Maximal Cohen-Macaulay Modules Research Articles

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).

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.

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 > 0 p >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.

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.

Non-Commutative Resolutions for the Discriminant of the Complex Reflection Group G(m, p, 2)

We show that for the family of complex reflection groups G = G(m, p,2) appearing in the Shephard–Todd classification, the endomorphism ring of the reduced hyperplane arrangement A(G) is a non-commutative resolution for the coordinate ring of the discriminant Δ of G. This furthers the work of Buchweitz, Faber and Ingalls who showed that this result holds for any true reflection group. In particular, we construct a matrix factorization for Δ from A(G) and decompose it using data from the irreducible representations of G. For G(m, p,2) we give a full decomposition of this matrix factorization, including for each irreducible representation a corresponding maximal Cohen–Macaulay module. The decomposition concludes that the endomorphism ring of the reduced hyperplane arrangement A(G) will be a non-commutative resolution. For the groups G(m,1,2), the coordinate rings of their respective discriminants are all isomorphic to each other. We also calculate and compare the Lusztig algebra for these groups.

On the blow-up of a normal singularity at maximal Cohen–Macaulay modules

On the blow-up of a normal singularity at maximal Cohen–Macaulay modules

Twisted Segre products

We introduce the notion of the twisted Segre product A∘ψB of Z-graded algebras A and B with respect to a twisting map ψ. It is proved that if A and B are noetherian Koszul Artin-Schelter regular algebras and ψ is a twisting map such that the twisted Segre product A∘ψB is noetherian, then A∘ψB is a noncommutative graded isolated singularity. To prove this result, the notion of densely (bi-)graded algebras is introduced. Moreover, we show that the twisted Segre product A∘ψB of A=k[u,v] and B=k[x,y] with respect to a diagonal twisting map ψ is a noncommutative quadric surface (so in particular it is noetherian), and we compute the stable category of graded maximal Cohen-Macaulay modules over it.

On 2-Representation Infinite Algebras Arising From Dimer Models

AbstractThe Jacobian algebra arising from a consistent dimer model is a bimodule 3-Calabi–Yau algebra, and its center is a 3-dimensional Gorenstein toric singularity. A perfect matching (PM) of a dimer model gives the degree, making the Jacobian algebra $\mathbb{Z}$-graded. It is known that if the degree zero part of such an algebra is finite dimensional, then it is a 2-representation infinite algebra that is a generalization of a representation infinite hereditary algebra. Internal PMs, which correspond to toric exceptional divisors on a crepant resolution of a 3-dimensional Gorenstein toric singularity, characterize the property that the degree zero part of the Jacobian algebra is finite dimensional. Combining this characterization with the theorems due to Amiot–Iyama–Reiten, we show that the stable category of graded maximal Cohen–Macaulay modules admits a tilting object for any 3-dimensional Gorenstein toric isolated singularity. We then show that all internal PMs corresponding to the same toric exceptional divisor are transformed into each other using the mutations of PMs, and this induces derived equivalences of 2-representation infinite algebras.

Noncommutative Knörrer’s periodicity theorem and noncommutative quadric hypersurfaces

Noncommutative hypersurfaces, in particular, noncommutative quadric hypersurfaces are major objects of study in noncommutative algebraic geometry. In the commutative case, Kn\"orrer's periodicity theorem is a powerful tool to study Cohen-Macaulay representation theory since it reduces the number of variables in computing the stable category $\underline{\operatorname{CM}}(A)$ of maximal Cohen-Macaulay modules over a hypersurface $A$. In this paper, we prove a noncommutative graded version of Kn\"orrer's periodicity theorem. Moreover, we prove another way to reduce the number of variables in computing the stable category ${\underline{\operatorname{CM}}}^{\mathbb Z}(A)$ of graded maximal Cohen-Macaulay modules if $A$ is a noncommutative quadric hypersurface. Under high rank property defined in this paper, we also show that computing ${\underline{\operatorname{CM}}}^{\mathbb Z}(A)$ over a noncommutative smooth quadric hypersurface $A$ in up to six variables can be reduced to one or two variables cases. In addition, we give a complete classification of ${\underline{\operatorname{CM}}}^{\mathbb Z}(A)$ over a smooth quadric hypersurface $A$ in a skew $\mathbb P^{n-1}$, where $n \leq 6$, without high rank property using graphical methods.

