Indecomposable Maximal Cohen Macaulay Modules Research Articles

Preresolutions of noncommutative isolated singularities

We introduce the notion of right pre-resolutions (quasi-resolutions) for noncommutative isolated singularities, which is a weaker version of quasi-resolutions introduced by Qin-Wang-Zhang. We prove that right quasi-resolutions for noetherian bounded below and locally finite graded algebra with right injective dimension 2 are always Morita equivalent. When we restrict to noncommutative quadric hypersurfaces, we prove that a noncommutative quadric hypersurface, which is a noncommutative isolated singularity, always admits a right pre-resolution. Besides, we provide a method to verify whether a noncommutative quadric hypersurface is an isolated singularity. An example of noncommutative quadric hypersurfaces with detailed computations of indecomposable maximal Cohen-Macaulay modules and right pre-resolutions is included as well.

Cohen–Macaulay test ideals over rings of finite and countable Cohen–Macaulay type

R.G. and Perez proved that under certain conditions the test ideal of a module closure agrees with the trace ideal of the module closure. We use this fact to compute the test ideals of various rings with respect to the closures coming from their indecomposable maximal Cohen–Macaulay modules. We also give an easier way to compute the test ideal of a hypersurface ring in three variables coming from a module with a particular type of matrix factorization.

The first Brauer–Thrall conjecture over one-dimensional local rings

Let [Formula: see text] be a one-dimensional commutative Noetherian complete local ring. Assume that the cohomology annihilator [Formula: see text] is [Formula: see text]-primary. We use the notion of Gabriel–Roiter (co)measure in the category of maximal Cohen–Macaulay [Formula: see text]-modules and prove that, if there is an infinite set [Formula: see text] of indecomposable maximal Cohen–Macaulay [Formula: see text]-modules of bounded multiplicity, then there are indecomposable maximal Cohen–Macaulay modules of arbitrary large multiplicity which are cogenerated by modules in [Formula: see text]. This, in particular, guarantees the validity of the first Brauer–Thrall-type theorem for the category of maximal Cohen–Macaulay [Formula: see text]-modules.

Maximal Cohen–Macaulay modules that are not locally free on the punctured spectrum

We say that a Cohen–Macaulay local ring has finite CM+-representation type if there exist only finitely many isomorphism classes of indecomposable maximal Cohen–Macaulay modules that are not locally free on the punctured spectrum. In this paper, we consider finite CM+-representation type from various points of view, relating it with several conjectures on finite/countable Cohen–Macaulay representation type. We prove in dimension one that the Gorenstein local rings of finite CM+-representation type are exactly the local hypersurfaces of countable CM-representation type, that is, the hypersurfaces of type (A∞) and (D∞). We also discuss the closedness and dimension of the singular locus of a Cohen–Macaulay local ring of finite CM+-representation type.

Maximal Cohen–Macaulay Modules Over Non–Isolated Surface Singularities and Matrix Problems

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 $k\llbracket x,y,z\rrbracket/(xyz)$ 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.

On two dimensional mixed characteristic rings of finite Cohen Macaulay type

In this paper we give a bountiful number of examples of two dimensional mixed characteristic rings of finite Cohen Macaulay type. For a large subclass of these examples we give a complete description of its indecomposable maximal Cohen–Macaulay modules and we also compute its AR-quiver.

Ranks of Indecomposable Modules over Rings of Infinite Cohen-Macaulay Type

Let (R, 𝔪, k) be a one-dimensional analytically unramified local ring with minimal prime ideals P 1, …, P s . Our ultimate goal is to study the direct-sum behavior of maximal Cohen-Macaulay modules over R. Such behavior is encoded by the monoid ℭ(R) of isomorphism classes of maximal Cohen-Macaulay R-modules: the structure of this monoid reveals, for example, whether or not every maximal Cohen-Macaulay module is uniquely a direct sum of indecomposable modules; when uniqueness does not hold, invariants of this monoid give a measure of how badly this property fails. The key to understanding the monoid ℭ(R) is determining the ranks of indecomposable maximal Cohen-Macaulay modules. Our main technical result shows that if R/P 1 has infinite Cohen-Macaulay type and the residue field k is infinite, then there exist |k| pairwise non-isomorphic indecomposable maximal Cohen-Macaulay R-modules of rank (r 1, ⋅, r s ) provided r 1 ≥ r i for all i ∈ {1,…, s}. This result allows us to describe the monoid ℭ(R) when has infinite Cohen-Macaulay type for every minimal prime ideal Q of the 𝔪-adic completion .

