On canonical Cohen–Macaulay modules

The Cohen-Macaulay locus of any finite module over a noetherian local ring A is studied and it is shown that it is a Zariski-open subset of Spec A in certain cases. In this connection, the rings whose formal fibres over certain prime ideals are Cohen-Macaulay are studied.

Journal of Pure and Applied Algebra, 2000

Let R be a Gorenstein complete local ring. We say that ÿnitely generated modules M and N are linked if Hom R= R (M; R= R) ∼ = 1 R= R (N ), where is a regular sequence contained in both of the annihilators of M and N . We shall show that the Cohen-Macaulay approximation functor gives rise to a map r from the set of even linkage classes of Cohen-Macaulay modules of codimension r to the set of isomorphism classes of maximal Cohen-Macaulay modules. When r = 1, we give a condition for two modules to have the same image under the map 1. If r = 2 and if R is a normal domain of dimension two, then we can show that 2 is a surjective map if and only if R is a unique factorization domain. Several explicit computations for hypersurface rings are also given.

2016

Let (R, m) be a commutative Noetherian local ring, and M be a non-zero finitely-generated R-module. We show that if R is almost Cohen-Macaulay and M is perfect with finite projective dimension, then M is an almost Cohen-Macaulay module. Also, we give some necessary and sufficient conditions on M to be an almost Cohen-Macaulay module, by using Ext functors.

Nagoya mathematical journal

Introduction Throughout this paper, A denotes a noetherian local ring with maximal ideal m and M a finitely generated A-module with d: = DEFINITION. M is called a generalized Cohen-Macaulay (abbr. C-M) module if l(HUM)) < oo for i = 0, •••, d -1, where / denotes the length and H τ m (M) the ith local cohomology module of M with respect to m. The notion of generalized C-M modules was introduced in [6]. It has its roots in a problem of D.A. Buchsbaum. Roughly speaking, this problem says that the difference I(q; M) := l(M/qM) -β(q; M) takes a constant value for all parameter ideals q of M, where e(q; M) denotes the multiplicity of M relative to q [5]. In general, that is not true [30]. However, J. Stύckrad and W. Vogel found that modules satisfying this problem enjoy many interesting properties which are similar to the ones of C-M modules and gave them the name Buchsbaum modules [22], [23]. That led in [6] to the study of modules M with the property I(M):= sup/(q;M) < oo where q runs through all parameter ideals of M, and it turned out that they are just generalized C-M modules. The class of generalized C-M module is rather large. For instance, most of the considered geometric local rings such as the ones of isolated singularities or of the vertices of affine cones over projective curves are Received February 15, 1983. 2 NGO VIET TRUNG generalized C-M rings. So it would be of interest to establish a theory of generalized C-M modules. Although the theory of Buchsbaum modules has been rapidly developed by works of S. Goto, P. Schenzel, J. Stύckrad, W. Vogel (see the monograph [20], little is known about generalized C-M modules. Besides, it lacks something which connects both kinds of modules together. If one is acquainted enough with the few references on generalized C-M modules [6], [11], [18], one might have the notice that almost all properties of systems of parameters (abbr. s.o.p.'s) of Buchsbaum modules also hold for s.o.p.'s of generalized C-M modules which are contained in a large power of the maximal ideal. For instance, if M is a generalized C-M module, there exists a positive integer n such that for all parameter ideals qcim n of M. So, with regard to the origin of generalized C-M modules, one should try to explain the above phenomenon in studying s.o.p.'s a ly , a d of M with the property Such s.o.p.'s will be called standard.

Journal of Algebra, 1998

Let (R, m) be a local Cohen-Macaulay ring whose m-adic completion R has an isolated singularity. We verify the following conjecture of F.-O. Schreyer: R has finite Cohen-Macaulay type if and only if R has finite Cohen-Macaulay type. We show also that the hypersurface k[[x 0 ,. .. , x d ]]/(f) has finite Cohen-Macaulay type if and only if k s [[x 0 ,. .. , x d ]]/(f) has finite Cohen-Macaulay type, where k s is the separable closure of the field k.

2019

A finitely generated module M over a commutative Noetherian ring R is called an I-Cohen Macaulay module, if (I,M) + (M/IM)= (M), where I is a proper ideal of R. The aim of this paper is to study the structure of this class of modules. It is discovered that I-Cohen Macaulay modules enjoy many interesting properties which are analogous to those of Cohen Macaulay modules. Also, various characterizations of I-Cohen Macaulay modules are presented here.