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.

Auslander's formula and correspondence for exact categories

The Auslander correspondence is a fundamental result in Auslander-Reiten theory. In this paper we introduce the category modadm(E) of admissibly finitely presented functors and use it to give a version of Auslander correspondence for any exact category E. An important ingredient in the proof is the localization theory of exact categories. We also investigate how properties of E are reflected in modadm(E), for example being (weakly) idempotent complete or having enough projectives or injectives. Furthermore, we describe modadm(E) as a subcategory of mod(E) when E is a resolving subcategory of an abelian category. This includes the category of Gorenstein projective modules and the category of maximal Cohen-Macaulay modules as special cases. Finally, we use modadm(E) to give a bijection between exact structures on an idempotent complete additive category C and certain resolving subcategories of mod(C).

On Some Modules Supported in the Chow Variety

The study of Chow varieties of decomposable forms lies at the confluence of algebraic geometry, commutative algebra, representation theory and combinatorics. There are many open questions about homological properties of Chow varieties and interesting classes of modules supported on them. The goal of this note is to survey some fundamental constructions and properties of these objects, and to propose some new directions of research. Our main focus will be on the study of certain maximal Cohen-Macaulay modules of covariants supported on Chow varieties, and on defining equations and syzygies. We also explain how to assemble Tor groups over Veronese subalgebras into modules over a Chow variety, leading to a result on the polynomial growth of these groups.

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.

On the existence of birational maximal Cohen-Macaulay modules over biradical extensions in mixed characteristic

Let $S$ be an unramified regular local ring of mixed characteristic $p\geq 3$ and $S^p$ the subring of $S$ obtained by lifting to $S$ the image of the Frobenius map on $S/pS$. Let $R$ be the integral closure of $S$ in a biradical extension of degree $p^2$ of its quotient field obtained by adjoining $p$-th roots of sufficiently general square free elements $f,g\in S^p$. We show that $R$ admits a birational maximal Cohen-Macaulay module. It is noted that $R$ is not automatically Cohen-Macaulay.

Geometry of varieties for graded maximal Cohen–Macaulay modules

We study a variety for graded maximal Cohen–Macaulay modules, which was introduced by Dao and Shipman. The main result of this paper asserts that there are only a finite number of isomorphism classes of graded maximal Cohen–Macaulay modules with fixed Hilbert series over Cohen–Macaulay algebras of graded countable representation type.

Grothendieck groups, convex cones and maximal Cohen–Macaulay points

Let R be a commutative noetherian ring. Let $$\mathsf {H}(R)$$ be the quotient of the Grothendieck group of finitely generated R-modules by the subgroup generated by pseudo-zero modules. Suppose that the $$\mathbb {R}$$ -vector space $$\mathsf {H}(R)_\mathbb {R}=\mathsf {H}(R)\otimes _\mathbb {Z}\mathbb {R}$$ has finite dimension. Let $$\mathsf {C}(R)$$ (resp. $$\mathsf {C}_r(R)$$ ) be the convex cone in $$\mathsf {H}(R)_\mathbb {R}$$ spanned by maximal Cohen–Macaulay R-modules (resp. maximal Cohen–Macaulay R-modules of rank r). We explore the interior, closure and boundary, and convex polyhedral subcones of $$\mathsf {C}(R)$$ . We provide various equivalent conditions for R to have only finitely many rank r maximal Cohen–Macaulay points in $$\mathsf {C}_r(R)$$ in terms of topological properties of $$\mathsf {C}_r(R)$$ . Finally, we consider maximal Cohen–Macaulay modules of rank one as elements of the divisor class group $${\text {Cl}}(R)$$ .