Let (R, m) be a Cohen-Macaulay local ring of dimension d > 0, I an m-primary ideal of R and K an ideal containing I. When depth G(I) ≥ d− 1 and r(I|K) < ∞, we present an upper bound on the second fiber coefficient f2(I, K) of the fiber cones FK(I), and also provide a characterization, in terms of f2(I, K), of the condition depth FK(I) ≥ d− 2.

Rank 2 indecomposable arithmetically Cohen-Macaulay bundles E on a nonsingular cubic surface X in P are classified, by means of the possible forms taken by the minimal graded free resolution of E over P. The admissible values of the Chern classes of E are listed and the vanishing locus of a general section of E is studied. Properties of E such as slope (semi) stability and simplicity are invest...

‎In this paper‎, ‎we give some necessary conditions for an $r$-partite graph such that the edge ring of the graph is Cohen-Macaulay‎. ‎It is proved that if there exists a cover of an $r$-partite Cohen-Macaulay graph by disjoint cliques of size $r$‎, ‎then such a cover is unique‎.

Let G be a simple (i.e., no loops and no multiple edges) graph. We investigate the question of how to modify G combinatorially to obtain a sequentially CohenMacaulay graph. We focus on modifications given by adding configurations of whiskers to G, where to add a whisker one adds a new vertex and an edge connecting this vertex to an existing vertex in G. We give various sufficient conditions and...

In this article, we introduce an invariant of Cohen-Macaulay local rings in terms the reduction number canonical ideals. The can be defined arbitrary and it measures how close to being Gorenstein. First, clarify relation between almost Gorenstein nearly by using dimension one. We next characterize idealization trace ideals over invariant. It provides better prospects for a result on property id...

The theorem of Hochster and Roberts says that for any module V of a linearly reductive group G over a eld K the invariant ring KV ] G is Cohen-Macaulay. We prove the following converse: if G is a reductive group and KV ] G is Cohen-Macaulay for any module V , then G is linearly reductive.

Let G be a finite p-group which does not contain a rank two elementary abelian p-group as a direct factor. Then the ideal of essential classes in the mod-p cohomology ring of G is a Cohen–Macaulay module whose Krull dimension is the p-rank of the centre of G. This basically answers in the affirmative a question posed by J. F. Carlson (Question 5.4 in [7]).

Associated to a simple undirected graph G is a simplicial complex ∆G whose faces correspond to the independent sets of G. We call a graph G shellable if ∆G is a shellable simplicial complex in the non-pure sense of Björner-Wachs. We are then interested in determining what families of graphs have the property that G is shellable. We show that all chordal graphs are shellable. Furthermore, we cla...

This article concerns linear parts of minimal resolutions of finitely generated modules over commutative local, or graded rings. The focus is on the linearity defect of a module, which marks the point after which the linear part of its minimal resolution is acyclic. The results established track the change in this invariant under some standard operations in commutative algebra. As one of the ap...