arXiv (Cornell University), 2010

In this article we develop a new method to deal with maximal Cohen-Macaulay modules over non-isolated surface singularities. In particular, we give a negative answer on an old question of Schreyer about surface singularities with only countably many indecomposable maximal Cohen-Macaulay modules. Next, we prove that the degenerate cusp singularities have tame Cohen-Macaulay representation type. Our approach is illustrated on the case of x, y, z /(xyz) as well as several other rings. This study of maximal Cohen-Macaulay modules over non-isolated singularities leads to a new class of problems of linear algebra, which we call representations of decorated bunches of chains. We prove that these matrix problems have tame representation type and describe the underlying canonical forms. Contents 1. Introduction, motivation and historical remarks 2. Generalities on maximal Cohen-Macaulay modules 2.1. Maximal Cohen-Macaulay modules over surface singularities 2.2. On the category CM lf (A) 3. Main construction 4. Serre quotients and proof of Main Theorem 5. Maximal Cohen-Macaulay modules over x, y, z /(x 2 + y 3 − xyz) 6. Representations of decorated bunches of chains-I 6.1. Notation 6.2. Bimodule problems 6.3. Definition of a decorated bunch of chains 6.4. Matrix description of the category Rep(X) 6.5. Strings and Bands 7. Representations of decorated bunches of chains-II 7.1. Idea of the proof 7.2. Reduction Cases 7.3. Decorated Kronecker problem 8. Maximal Cohen-Macaulay modules over degenerate cusps-I 8.1. Maximal Cohen-Macaulay modules over x, y, z /(xyz) 8.2. Maximal Cohen-Macaulay modules over x, y, u, v /(xy, uv) 8.3. Maximal Cohen-Macaulay modules over x, y, z, u, v /(xz, xu, yu, yv, zv) 9. Singularities obtained by gluing cyclic quotient singularities 9.1. Non-isolated surface singularities obtained by gluing normal rings 9.2. Generalities about cyclic quotient singularities 9.3. Degenerate cusps and their basic properties 2. Generalities on maximal Cohen-Macaulay modules Let (A, m) be a Noetherian local ring, = A/m its residue field and d = kr. dim(A) its Krull dimension. Throughout the paper A−mod denotes the category of Noetherian (i.e. finitely generated) A-modules, whereas A−Mod stands for the category of all A-modules, Q = Q(A) is the total ring of fractions of A and P is the set of prime ideals of height 1. Definition 2.1. A Noetherian A-module M is called maximal Cohen-Macaulay if Ext i A (, M) = 0 for all 0 ≤ i < d. 2.1. Maximal Cohen-Macaulay modules over surface singularities. In this article we focus on the study of maximal Cohen-Macaulay modules over Noetherian rings of Krull dimension two, also called surface singularities. This case is actually rather special because of the following well-known lemma. Lemma 2.2. Let (A, m) be a surface singularity, N be a maximal Cohen-Macaulay Amodule and M a Noetherian A-module. Then the A-module Hom A (M, N) is maximal Cohen-Macaulay. Proof. From a free presentation A n ϕ → A m → M → 0 of M we obtain an exact sequence: 0 −→ Hom A (M, N) −→ N m ϕ * −→ N n −→ coker(ϕ *) −→ 0. Proof. From the canonical short exact sequence 0 → tor(M) → M → M/ tor(M) → 0 we get the isomorphism M/ tor(M) ∨ → M ∨. Since M/ tor(M) is a maximal Cohen-Macaulay A-module and ∨ is a dualizing functor, we get two natural isomorphisms M/ tor(M)

Nagoya Mathematical Journal, 1986

What we call the homological conjectures on commutative Noetherian local rings were first collected and partially settled by C. Peskine and L. Szpiro [PS1]. The subsequent remarkable progress was made by M. Hochster [H1] who conjectured the existence of big Cohen-Macaulay modules and solved it in the affirmative for equicharacteristic local rings. It is, however, still open in general setting.

Journal of Algebra, 2011

As a stable analogue of degenerations, we introduce the notion of stable degenerations for Cohen-Macaulay modules over a Gorenstein local algebra. We shall give several necessary and/or sufficient conditions for the stable degeneration. These conditions will be helpful to see when a Cohen-Macaulay module degenerates to another. R corresponding to an R-module M. Then we say that M degenerates to N if O(N) is 0 2000 Mathematics Subject Classification. Primary 13C14; Secondary 13D10. 0

Mathematische Zeitschrift, 2010

In this paper we completely classify all the special Cohen-Macaulay (=CM) modules corresponding to the exceptional curves in the dual graph of the minimal resolutions of all two dimensional quotient singularities. In every case we exhibit the specials explicitly in a combinatorial way. Our result relies on realizing the specials as those CM modules whose first Ext group vanishes against the ring R, thus reducing the problem to combinatorics on the AR quiver; such possible AR quivers were classified by Auslander and Reiten. We also give some general homological properties of the special CM modules and their corresponding reconstruction algebras. Lemma 2.3. We have the following commutative diagram whose rows are equivalences and columns are dualities:

Manuscripta Mathematica, 1994

Algebras and Representation Theory, 2005

Let (R, m, k) be a local Cohen-Macaulay (CM) ring of dimension one. It is known that R has finite CM type if and only if R is reduced and has bounded CM type. Here we study the one-dimensional rings of bounded but infinite CM type. We will classify these rings up to analytic isomorphism (under the additional hypothesis that the ring contains an infinite field). In the first section we deal with the complete case, and in the second we show that bounded CM type ascends to and descends from the completion. In the third section we study ascent and descent in higher dimensions and prove a Brauer-Thrall theorem for excellent rings.

