Macaulay Modules Research Articles

Hankel determinantal rings have rational singularities

Hankel determinantal rings, i.e., determinantal rings defined by minors of Hankel matrices of indeterminates, arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves; they may also be viewed as linear specializations of generic determinantal rings. We prove that, over fields of characteristic zero, Hankel determinantal rings have rational singularities; in the case of positive prime characteristic, we prove that they are F-pure. Independent of the characteristic, we give a complete description of the divisor class groups of these rings, and show that each divisor class group element is the class of a maximal Cohen–Macaulay module.

Conductors in mixed characteristic

In this paper we study conductors that occur in an integral extension of an unramified regular local ring S of mixed characteristic p>0 obtained by adjoining a pnth root ω of an element of S. We calculate the primary decomposition of the conductor of the integral closure of S[ω] and give some applications of this calculation. In particular, we show that under certain special circumstances, the integral closure of S[ω] admits a finitely generated, maximal Cohen–Macaulay module.

Maximal Cohen–Macaulay modules over certain Segre products

We prove some results on the non-existence of rank one maximal Cohen–Macaulay modules over certain Segre product rings. As an application we show that over these Segre product rings there do not exist maximal Cohen–Macaulay modules with multiplicity less than or equal to the parameter degree of the ring. This disproves a conjecture of Schoutens [14].

Axiomatic closure operations, phantom extensions, and solidity

In this article, we generalize a previously defined set of axioms for a closure operation that induces balanced big Cohen–Macaulay modules. While the original axioms were only defined in terms of finitely generated modules, these new ones will apply to all modules over a local domain. The new axioms will lead to a notion of phantom extensions for general modules, and we will prove that all modules that are phantom extensions can be modified into balanced big Cohen–Macaulay modules and are also solid modules. As a corollary, if R has characteristic p>0 and is F-finite, then all solid algebras are phantom extensions. If R also has a big test element (e.g., if R is complete), then solid algebras can be modified into balanced big Cohen–Macaulay modules. (Hochster and Huneke have previously demonstrated that there exist solid algebras that cannot be modified into balanced big Cohen–Macaulay algebras.) We also point out that tight closure over local domains in characteristic p generally satisfies the new axioms and that the existence of a big Cohen–Macaulay module induces a closure operation satisfying the new axioms.

On the sequential polynomial type of modules

Let $M$ be a finitely generated module over a Noetherian local ring $R$. The sequential polynomial type $\mathrm{sp}(M)$ of $M$ was recently introduced by Nhan, Dung and Chau, which measures how far the module $M$ is from the class of sequentially Cohen–Macaulay modules. The present paper purposes to give a parametric characterization for $M$ to have $\mathrm{sp}(M)\le s$, where $s\ge -1$ is an integer. We also study the sequential polynomial type of certain specific rings and modules. As an application, we give an inequality between $\mathrm{sp}(S)$ and $\mathrm{sp}(S^G) $, where $S$ is a Noetherian local ring and $G$ is a finite subgroup of $\mathrm{Aut}S$ such that the order of $G$ is invertible in $S$.

Tilting and Cluster Tilting for Preprojective Algebras and Coxeter Groups

AbstractWe study the stable category of the graded Cohen–Macaulay modules of the factor algebra of the preprojective algebra associated with an element $w$ of the Coxeter group of a quiver. We show that there exists a silting object $M(\boldsymbol{w})$ of this category associated with each reduced expression $\boldsymbol{w}$ of $w$ and give a sufficient condition on $\boldsymbol{w}$ such that $M(\boldsymbol{w})$ is a tilting object. In particular, the stable category is triangle equivalent to the derived category of the endomorphism algebra of $M(\boldsymbol{w})$. Moreover, we compare it with a triangle equivalence given by Amiot–Reiten–Todorov for a cluster category.

On the stable hom relation and stable degenerations of Cohen–Macaulay modules

We study the stable hom relation for Cohen–Macaulay modules over Gorenstein local algebras. We give the sufficient condition to make the stable hom relation a partial order when the base algebra is of finite representation type. As an application, we give the description of stable degenerations of Cohen–Macaulay modules over simple singularities of several types by using the stable hom relation.

Closure operations that induce big Cohen–Macaulay algebras

We study closure operations over a local domain R that satisfy a set of axioms introduced by Geoffrey Dietz. The existence of a closure operation satisfying the axioms (called a Dietz closure) is equivalent to the existence of a big Cohen–Macaulay module for R. When R is complete and has characteristic p>0, tight closure and plus closure satisfy the axioms.We give an additional axiom (the Algebra Axiom), such that the existence of a Dietz closure satisfying this axiom is equivalent to the existence of a big Cohen–Macaulay algebra. We prove that many closure operations satisfy the Algebra Axiom, whether or not they are Dietz closures. We discuss the smallest big Cohen–Macaulay algebra closure on a given ring, and show that every Dietz closure satisfying the Algebra Axiom is contained in a big Cohen–Macaulay algebra closure. This leads to proofs that in rings of characteristic p>0, every Dietz closure satisfying the Algebra Axiom is contained in tight closure, and there exist Dietz closures that do not satisfy the Algebra Axiom.

The Ziegler Spectrum and Ringel’s Quilt of the A-infinity Plane Curve Singularity

We describe the Cohen–Macaulay part of the Ziegler spectrum and calculate Ringel’s quilt of the category of finitely generated Cohen–Macaulay modules over the A-infinity plane curve singularity.

Relative derived dimensions for cotilting modules

For a Noetherian ring R and a cotilting R-module T of injective dimension at least 1, we prove that the derived dimension of R with respect to the category XT is precisely the injective dimension of T by applying Auslander–Buchweitz theory and Ghost Lemma. In particular, when R is a commutative Noetherian Cohen–Macaulay local ring with a canonical module ωR and dim⁡R≥1, the derived dimension of R with respect to the category of maximal Cohen–Macaulay modules is precisely dim⁡R.

