Cohen-Macaulay Modules Research Articles

Krull–Gabriel dimension of Cohen–Macaulay modules over hypersurfaces of countable Cohen-Macaulay representation type

We calculate the Krull–Gabriel dimension of the functor category of the (stable) category of maximal Cohen–Macaulay modules over complete Cohen-Macaulay local rings of countable Cohen-Macaulay representation type. We show that the Krull–Gabriel dimension is 0 if and only if the ring is of finite Cohen–Macaulay representation type and that is 2 if the ring is a hypersurface of countable but not finite Cohen–Macaulay representation type.

On Gorenstein algebras of finite Cohen-Macaulay type: Dimer tree algebras and their skew group algebras

Dimer tree algebras are a class of non-commutative Gorenstein algebras of Gorenstein dimension 1. In previous work we showed that the stable category of Cohen-Macaulay modules of a dimer tree algebra A is a 2-cluster category of Dynkin type A. Here we show that, if A has an admissible action by the group G with two elements, then the stable Cohen-Macaulay category of the skew group algebra AG is a 2-cluster category of Dynkin type D. This result is reminiscent of and inspired by a result by Reiten and Riedtmann, who showed that for an admissible G-action on the path algebra of type A the resulting skew group algebra is of type D. Moreover, we provide a geometric model of the syzygy category of AG in terms of a punctured polygon P with a checkerboard pattern in its interior, such that the 2-arcs in P correspond to indecomposable syzygies in AG and 2-pivots correspond to morphisms. In particular, the dimer tree algebras and their skew group algebras are Gorenstein algebras of finite Cohen-Macaulay type A and D respectively. We also provide examples of types E6,E7, and E8.

An almost p-standard system of parameters and approximately Cohen–Macaulay modules

An almost p-standard system of parameters and approximately Cohen–Macaulay modules

Quasi-pure resolutions and some lower bounds of Hilbert coefficients of Cohen-Macaulay modules

Let (A,m) be a Noetherian local ring and let M be a finitely generated Cohen Macaulay A module. Let G(A)=⨁n≥0mn/mn+1 be the associated graded ring of A and G(M)=⨁n≥0mnM/mn+1M be the associated graded module of M. If A is regular and if G(M) has a quasi-pure resolution then we show that G(M) is Cohen-Macaulay. If A is Gorenstein and G(A) is Cohen-Macaulay and if M has finite projective dimension then we give lower bounds on e0(M) and e1(M). Finally let A=Q/(f1,…,fc) be a strict complete intersection with ord(fi)=s for all i, where Q is a regular local ring. Let M be an Cohen-Macaulay module with complexity cxA(M)=r<c. We give lower bounds on e0(M) and e1(M).

On Cohen–Macaulay modules over the Weyl algebra

We investigate a definition of Cohen–Macaulay modules over the Weyl algebra [Formula: see text] and give a sufficient condition for a GKZ [Formula: see text]-hypergeometric [Formula: see text]-module to be Cohen–Macaulay.

Clifford Deformations of Koszul Frobenius Algebras and Noncommutative Quadrics

A Clifford deformation of a Koszul Frobenius algebra [Formula: see text] is a finite dimensional [Formula: see text]-graded algebra [Formula: see text], which corresponds to a noncommutative quadric hypersurface [Formula: see text] for some central regular element [Formula: see text]. It turns out that the bounded derived category [Formula: see text] is equivalent to the stable category of the maximal Cohen-Macaulay modules over [Formula: see text] provided that [Formula: see text] is noetherian. As a consequence, [Formula: see text] is a noncommutative isolated singularity if and only if the corresponding Clifford deformation [Formula: see text] is a semisimple [Formula: see text]-graded algebra. The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius algebra are related to Knörrer's periodicity theorem for quadric hypersurfaces. As an application, we recover Knörrer's periodicity theorem without using matrix factorizations.

