We consider the poset P (N ;A1, A2, . . . , Am) consisting of all subsets of a finite set N which do not contain any of the Ai’s, where the Ai’s are mutually disjoint subsets of N . The elements of P are ordered by inclusion. We show that P belongs to the class of Macaulay posets, i.e. we show a Kruskal-Katona type theorem for P . For the case that the Ai’s form a partition of N , the dual P ∗ ...

The aim of this paper is to investigate properties of the local cohomology of rings of mixed characteristic that are analogous to Segre products of rings de ned over a eld. The main question is whether the local cohomology can be almost killed in a nite extension (we de ne what this means below). There are two reasons for considering this type of ring. First, there are special properties of the...

Let W be a Weyl group corresponding to the root system An−1 or Bn. We define a simplicial complex ∆mW in terms of polygon dissections for such a group and any positive integer m. For m = 1, ∆ W is isomorphic to the cluster complex corresponding to W , defined in [9]. We enumerate the faces of ∆ W and show that the entries of its h-vector are given by the generalized Narayana numbers N W (i), de...

A few years ago, Huneke and Leuschke proved a theorem which solved a conjecture of Schreyer. It asserts that an excellent Cohen-Macaulay local ring of countable Cohen-Macaulay type which is complete or has uncountable residue field has at most a one-dimensional singular locus. In this paper, it is verified that the assumption of the excellent property can be removed, and the theorem is consider...

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 im...

We provide the regularity and Cohen–Macaulay type of binomial edge ideals cones, we show extremal Betti numbers some classes ideals: bipartite fan graphs. In addition, compute Hilbert–Poincaré series

A canonical construction of a balanced big Cohen-Macaulay algebra for a domain of finite type over C is obtained by taking ultraproducts of absolute integral closures in positive characteristic. Among the applications are a new tight closure characterization of rational singularities in characteristic zero, and a necessary condition for Q-Gorenstein logterminal singularities. In particular, it ...

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...