In this paper we study simplicial complexes as higher dimensional graphs in order to produce algebraic statements about their facet ideals. We introduce a large class of square-free monomial ideals with Cohen-Macaulay quotients, and a criterion for the Cohen-Macaulayness of facet ideals of simplicial trees. Along the way, we generalize several concepts from graph theory to simplicial complexes.

The cut sets of a graph are special vertices whose removal disconnects the graph. They fundamental in study binomial edge ideals, since they encode their minimal primary decomposition. We introduce class accessible graphs as with unmixed ideal and form an set system. prove that is Cohen-Macaulay we conjecture converse holds. settle for large classes graphs, including chordal traceable providing...

Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5-chordal graphs with no chordless 4-cycles are shellable and sequentially Cohen-Macaulay. We use this result to characterize the obstructions to shellability in flag complexes, exte...

We describe a combinatorial condition on graphwhich guarantees that all powers of its vertex cover ideal are componentwise linear. Then motivated by Eagon and Reiner’s Theorem we study whether the Cohen-Macaulay graph have linear free resolutions. After giving complete characterization cactus graphs (i.e., connected in which each edge belongs to at most one cycle) show their ideals

In this article we study the structure of residual intersections via constructing a finite complex which is acyclic under some sliding depth conditions on the cycles of the Koszul complex. This complex provides information on an ideal which coincides with the residual intersection in the case of geometric residual intersection; and is closely related to it in general. A new success obtained thr...

We discuss Matijevic–Roberts type theorem on strong F -regularity, F -purity, and Cohen–Macaulay F -injective (CMFI for short) property. Related to this problem, we also discuss the base change problem and the openness of loci of these properties. In particular, we define the notion of F -purity of homomorphisms using Radu–André homomorphisms, and prove basic properties of it. We also discuss a...

Let be a commutative Noetherian ring and let I be a proper ideal of . D’Anna and Fontana in [6] introduced a new construction of ring,  named amalgamated duplication of along I. In this paper by considering the ring homomorphism , it is shown that if , then , also it is proved that if , then there exists  such that . Using this result it is shown that if is generically Cohen-Macaulay (resp. gen...

Let  be a local Cohen-Macaulay ring with infinite residue field,  an Cohen - Macaulay module and  an ideal of  Consider  and , respectively, the Rees Algebra and associated graded ring of , and denote by  the analytic spread of  Burch’s inequality says that  and equality holds if  is Cohen-Macaulay. Thus, in that case one can compute the depth of associated graded ring of  as  In this paper we ...

We consider a class of graphs G such that the height of the edge ideal I(G) is half of the number ♯V (G) of the vertices. We give Cohen-Macaulay criteria for such graphs.

In a recent paper, E. Steingŕımsson associated to each simple graph G a simplicial complex ∆G denoted as the coloring complex of G. Certain nonfaces of ∆G correspond in a natural manner to proper colorings of G. Indeed, the h-vector is an affine transformation of the chromatic polynomial χG of G, and the reduced Euler characteristic is, up to sign, equal to |χG(−1)| − 1. We show that ∆G is cons...