The Castelnuovo-Mumford regularity r of a variety V ⊆ Pn C is an upper bound for the degrees of the hypersurfaces necessary to cut out V . In this note we give a bound for r when V is left invariant by a vector field on

D. J. Benson conjectures that the Castelnuovo-Mumford regularity of the cohomology ring of a finite group is always zero. More generally he conjectures that there is always a very strongly quasi-regular system of parameters. Computer calculations show that the second conjecture holds for all groups of order less than 256.

We consider the possibility of characterizing Buchsbaum and some special generalized Cohen-Macaulay rings by systems of parameters having certain properties of regular sequences. As an application, we give a bound on Castelnuovo-Mumford regularity of so-called (k, d)-Buchsbaum graded Kalgebras.

Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially Cohen-Macaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the Castelnuovo-Mumford regularity of R/I(G) can be determined from the invariants of G.

let $l$ be a lattice in $zz^n$ of dimension $m$. we prove that there exist integer constants $d$ and $m$ which are basis-independent such that the total degree of any graver element of $l$ is not greater than $m(n-m+1)md$. the case $m=1$ occurs precisely when $l$ is saturated, and in this case the bound is a reformulation of a well-known bound given by several authors. as a corollary, we show t...

The Castelnuovo-Mumford regularity of a graded ring is an important invariant in computational commutative algebra, and there is increasing interest in multigraded generalizations. We study connections between two recent definitions of multigraded regularity with a view towards a better understanding of the multigraded Hilbert function of fat point schemes in P n1 × · · · × P k . Introduction L...

Dave Benson, in [2], conjectured that for any finite group G and any prime p the Castelnuovo-Mumford regularity of the cohomology ring, H(G,Fp), is zero. He showed that reg(H(G,Fp)) ≥ 0 and succeeded in proving equality when the difference between the dimension and the depth is at most two. The purpose of this paper is to prove Benson’s Regularity Conjecture as a corollary of the following result.

Let S be a standard N k-graded polynomial ring over a field K, let I be a multigraded homogeneous ideal of S, and let M be a finitely generated Z k-graded S-module. We prove that the resolution regularity, a multigraded variant of Castelnuovo-Mumford regularity, of I n M is asymptotically a linear function. This shows that the well known Z-graded phenomenon carries to multigraded situation.