Generalized local cohomology and regularity of Ext modules

Abstract

Let R be a Noetherian standard graded ring, and M and N two finitely generated graded R-modules. We introduce reg R ( M , N ) by using the notion of generalized local cohomology instead of local cohomology, in the definition of regularity. We prove that reg R ( M , N ) is finite in several cases. In the case that the base ring is a field, we show that reg R ( M , N ) = reg ( N ) − indeg ( M ) . This formula, together with a graded version of duality for generalized local cohomology, gives a formula for the minimum of the initial degrees of some Ext modules (in the case R is Cohen–Macaulay), of which the three usual definitions of regularity are special cases. Bounds for regularity of certain Ext modules are obtained, using the same circle of ideas.