Let G be a finite simple graph on a vertex set V (G) = {x11, . . . , xn1}. Also let m1, . . . , ,mn ≥ 2 be integers and G1, . . . , Gn be connected simple graphs on the vertex sets V (Gi) = {xi1, . . . , ximi}. In this paper, we provide necessary and sufficient conditions on G1, . . . , Gn for which the graph obtained by attaching the Gi to G is unmixed or vertex decomposable. Then we character...

Let C ⊂ N be an affine semigroup, and R = K[C] its semigroup ring. This paper is a collection of various results on “C-graded” R-modules M = ⊕ c∈C Mc, especially, monomial ideals of R. For example, we show the following: If R is normal and I ⊂ R is a radicalmonomial ideal (i.e., R/I is a generalization of Stanley-Reisner rings), then the sequentially Cohen-Macaulay property of R/I is a topologi...

Generalizing the notion of a Koszul algebra, a graded kalgebra A is K2 if its Yoneda algebra ExtA(k, k) is generated as an algebra in cohomology degrees 1 and 2. We prove a strong theorem about K2 factor algebras of Koszul algebras and use that theorem to show the Stanley-Reisner face ring of a simplicial complex ∆ is K2 whenever the Alexander dual simplicial complex ∆∗ is (sequentially) Cohen-...

We introduce the notion of E-depth graded modules over polynomial rings to measure depth certain Ext modules. First, we characterize with (sufficiently) large as those whose partial general initial submodules preserve Hilbert function local cohomology supported at irrelevant maximal ideal, extending a result Herzog and Sbarra on sequentially Cohen-Macaulay Second, describe cone tables sufficien...

Let HilbpK be the Hilbert scheme parametrizing the closed subschemes of Pn K with Hilbert polynomial p ∈ Q[t] over a field K of characteristic zero. By bounding below the cohomological Hilbert functions of the points of HilbpK we define locally closed subspaces of the Hilbert scheme. The aim of this thesis is to show that some of these subspaces are connected. For this we exploit the binomial i...

let $(r,m)$ be a commutative noetherian local ring, $m$ a finitely generated $r$-module of dimension $d$, and let $i$ be an ideal of definition for $m$. in this paper, we extend cite[corollary 10(4)]{p} and also we show that if $m$ is a cohen-macaulay $r$-module and $d=2$, then $lambda(frac{widetilde{i^nm}}{jwidetilde{i^{n-1}m}})$ does not depend on $j$ for all $ngeq 1$, where $j$ is a minimal ...

In recent years our studies of experimental syphilis have centred in the use of the chimpanzee as an animal model for the disease. Chimpanzees have been infected with Treponema pallidum, T. carateum, and T. cuniculi. The characteristics of these infections have been described in part or in whole elsewhere (Brown, Kuhn, Tolliver, and Norins, 1970; Chandler, Kaufmann, and Kuhn, 1972; Kuhn 1970, 1...

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...

A recent framework for generalizing the Erdős-KoRado Theorem, due to Holroyd, Spencer, and Talbot, defines the Erdős-Ko-Rado property for a graph in terms of the graph’s independent sets. Since the family of all independent sets of a graph forms a simplicial complex, it is natural to further generalize the Erdős-Ko-Rado property to an arbitrary simplicial complex. An advantage of working in sim...