RESOLUTIONS AND COHOMOLOGIES OF TORIC SHEAVES: THE AFFINE CASE

We study equivariant resolutions and local cohomologies of toric sheaves for affine toric varieties, where our focus is on the construction of new examples of indecomposable maximal Cohen–Macaulay modules of higher rank. A result of Klyachko states that the category of reflexive toric sheaves is equivalent to the category of vector spaces together with a certain family of filtrations. Within this setting, we develop machinery which facilitates the construction of minimal free resolutions for the smooth case as well as resolutions which are acyclic with respect to local cohomology functors for the general case. We give two main applications. First, over the polynomial ring, we determine in explicit combinatorial terms the ℤn-graded Betti numbers and local cohomology of reflexive modules whose associated filtrations form a hyperplane arrangement. Second, for the nonsmooth, simplicial case in dimension d ≥ 3, we construct new examples of indecomposable maximal Cohen–Macaulay modules of rank d – 1.

On two examples by Iyama and Yoshino

AbstractIn a recent paper, Iyama and Yoshino considered two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal Cohen–Macaulay modules in terms of linear algebra data. In this paper, we present two new approaches to these examples. In the first approach we give a relation with cluster categories. In the second approach we use Orlov’s result on the graded singularity category.

On the Structure of Maximal Cohen–Macaulay Modules over the Ring $\boldsymbol{K[[x, y]]/(x^n)}$

We study maximal Cohen–Macaulay modules over the hypersurface ring \(K[[x, y]]/(x^n),\ n\geq 3,\) K being a field. Infinite families of non-isomorphic indecomposable maximal Cohen–Macaulay modules of arbitrary number of minimal generators or of arbitrary rank are constructed.

Chem classes and Lie-Rinehart algebras

Let A be a F-algebra where F is a field, and let W be an A-module of finite presentation. We use the linear Lie-Rinehart algebra V W of W to define the first Chern-class c 1( W) in H 2 ( V W | U , O U ) , where U in Spec( A) is the open subset where W is locally free. We compute explicitly algebraic V W -connections on maximal Cohen-Macaulay modules W on the hypersurface-singularities B mn2 = x m + y n + z 2, and show that these connections are integrable, hence the first Chern-class c 1( W) vanishes. We also look at indecomposable maximal Cohen-Macaulay modules on quotient-singularities in dimension 2, and prove that their first Chern-class vanish.

Hypersurfaces of bounded Cohen–Macaulay type

Let R = k [ [ x 0 , … , x d ] ] / ( f ) , where k is a field and f is a non-zero non-unit of the formal power series ring k [ [ x 0 , … , x d ] ] . We investigate the question of which rings of this form have bounded Cohen–Macaulay type, that is, have a bound on the multiplicities of the indecomposable maximal Cohen–Macaulay modules. As with finite Cohen–Macaulay type, if the characteristic is different from two, the question reduces to the one-dimensional case: The ring R has bounded Cohen–Macaulay type if and only if R ≅ k [ [ x 0 , … , x d ] ] / ( g + x 2 2 + ⋯ + x d 2 ) , where g ∈ k [ [ x 0 , x 1 ] ] and k [ [ x 0 , x 1 ] ] / ( g ) has bounded Cohen–Macaulay type. We determine which rings of the form k [ [ x 0 , x 1 ] ] / ( g ) have bounded Cohen–Macaulay type.

Steps in the Classification of Cohen–Macaulay Modules Over Singularities of Type xt + y3

Let k be an algebraically closed field of characteristic ≠ 3 and i, j, t some positive integers such that 1 ≤ i < j < t, i + j ≠ t. Then there exist a finite number of nonisomorphic indecomposable maximal Cohen–Macaulay modules N over k[[x, y]] /(xt + y3) such that N / y N is a direct sum of copies of k[[x]] /(xi), k[[x]] /(xj) and we describe them completely.

Indecomposable Cohen-Macaulay modules and their multiplicities

The main aim of this paper is to find a large class of rings for which there are indecomposable maximal Cohen-Macaulay modules of arbitrary high multiplicity (or rank in the case of domains).

The classification of homogeneous Cohen-Macaulay rings of finite representation type

In this paper we classify the homogeneous Cohen-Macaulay rings which are of finite representation type, that is, the Cohen-Macaulay rings which are positively graded and generated in degree 1, with an algebraically closed field of characteristic 0 in degree 0, and which have, up to isomorphisms and shifts in the grading, only a finite number of indecomposable maximal Cohen-Macaulay modules (MCM-modules). Our main contribution is to show that a homogeneous Cohen-Macaulay ring of finite representation type and dimension >__ 2 must have minimal multiplicity. Putting this together with previous results of Auslander, Auslander-Reiten, Buchweitz-Greuel-Schreyer, Greuel-Kn6rrer, and Solberg, we obtain:

Indecomposable syzygy-modules of high rank over hypersurface rings

In this paper we construct certain indecomposable syzygy modules over graded hypersurface rings and give a lower bound of their rank. Thereby we show that there is no upper bound for the ranks of the indecomposable maximal Cohen-Macaulay modules over a graded hypersurface ring of dimension at most two and of multiplicity at most three.