Matrix factorizations of the discriminant of Sn

Consider the symmetric group Sn acting as a reflection group on the polynomial ring k[x1,…,xn] where k is a field, such that Char(k) does not divide n!. We use Higher Specht polynomials to construct matrix factorizations of the discriminant of this group action: these matrix factorizations are indexed by partitions of n and respect the decomposition of the coinvariant algebra into isotypical components. The maximal Cohen–Macaulay modules associated to these matrix factorizations give rise to a noncommutative resolution of the discriminant and they correspond to the nontrivial irreducible representations of Sn. All our constructions are implemented in Macaulay2 and we provide several examples. We also discuss extensions of these results to Young subgroups of Sn and indicate how to generalize them to the reflection groups G(m,1,n).

Some characterizations of relative sequentially Cohen–Macaulay and relative Cohen–Macaulay modules

Let [Formula: see text] be an [Formula: see text]-module over a Noetherian ring [Formula: see text] and [Formula: see text] an ideal of [Formula: see text] with c = [Formula: see text] ([Formula: see text], R). First, over an [Formula: see text]-relative Cohen–Macaulay local ring [Formula: see text], we provide a characterization of the [Formula: see text]-relative sequentially Cohen–Macaulay modules [Formula: see text] in terms of [Formula: see text]-relative Cohen–Macaulayness of the [Formula: see text]-modules [Formula: see text] for all [Formula: see text], where [Formula: see text]. Next, we prove that [Formula: see text] is finite [Formula: see text]-relative Cohen–Macaulay if and only if [Formula: see text] for all [Formula: see text] and [Formula: see text]. Finally, we provide another characterization of [Formula: see text]-relative sequentially Cohen–Macaulay modules [Formula: see text] in terms of vanishing of the local homology modules [Formula: see text] for all [Formula: see text] and for all [Formula: see text].

A Note on the Global Dimension of Shifted Orders

We consider the dominant dimension of an order over a Cohen-Macaulay ring in the category of centrally Cohen-Macaulay modules. There is a canonical tilting module in the case of positive dominant dimension and we give an upper bound on the global dimension of its endomorphism ring.

On associated graded modules of maximal Cohen–Macaulay modules over hypersurface rings

Let [Formula: see text] where [Formula: see text] be a complete regular local ring of dimension [Formula: see text], [Formula: see text] for some [Formula: see text] and [Formula: see text] an MCM [Formula: see text]-module with [Formula: see text] then we prove that [Formula: see text]. If [Formula: see text] is a complete hypersurface ring of dimension [Formula: see text] with infinite residue field and [Formula: see text], let [Formula: see text] be an MCM [Formula: see text]-module with [Formula: see text] or [Formula: see text] then we prove that [Formula: see text]. Our paper is the first systematic study of depth of associated graded modules of MCM modules over hypersurface rings.

Remarks on the small Cohen-Macaulay conjecture and new instances of maximal Cohen-Macaulay modules

We show that any quasi-Gorenstein deformation of a 3-dimensional quasi-Gorenstein Buchsbaum local ring with I-invariant 1 admits a maximal Cohen-Macaulay module, provided it is a quotient of a Gorenstein ring. Such a class of rings includes two instances of unique factorization domains constructed by Marcel-Schenzel and by Imtiaz-Schenzel, respectively. Apart from this result, motivated by the small Cohen-Macaulay conjecture in prime characteristic, we examine a question about when the Frobenius pushforward F⁎e(M) of an R-module M comprises a maximal Cohen-Macaulay direct summand in both local and graded cases.

On an instance of the small Cohen-Macaulay conjecture

We provide a simplified proof of a theorem proved by Tavanfar and Shimomoto which states that a quasi-Gorenstein deformation of a 3-dimensional quasi-Gorenstein local ring (A,m,k) with Hm2(A)=k admits a small Cohen-Macaulay module.

Upper bounds on two Hilbert coefficients

New upper bounds on the first and the second Hilbert coefficients of a Cohen-Macaulay module over a local ring are given. Characterizations are provided for some upper bounds to be attained. The characterizations are given in terms of Hilbert series as well as in terms of the Castelnuovo-Mumford regularity of the associated graded module.

