In this paper, we explore the relation between index of reducibility and Hilbert coefficients in local rings. Consequently, main result study provides a characterization sequentially Cohen-Macaulay ring terms its non-parameter ideals. As corollaries to theorem, obtain characterizations Gorenstein/Cohen-Macaulay Chern

Let I ⊂ R be a graded ideal in the polynomial ring R = K[x1, . . . , xn] where K is a field, and fix a term order <. It has been shown in [17] that the Hilbert functions of the local cohomology modules of R/I are bounded by those of R/ in(I), where in(I) denotes the initial ideal of I with respect to <. In this note we study the question when the local cohomology modules of R/I and R/ in(I) hav...

Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially Cohen-Macaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the Castelnuovo-Mumford regularity of R/I(G) can be determined from the invariants of G.

We characterize unmixed and Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs. also provide examples oriented graphs that have non-Cohen-Macaulay vertex-weighted ideals, while the ideal their underlying graph is Cohen-Macaulay. This disproves a conjecture posed by Pitones, Reyes, Toledo.

We extend the definition of chordal from graphs to clutters. The resulting family generalizes both chordal graphs and matroids, and obeys many of the same algebraic and geometric properties. Specifically, the independence complex of a chordal clutter is shellable, hence sequentially Cohen-Macaulay; and the circuit ideal of a certain complement to such a clutter has a linear resolution. Minimal ...