Abstract:
We study unmixed and Cohen-Macaulay properties of the binomial edge ideal of some classes of graphs. We compute the depth of the binomial edge ideal of a generalized block graph. We also characterize all generalized block graphs whose binomial edge ideals are Cohen-Macaulay and unmixed. So that we generalize the results of Ene, Herzog and Hibi on block graphs. Moreover, we study unmixedness and Cohen-Macaulayness of the binomial edge ideal of some graph products such as the join and corona of two graphs with respect to the original graphs'.

Abstract:
In this paper we study the finitely generated bigraded modules over a standard bigraded polynomial ring which are relative Cohen-Macaulay or relative unmixed with respect to one of the irrelevant bigraded ideals. A generalization of Reisner's criterion for Cohen-Macaulay simplicial complexes is considered.

Abstract:
We study weighted graphs and their "edge ideals" which are ideals in polynomial rings that are defined in terms of the graphs. We provide combinatorial descriptions of m-irreducible decompositions for the edge ideal of a weighted graph in terms of the combinatorics of "weighted vertex covers". We use these, for instance, to say when these ideals are m-unmixed. We explicitly describe which weighted cycles, suspensions, and trees are unmixed and which ones are Cohen-Macaulay, and we prove that all weighted complete graphs are Cohen-Macaulay.

Abstract:
This paper was motivated by a problem left by Herzog and Hibi, namely to classify all unmixed polymatroidal ideals. In the particular case of polymatroidal ideals corresponding to discrete polymatroids of Veronese type, i.e ideals of Veronese type, we give a complete description of the associated prime ideals and then, we show that such an ideal is unmixed if and only if it is Cohen-Macaulay. We also give for this type of ideals equivalent characterizations for being equidimensional.

Abstract:
Let be a commutative Noetherian local ring and let be a finitely generated -module of dimension . Then the following statements hold: (a) if width for all with , then is co-Cohen-Macaulay of Noetherian dimension ; (b) if is an unmixed -module and depth , then is co-Cohen-Macaulay of Noetherian dimension if and only if is either zero or co-Cohen-Macaulay of Noetherian dimension . As consequence, if is co-Cohen-Macaulay of Noetherian dimension for all with , then is co-Cohen-Macaulay of Noetherian dimension . 1. Introduction Throughout this paper, let be a commutative Noetherian local ring and let be a finitely generated -module of dimension . We denote the th local cohomology module of with respect to by . It is well known that is Artinian for all (cf. [1]). The Noetherian dimension of an Artinian -module , denoted by , is defined inductively as follows: when , put . Then by induction, for any integer , put if is false, and for any ascending chain of submodules of there exists an integer such that for all . Therefore if and only if is a nonzero Noetherian module. Moreover, if is an exact sequence of Artinian modules, then . Let . is an -coregular sequence if is surjective for and . The width of , denoted by , is the length of any maximal -coregular sequence in . For any -coregular element , we have that and . Details about and can be found in Roberts [2], Kirby [3], and Ooishi [4]; there is a general fact: for any Artinian -module holds and is co-Cohen-Macaulay if and only if holds (cf. [5–7]). Tang [8] has shown that if either or is Cohen-Macaulay, then is co-Cohen-Macaulay (see also [9]). Following Nagata [10], is unmixed if for all . The main aim of this paper is to prove the following theorem. Theorem 1. The following statements are true.(a)If for all with , then is co-Cohen-Macaulay of Noetherian dimension . (b)If is an unmixed -module and , then is co-Cohen-Macaulay of Noetherian dimension if and only if is either zero or co-Cohen-Macaulay of Noetherian dimension . 2. The Results Following Macdonald [11], every Artinian -module has minimal secondary representation , where is secondary. The set is independent of the choice of the minimal secondary representation of . This set is called the set of attached prime ideals of and denoted by . The set of all minimal elements of is exactly the set of all minimal elements of . A sequence of elements in is called a strict f-sequence of if for all for all . This notion was introduced in [12]. Lemma 2 (see [9]). For all integer , one has and . Lemma 3. Let be a strict -sequence of . Then the following

Abstract:
The Grothendieck-Serre formula for the difference between the Hilbert function and Hilbert polynomial of a graded algebra is generalized for bigraded standard algebras. This is used to get a similar formula for the difference between the Bhattacharya function and Bhattacharya polynomial of two m-primary ideals I and J in a local ring (A,m) in terms of local cohomology modules of Rees algebras of I and J. The cohomology of a variation of the Kirby-Mehran complex for bigraded Rees algebras is studied which is used to characterize the Cohen-Macaulay property of bigraded Rees algebra of I and J for two dimensional Cohen-Macaulay local rings.

Abstract:
All Cohen--Macaulay polymatroidal ideals are classified. The Cohen--Macaulay polymatroidal ideals are precisely the principal ideals, the Veronese ideals, and the squarefree Veronese ideals.

Abstract:
The Bernstein-Sato polynomial (or global b-function) is an important invariant in singularity theory, which can be computed using symbolic methods in the theory of D-modules. After surveying algorithms for computing the global b-function, we develop a new method to compute the local b-function for a single polynomial. We then develop algorithms that compute generalized Bernstein-Sato polynomials of Budur-Mustata-Saito and Shibuta for an arbitrary polynomial ideal. These lead to computations of log canonical thresholds, jumping coefficients, and multiplier ideals. Our algorithm for multiplier ideals simplifies that of Shibuta and shares a common subroutine with our local b-function algorithm. The algorithms we present have been implemented in the D-modules package of the computer algebra system Macaulay2.

Abstract:
Let $R=k[x_{1},\ldots,x_{n}]$, where $k$ is a field. The path ideal (of length $t\geq 2$) of a directed graph $G$ is the monomial ideal, denoted by $I_{t}(G)$, whose generators correspond to the directed paths of length $t$ in $G$. Let $\Gamma$ be a directed rooted tree. We characterize all such trees whose path ideals are unmixed and Cohen-Macaulay. Moreover, we show that $R/I_{t}(\Gamma)$ is Gorenstein if and only if the Stanley-Reisner simplicial complex of $I_{t}(\Gamma)$ is a matroid.