Frobenius splitting, strong F-regularity, and small Cohen-Macaulay modules

Let M M be a finitely generated module over an (F-finite local) ring R R of prime characteristic p &gt; 0 p &gt;0 . Let e M {}^e\!M denote the result of restricting scalars using the map F e : R → R F^e\colon R \to R , the e e\, th iteration of the Frobenius endomorphism. Motivated in part by the fact that in certain circumstances the splitting of e M {}^e\!M as e e grows can be used to prove the existence of small (i.e., finitely generated) maximal Cohen-Macaulay modules, we study splitting phenomena for e M {}^e\!M from several points of view. In consequence, we are able to prove new results about when one has such splittings that generalize results previously known only in low dimension, we give new characterizations of when a ring is strongly F-regular, and we are able to prove new results on the existence of small maximal Cohen-Macaulay modules in the multi-graded case. In addition, we study certain corresponding questions when the ring is no longer assumed F-finite and purity is considered in place of splitting. We also answer a question, raised by Datta and Smith, by showing that a regular Noetherian domain, even in dimension 2, need not be very strongly F-regular.

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.

Reductions towards a characteristic free proof of the Canonical Element Theorem

We reduce Hochster's Canonical Element Conjecture (theorem since 2016) to a localization problem in a characteristic free way. We prove the validity of a new variant of the Canonical Element Theorem (CET) and explain how a characteristic free deduction of the new variant from the original CET would provide us with a characteristic free proof of the CET. We also show that the Balanced Big Cohen-Macaulay Module Theorem can be settled by a characteristic free proof if the big Cohen-Macaulayness of Hochster's modification module can be deduced from the existence of a maximal Cohen-Macaulay complex in a characteristic free way.

The $$\mathfrak a$$-filter Grade of an Ideal $$\mathfrak b$$ and $$(\mathfrak a,\mathfrak b)$$-f-modules

Let $$\mathfrak a,\mathfrak b$$ be two ideals of a commutative noetherian ring R and M a finitely generated R-module. We continue to study f-gradR( $$\mathfrak a,\mathfrak b$$ ,M) which was introduced in [Bull. Malays. Math. Sci. Soc., 2015, 38(2): 467–482]. Some computations and bounds of f-gradR( $$\mathfrak a,\mathfrak b$$ ,M) are provided. We also give the structure of ( $$\mathfrak a,\mathfrak b$$ )-f-modules. Some properties which are analogous to those of Cohen-Macaulay modules are discovered.

Categories for Grassmannian Cluster Algebras of Infinite Rank

Abstract We construct Grassmannian categories of infinite rank, providing an infinite analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. Each Grassmannian category of infinite rank is given as the category of graded maximal Cohen–Macaulay modules over a certain hypersurface singularity. We show that generically free modules of rank $1$ in a Grassmannian category of infinite rank are in bijection with the Plücker coordinates in an appropriate Grassmannian cluster algebra of infinite rank. Moreover, this bijection is structure preserving, as it relates rigidity in the category to compatibility of Plücker coordinates. Along the way, we develop a combinatorial formula to compute the dimension of the $\textrm {Ext}^{1}$-spaces between any two generically free modules of rank $1$ in the Grassmannian category of infinite rank.

Skew graded (A ∞ ) hypersurface singularities

For a skew version of a graded (A ∞ ) hypersurface singularity A, we study the stable category of graded maximal Cohen-Macaulay modules over A. As a consequence, we see that A has countably infinite Cohen–Macaulay representation type and is not a noncommutative graded isolated singularity.

Generalized local duality, canonical modules, and prescribed bound on projective dimension

We present various approaches to J. Herzog's theory of generalized local cohomology and explore its main aspects, e.g., (non-)vanishing results as well as a general local duality theorem which extends, to a much broader class of rings, previous results by Herzog-Zamani and Suzuki. As an application, we establish a prescribed upper bound for the projective dimension of a module satisfying suitable cohomological conditions, and we derive some freeness criteria and questions of Auslander-Reiten type. Along the way, we prove a new characterization of Cohen-Macaulay modules which truly relies on generalized local cohomology, and in addition we introduce and study a generalization of the notion of canonical module.