Generalized local cohomology and regularity of Ext modules

Let R be a Noetherian standard graded ring, and M and N two finitely generated graded R  -modules. We introduce regR(M,N)regR(M,N) by using the notion of generalized local cohomology instead of local cohomology, in the definition of regularity. We prove that regR(M,N)regR(M,N) is finite in several cases. In the case that the base ring is a field, we show thatregR(M,N)=reg(N)−indeg(M).regR(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.