D. Bayer and M. Stillman showed that Gröbner bases can be used to compute the Castelnuovo-Mumford regularity which is a measure for the vanishing of graded local cohomology modules. The aim of this paper is to show that the same method can be applied to study other cohomological invariants as well as the reduction number.

We study the initial ideal of binomial edge ideal in degree 2 ([in<(JG)]2), associated to a graph G. We computed dimension, depth, Castelnuovo-Mumford regularity, Hilbert function and Betti numbers of [in<(JG)]2 for some classes of graphs. AMS Mathematics Subject Classification (2010): 05E40, 16E30

In this paper we show how, given a complex of graded modules and knowing some partial Castelnuovo-Mumford regularities for all the modules in the complex and for all the positive homologies, it is possible to get a bound on the regularity of the zero homology. We use this to prove that if dimTor 1 (M, N) ≤ 1 then reg(M⊗N) ≤ reg(M)+reg(N), generalizing results of Chandler, Conca and Herzog, and ...

Let ρC be the regularity of the Hilbert function of a projective space curve C over an algebraically closed field, αI be the initial degree of the defining ideal I of C and βI the least degree t such that the depth of the ideal generated by I≤t is depth(I). Then the Castelnuovo-Mumford regularity reg(C) of C is upper bounded by max{ρC +1, αI +βI −1}. We study in deep and refine the above bound ...

There are two motivating questions in [M. Mahmoudi, A. Mousivand, M. Crupi, G. Rinaldo, N. Terai and S. Yassemi, arXiv:1006.1087v1] J. Pure Appl. Algebra, 215(10) (2011), 2473-2480] about Castelnuovo-Mumford regularity vertex decomposability of simple graphs. In this paper, we give negative answers to the by providing counterexamples.

Abstract Let I be a homogeneous ideal in polynomial ring S . In this paper, we extend the study of asymptotic behavior minimum distance function δ and give bounds for its stabilization point, r , when is an F -pure or square-free monomial ideal. These are related with dimension Castelnuovo–Mumford regularity

We study the topology of the lcm-lattice of edge ideals and derive upper bounds on the Castelnuovo-Mumford regularity of the ideals. In this context it is natural to restrict to the family of graphs with no induced 4-cycle in their complement. Using the above method we obtain sharp upper bounds on the regularity when the complement is a chordal graph, or a cycle, or when the original graph is c...

Let $$\mathcal {D}$$ be a weighted oriented graph and let $$I(\mathcal {D})$$ its edge ideal in polynomial ring R. We give the formula of Castelnuovo–Mumford regularity $$R/I(\mathcal when is path or cycle such that edges are one direction. Additionally, we compute projective dimension for this class graphs.

We study the topology of the lcm-lattice of edge ideals and derive upper bounds on the Castelnuovo-Mumford regularity of the ideals. In this context it is natural to restrict to the family of graphs with no induced 4-cycle in their complement. Using the above method we obtain sharp upper bounds on the regularity when the complement is a chordal graph, or a cycle, or when the primal graph is cla...

We investigate the behavior of Castelnuovo-Mumford regularity with respect to some classical functors : Tor, the Frobenius functor in positive characteristic, taking a power or a product (on ideals). These generalizes and refines previous results on these issues by several authors. As an application we provide results on the regularity of an intersection of subschemes of a projective scheme, un...