We study bounds for the Castelnuovo–Mumford regularity of homogeneous ideals in a polynomial ring in terms of the number of variables and the degree of the generators. In particular, our aim is to give a positive answer to a question posed by Bayer and Mumford in What can be computed in algebraic geometry? (Computational algebraic geometry and commutative algebra, Symposia Mathematica, vol. XXX...

Let R be a Noetherian ring and I an ideal in R. Then there exists an integer k such that for all n 1 there exists a primary decomposition I n = q 1 \ \ q s such that for all i, p q i nk q i. Also, for each homogeneous ideal I in a polynomial ring over a eld there exists an integer k such that the Castelnuovo-Mumford regularity of I n is bounded above by kn. The regularity part follows from the ...

Here we define the concept of Qregularity for coherent sheaves on quadrics. In this setting we prove analogs of some classical properties. We compare the Qregularity of coherent sheaves on Qn ⊂ P n+1 with the Castelnuovo-Mumford regularity of their extension by zero in P. We also classify the coherent sheaves with Qregularity −∞. We use our notion of Qregularity in order to prove an extension o...

We prove two recent conjectures on some upper bounds for the Castelnuovo-Mumford regularity of the binomial edge ideals of some different classes of graphs. We prove the conjecture of Matsuda and Murai for chordal graphs. We also prove the conjecture due to the authors for a class of chordal graphs. We determine the regularity of the binomial edge ideal of the join of graphs in terms of the reg...

Introduction. The aim of this article is to show how one may use interesting properties of the canonical module for computational applications. It seems that the ideas we develop here are not used in computational algebraic geometry today. We do not claim any major improvement on the theoretical complexity of the problems we address, however we have the feeling that this alternative approach fo...

The algebra of basic covers of a graph G, denoted by Ā(G), was introduced by Herzog as a suitable quotient of the vertex cover algebra. In this paper we compute the Krull dimension of Ā(G) in terms of the combinatorics of G. As a consequence, we get new upper bounds on the arithmetical rank of monomial ideals of pure codimension 2. Furthermore, we show that if the graph is bipartite, then Ā(G) ...