  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.
