On pseudo symmetric monomial curves

We study monomial curves, toric ideals and monomial algebras associated to 4-generated pseudo symmetric numerical semigroups. Namely, we determine indispensable binomials of these toric ideals, give a characterization for these monomial algebras to have strongly indispensable minimal graded free resolutions. We also characterize when the tangent cones of these monomial curves at the origin are Cohen–Macaulay.     

Fiber cones of rational normal scrolls are Cohen–Macaulay

Journal of Algebraic Combinatorics - In this paper, we show that the fiber cones of rational normal scrolls are Cohen–Macaulay. As an application, we compute their Castelnuovo–Mumford...     

Maximal and Cohen–Macaulay projective monomial curves

In this paper we continue the investigation of Cohen–Macaulay projective monomial curves begun in [Les Reid, Leslie G. Roberts, Non-Cohen–Macaulay projective monomial curves, J. Algebra 291 (2005) 171–186]. In the process we introduce maximal curves. Cohen–Macaulay curves are maximal, but not conversely. We show that the number of all curves of degree d that are Cohen–Macaulay grows exponentially, but not as fast as the total number of curves, and also that maximal curves of degree d with sufficiently large embedding dimension relative to d are Cohen–Macaulay.     

GORENSTEIN MODULES,FINITE INDEX,AND FINITE COHEN–MACAULAY TYPE

ABSTRACT

A Gorenstein module over a local ring R is a maximal Cohen–Macaulay module of finite injective dimension. We use existence of Gorenstein modules to extend a result due to S. Ding: A Cohen–Macaulay ring of finite index, with a Gorenstein module, is Gorenstein on the punctured spectrum. We use this to show that a Cohen–Macaulay local ring of finite Cohen–Macaulay type is Gorenstein on the punctured spectrum. Finally, we show that for a large class of rings (including all excellent rings), the Gorenstein locus of a finitely generated module is an open set in the Zariski topology.     

(Bi-)Cohen–Macaulay Simplicial Complexes and Their Associated Coherent Sheaves

ABSTRACT

Via the BGG correspondence, a simplicial complex Δ on [n] is transformed into a complex of coherent sheaves on P ^n?1. We show that this complex reduces to a coherent sheaf ? exactly when the Alexander dual Δ* is Cohen–Macaulay.

We then determine when both Δ and Δ* are Cohen–Macaulay. This corresponds to ? being a locally Cohen–Macaulay sheaf.

Lastly, we conjecture for which range of invariants of such Δ's it must be a cone, and show the existence of such Δ's which are not cones outside of this range.     

Patch ideals and Peterson varieties

Patch ideals encode neighbourhoods of a variety in GL _n /B. For Peterson varieties we determine generators for these ideals and show they are complete intersections, and thus Cohen–Macaulay and Gorenstein. Consequently, we

— combinatorially describe the singular locus of the Peterson variety;

— give an explicit equivariant K-theory localization formula; and

— extend some results of [B. Kostant ‘96] and of D. Peterson to intersections of Peterson varieties with Schubert varieties.

We conjecture that the tangent cones are Cohen–Macaulay, and that their h-polynomials are nonnegative and upper-semicontinuous. Similarly, we use patch ideals to briey analyze other examples of torus invariant subvarieties of GL _n /B, including Richardson varieties and Springer fibers.     

When almost Cohen–Macaulay algebras map into big Cohen–Macaulay modules

In this article, we show that almost Cohen–Macaulay algebras are solid. Moreover, we seek for the conditions when (a) an almost Cohen–Macaulay algebra is a phantom extension and (b) when it maps into a balanced big Cohen–Macaulay module.     

Buchsbaum Stanley–Reisner Rings and Cohen–Macaulay Covers

First, we give a new criterion for Buchsbaum Stanley–Reisner rings to have linear resolutions. Next, we prove that every (d ? 1)-dimensional complex Δ of initial degree d is contained in the same dimensional Cohen–Macaulay complex whose (d ? 1)th reduced homology is isomorphic to that of Δ. We call such a simplicial complex a Cohen–Macaulay cover of Δ. And we also show that all the intermediate complexes between Δ and its Cohen–Macaulay cover are Buchsbaum provided that Δ is Buchsbaum. As an application, we determine the h-vectors of the 3-dimensional Buchsbaum Stanley–Reisner rings with initial degree 3.     

Vanishing of cohomology over Cohen–Macaulay rings

A 2003 counterexample to a conjecture of Auslander brought attention to a family of rings—colloquially called AC rings—that satisfy a natural condition on vanishing of cohomology. Several results attest to the remarkable homological properties of AC rings, but their definition is barely operational, and it remains unknown if they form a class that is closed under typical constructions in ring theory. In this paper, we study transfer of the AC property along local homomorphisms of Cohen–Macaulay rings. In particular, we show that the AC property is preserved by standard procedures in local algebra. Our results also yield new examples of Cohen–Macaulay AC rings.     

Modules with pure resolutions

Let R a standard graded algebra over a field k. In this paper, we give a relation in terms of graded Betti numbers, called the Herzog–Kühl equations, for a pure R-module M to satisfy the condition dim(R)?depth(R) = dim(M)?depth(M). When R is Cohen–Macaulay, we prove an analogous result characterizing all graded Cohen–Macaulay R-modules of finite projective dimension. Finally, as an application, we show that the property of R being Cohen–Macaulay is characterized by the existence of pure Cohen–Macaulay R-modules corresponding to any degree sequence of length at most depth(R).     

The AS–Cohen–Macaulay property for quantum flag manifolds of minuscule weight

It is shown that quantum homogeneous coordinate rings of generalised flag manifolds corresponding to minuscule weights, their Schubert varieties, big cells, and determinantal varieties are AS–Cohen–Macaulay. The main ingredient in the proof is the notion of a quantum graded algebra with a straightening law, introduced by T.H. Lenagan and L. Rigal [T.H. Lenagan, L. Rigal, Quantum graded algebras with a straightening law and the AS–Cohen–Macaulay property for quantum determinantal rings and quantum Grassmannians, J. Algebra 301 (2006) 670–702]. Using Stanley's Theorem it is moreover shown that quantum generalised flag manifolds of minuscule weight and their big cells are AS–Gorenstein.     

Hilbert functions of Gorenstein monomial curves

It is a conjecture due to M. E. Rossi that the Hilbert function of a one-dimensional Gorenstein local ring is non-decreasing. In this article, we show that the Hilbert function is non-decreasing for local Gorenstein rings with embedding dimension four associated to monomial curves, under some arithmetic assumptions on the generators of their defining ideals in the non-complete intersection case. In order to obtain this result, we determine the generators of their tangent cones explicitly by using standard basis computations under these arithmetic assumptions and show that the tangent cones are Cohen-Macaulay. In the complete intersection case, by characterizing certain families of complete intersection numerical semigroups, we give an inductive method to obtain large families of complete intersection local rings with arbitrary embedding dimension having non-decreasing Hilbert functions.

     

Column Invariants Over Cohen–Macaulay Local Rings

Abstract

We define some numerical invariants over Cohen–Macaulay local rings. These invariants are related to columns of the presenting matrices of maximal Cohen–Macaulay modules and syzygy modules without free summands. We study the relationship between these invariants, and the invariant col(A).

     

Tilting and cluster tilting for quotient singularities

We shall show that the stable categories of graded Cohen–Macaulay modules over quotient singularities have tilting objects. In particular, these categories are triangle equivalent to derived categories of finite dimensional algebras. Our method is based on higher dimensional Auslander–Reiten theory, which gives cluster tilting objects in the stable categories of (ungraded) Cohen–Macaulay modules.     

Zero-Dimensional Imbeddability

In this article, we try to understand which generically complete intersection monomial ideals with fixed radical are Cohen–Macaulay. We are able to give a complete characterization for a special class of simplicial complexes, namely the Cohen–Macaulay complexes without cycles in codimension 1. Moreover, we give sufficient conditions when the square-free monomial ideal has minimal multiplicity.     

Non-Noetherian Cohen–Macaulay rings

In this paper we investigate a property for commutative rings with identity which is possessed by every coherent regular ring and is equivalent to Cohen–Macaulay for Noetherian rings. We study the behavior of this property in the context of ring extensions (of various types) and rings of invariants.     

Cohen–Macaulay Auslander algebras of skewed-gentle algebras

For any skewed-gentle algebra, we characterize its indecomposable Gorenstein projective modules explicitly and describe its Cohen–Macaulay Auslander algebra. We prove that skewed-gentle algebras are always Gorenstein, which is independent of the characteristic of the ground field, and the Cohen–Macaulay Auslander algebras of skewed-gentle algebras are also skewed-gentle algebras.     

On the Stanley–Reisner ideal of an expanded simplicial complex

Let Δ be a simplicial complex. We study the expansions of Δ mainly to see how the algebraic and combinatorial properties of Δ and its expansions are related to each other. It is shown that Δ is Cohen–Macaulay, sequentially Cohen–Macaulay, Buchsbaum or k-decomposable, if and only if an arbitrary expansion of Δ has the same property. Moreover, some homological invariants like the regularity and the projective dimension of the Stanley–Reisner ideals of Δ and those of their expansions are compared.