Castelnuovo–Mumford Regularity and Cohomological Dimension

Abstract

Let be a standard graded ring, be the irrelevant ideal of R and 𝔞0 be an ideal of R 0. In this paper, as a generalization of the concept of Castelnouvo–Mumford regularity reg(M) of a finitely generated graded R-module M, we define the regularity of M with respect to 𝔞0 + R +, say reg𝔞0 + R + (M). We also study some relations of this new invariant with the classic one. To this end, we need to consider the cohomological dimension of some finitely generated R 0-modules. Also, we will express reg𝔞0 + R + (M) in terms of some invariants of the minimal graded free resolution of M and see that in a special case this invariant is independent of the choice of 𝔞0.