Syzygies of Cohen–Macaulay modules and Grothendieck groups

We study the converse of a theorem of Butler and Auslander–Reiten. We show that a Cohen–Macaulay local ring with an isolated singularity has only finitely many isomorphism classes of indecomposable summands of syzygies of Cohen–Macaulay modules if the Auslander–Reiten sequences generate the relation of the Grothendieck group of finitely generated modules. This extends a recent result of Hiramatsu, which gives an affirmative answer in the Gorenstein case to a conjecture of Auslander.

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.

Ulrich modules over cyclic quotient surface singularities

In this paper, we characterize Ulrich modules over cyclic quotient surface singularities using the notion of special Cohen–Macaulay modules. We also investigate the number of indecomposable Ulrich modules for a given cyclic quotient surface singularity, and show that the number of exceptional curves in the minimal resolution determines a boundary on the number of indecomposable Ulrich modules.

On Cohen–Macaulay modules over the plane curve singularity of type $$\varvec{T_{44}}$$ T 44

For a wide class of Cohen–Macaulay modules over the local ring of the plane curve singularity of type $$T_{44}$$ , we explicitly describe the corresponding matrix factorizations. The calculations are based on the technique of matrix problems, in particular, representations of bunches of chains.

Surfaces of minimal degree of tame representation type and mutations of Cohen–Macaulay modules

We provide two examples of smooth projective surfaces of tame CM type, by showing that the parameter space of isomorphism classes of indecomposable ACM bundles with fixed rank and determinant on a rational quartic scroll in P5 is either a single point or a projective line. These turn out to be the only smooth projective ACM varieties of tame CM type besides elliptic curves, [1].For surfaces of minimal degree and wild CM type, we classify rigid Ulrich bundles as Fibonacci extensions. For F0 and F1, embedded as quintic or sextic scrolls, a complete classification of rigid ACM bundles is given.

On the Auslander–Reiten conjecture for Cohen–Macaulay local rings

This paper studies vanishing of Ext modules over Cohen–Macaulay local rings. The main result of this paper implies that the Auslander–Reiten conjecture holds for maximal Cohen–Macaulay modules of rank one over Cohen–Macaulay normal local rings. It also recovers a theorem of Avramov–Buchweitz–Şega and Hanes–Huneke, which shows that the Tachikawa conjecture holds for Cohen–Macaulay generically Gorenstein local rings.

Representations of quivers over the algebra of dual numbers

The representations of a quiver Q over a field k (the kQ-modules, where kQ is the path algebra of Q over k) have been studied for a long time, and one knows quite well the structure of the module category modkQ. It seems to be worthwhile to consider also representations of Q over arbitrary finite-dimensional k-algebras A. Here we draw the attention to the case when A=k[ϵ] is the algebra of dual numbers (the factor algebra of the polynomial ring k[T] in one variable T modulo the ideal generated by T2), thus to the Λ-modules, where Λ=kQ[ϵ]=kQ[T]/〈T2〉. The algebra Λ is a 1-Gorenstein algebra, thus the torsionless Λ-modules are known to be of special interest (as the Gorenstein-projective or maximal Cohen–Macaulay modules). They form a Frobenius category L, thus the corresponding stable category L_ is a triangulated category. As we will see, the category L is the category of perfect differential kQ-modules and L_ is the corresponding homotopy category. The category L_ is triangle equivalent to the orbit category of the derived category Db(modkQ) modulo the shift and the homology functor H:modΛ→modkQ yields a bijection between the indecomposables in L_ and those in modkQ. Our main interest lies in the inverse, it is given by the minimal L-approximation. Also, we will determine the kernel of the restriction of the functor H to L and describe the Auslander–Reiten quivers of L and L_.

Extensions between Cohen\u2013Macaulay modules of Grassmannian cluster categories

In this paper we study extensions between Cohen–Macaulay modules for algebras arising in the categorifications of Grassmannian cluster algebras. We prove that rank 1 modules are periodic, and we give explicit formulas for the computation of the period based solely on the rim of the rank 1 module in question. We determine mathrm{Ext}^i(L_I, L_J) for arbitrary rank 1 modules L_I and L_J. An explicit combinatorial algorithm is given for the computation of mathrm{Ext}^i(L_I, L_J) when i is odd, and when i even, we show that mathrm{Ext}^i(L_I, L_J) is cyclic over the centre, and we give an explicit formula for its computation. At the end of the paper we give a vanishing condition of mathrm{Ext}^i(L_I, L_J) for any i>0.

An alternating matrix and a vector, with application to Aluffi algebras

Let X be a generic alternating matrix, t be a generic row vector, and J be the ideal Pf4(X)+I1(tX). We prove that J is a perfect Gorenstein ideal of grade equal to the grade of Pf4(X) plus two. This result is used by Ramos and Simis in their calculation of the Aluffi algebra of the module of derivations of the homogeneous coordinate ring of a smooth projective hypersurface. We also prove that J defines a domain, or a normal ring, or a unique factorization domain if and only if the base ring has the same property. The main object of study in the present paper is the module N which is equal to the column space of X, calculated mod Pf4(X). The module N is a self-dual maximal Cohen–Macaulay module of rank two; furthermore, J is a Bourbaki ideal for N. The ideals which define the homogeneous coordinate rings of the Plücker embeddings of the Schubert subvarieties of the Grassmannian of planes are used in the study of the module N.

The Hilbert–Kunz functions of two-dimensional rings of type ADE

We compute the Hilbert–Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen–Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals.