The nilpotent endomorphisms over a finite free module over a domain with principal ideal are characterized. One may apply these results to the study of the maximal Cohen-Macaulay modules over the ring R := A[[x]]/(x), n ≥ 2, where A is a DVR. Subject Classification: 15A21, 13C14.

Let R be a local ring and C semidualizing module of R. We investigate the behavior certain classes generalized Cohen-Macaulay R-modules under Foxby equivalence between Auslander Bass with respect to C. In particular, we show that are invariant this if M is finitely generated R-module in class such C\otimes_RM surjective Buchsbaum, then also Buchsbaum.

We define very proper intersections of modules and projective subschemes. It turns out that equidimensional locally Cohen-Macaulay modules intersect very properly if and only if they intersect properly. We prove a Bezout theorem for modules which meet very properly. Furthermore, we show for equidimensional subschemes X and Y : If they intersect properly in an arithmetically Cohen-Macaulay subsc...

A certain inequality is shown to hold for the values of the Möbius function of the poset obtained by attaching a maximum element to a lower Eulerian Cohen– Macaulay poset. In two important special cases, this inequality provides partial results supporting Stanley’s nonnegativity conjecture for the toric h-vector of a lower Eulerian Cohen–Macaulay meet-semilattice and Adin’s nonnegativity conjec...

In this paper we consider bi-Cohen-Macaulay graphs, and give a complete classification of such graphs in the case they are bipartite or chordal. General biCohen-Macaulay graphs are classified up to separation. The inseparable bi-CohenMacaulay graphs are determined. We establish a bijection between the set of all trees and the set of inseparable bi-Cohen-Macaulay graphs.

In this paper we study topological properties of the poset of injective words and the lattice of classical non-crossing partitions. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This extends the well-known result that those posets are shellable. Both results rely on a new poset fiber theorem, for doubly homo...

Let S = K[x1, . . . ,xn] be a polynomial ring and R = S/I where I ⊂ S is a graded ideal. The Multiplicity Conjecture of Herzog, Huneke, and Srinivasan states that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S as well as bounded below by a function of the minimal shifts if R is Cohen–Macaulay. In this paper we study t...

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 th...

Let (R,m) denote an n-dimensional Gorenstein ring. For an ideal I ⊂ R of height c we are interested in the endomorphism ring B = HomR(H c I (R),H c I (R)). It turns out that B is a commutative ring. In the case of (R, m) a regular local ring containing a field B is a Cohen-Macaulay ring. Its properties are related to the highest Lyubeznik number l = dimk Ext d R(k, H c I (R)). In particular R ≃...