Following a construction of Stanley we consider toric face rings associated to rational pointed fans. This class of rings is a common generalization of the concepts of Stanley–Reisner and affine monoid algebras. The main goal of this article is to unify parts of the theories of Stanley–Reisnerand affine monoid algebras. We consider (nonpure) shellable fan’s and the Cohen–Macaulay property. More...

Let R be a local Cohen-Macaulay ring with canonical module ωR. We investigate the following question of Huneke: If the sequence of Betti numbers {β i (ωR)} has polynomial growth, must R be Gorenstein? This question is well-understood when R has minimal multiplicity. We investigate this question for a more general class of rings which we say are homologically of minimal multiplicity. We provide ...

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.

Let G be a simple undirected graph on n vertices, and let I(G) ⊆ R = k[x 1 ,. .. , x n ] denote its associated edge ideal. We show that all chordal graphs G are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of I(G) is componentwise linear. Our result complements Faridi's theorem that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay and He...

We study properties of a poset generating a Cohen-Macaulay algebra with straightening laws (ASL for short). We show that if a poset P generates a Cohen-Macaulay ASL, then P is pure and, if P is moreover Buchsbaum, then P is Cohen-Macaulay. Some results concerning a Rees algebra of an ASL defined by a straightening closed ideal are also established. And it is shown that if P is a Cohen-Macaulay ...