts that, for each non-zero -tuple of non-negative integers, there is an infinite family of pairwise non-isomorphic indecomposable finitely generated -modules satisfying for each .

     

Maximal Cohen-Macaulay modules over a noncommutative 2-dimensional singularity

We study properties of graded maximal Cohen-Macaulay modules over an \mathbb{N}-graded locally finite, Auslander Gorenstein, and Cohen-Macaulay algebra of dimension two. As a consequence, we extend a part of the McKay correspondence in dimension two to a more general setting.