Memoirs of the American Mathematical Society, 2017

In this article we develop a new method to deal with maximal Cohen-Macaulay modules over non-isolated surface singularities. In particular, we give a negative answer on an old question of Schreyer about surface singularities with only countably many indecomposable maximal Cohen-Macaulay modules. Next, we prove that the degenerate cusp singularities have tame Cohen-Macaulay representation type. Our approach is illustrated on the case of x, y, z /(xyz) as well as several other rings. This study of maximal Cohen-Macaulay modules over non-isolated singularities leads to a new class of problems of linear algebra, which we call representations of decorated bunches of chains. We prove that these matrix problems have tame representation type and describe the underlying canonical forms. Contents 1. Introduction, motivation and historical remarks 2. Generalities on maximal Cohen-Macaulay modules 2.1. Maximal Cohen-Macaulay modules over surface singularities 2.2. On the category CM lf (A) 3. Main construction 4. Serre quotients and proof of Main Theorem 5. Maximal Cohen-Macaulay modules over x, y, z /(x 2 + y 3 − xyz) 6. Representations of decorated bunches of chains-I 6.1. Notation 6.2. Bimodule problems 6.3. Definition of a decorated bunch of chains 6.4. Matrix description of the category Rep(X) 6.5. Strings and Bands 7. Representations of decorated bunches of chains-II 7.1. Idea of the proof 7.2. Reduction Cases 7.3. Decorated Kronecker problem 8. Maximal Cohen-Macaulay modules over degenerate cusps-I 8.1. Maximal Cohen-Macaulay modules over x, y, z /(xyz) 8.2. Maximal Cohen-Macaulay modules over x, y, u, v /(xy, uv) 8.3. Maximal Cohen-Macaulay modules over x, y, z, u, v /(xz, xu, yu, yv, zv) 9. Singularities obtained by gluing cyclic quotient singularities 9.1. Non-isolated surface singularities obtained by gluing normal rings 9.2. Generalities about cyclic quotient singularities 9.3. Degenerate cusps and their basic properties 2. Generalities on maximal Cohen-Macaulay modules Let (A, m) be a Noetherian local ring, = A/m its residue field and d = kr. dim(A) its Krull dimension. Throughout the paper A−mod denotes the category of Noetherian (i.e. finitely generated) A-modules, whereas A−Mod stands for the category of all A-modules, Q = Q(A) is the total ring of fractions of A and P is the set of prime ideals of height 1. Definition 2.1. A Noetherian A-module M is called maximal Cohen-Macaulay if Ext i A (, M) = 0 for all 0 ≤ i < d. 2.1. Maximal Cohen-Macaulay modules over surface singularities. In this article we focus on the study of maximal Cohen-Macaulay modules over Noetherian rings of Krull dimension two, also called surface singularities. This case is actually rather special because of the following well-known lemma. Lemma 2.2. Let (A, m) be a surface singularity, N be a maximal Cohen-Macaulay Amodule and M a Noetherian A-module. Then the A-module Hom A (M, N) is maximal Cohen-Macaulay. Proof. From a free presentation A n ϕ → A m → M → 0 of M we obtain an exact sequence: 0 −→ Hom A (M, N) −→ N m ϕ * −→ N n −→ coker(ϕ *) −→ 0. Proof. From the canonical short exact sequence 0 → tor(M) → M → M/ tor(M) → 0 we get the isomorphism M/ tor(M) ∨ → M ∨. Since M/ tor(M) is a maximal Cohen-Macaulay A-module and ∨ is a dualizing functor, we get two natural isomorphisms M/ tor(M)

2015

A generalization of the notion of depth of an ideal on a module is introduced by applying the concept of local cohomology modules with respect to a pair of ideals . The concept of (I;J)- Cohen{Macaulay modules is also introduced as a generalization of the concept of Cohen{Macaulay modules . This kind of modules is dierent from the Cohen{Macaulay modules, as shown in an example. Also an Artinian result is given for such modules.

arXiv: Commutative Algebra, 2020

Let a be a proper ideal of a commutative Noetherian ring R with identity. Using the notion of a-relative system of parameters, we introduce a relative variant of the notion of generalized Cohen-Macaulay modules. In particular, we examine relative analogues of quasi-Buchsbaum, Buchsbaum and surjective Buchsbaum modules. We reveal several interactions between these types of modules that extend some of the existing results in the classical theory to the relative one.