  In this article we extend a previous definition of Castelnuovo-Mumford
regularity for modules over an algebra graded by a finitely generated abelian
group.
  Our notion of regularity is based on Maclagan and Smith's definition, and is
extended first by working over any commutative base ring, and second by
considering local cohomology with support in an arbitrary finitely generated
graded ideal $B$, obtaining, for each $B$, a $B$-regularity region. The first
extension provides a natural approach for working with families of sheaves or
of graded modules, while the second opens new applications.
  We provide tools to transfer knowledge in two directions. First to deduce
some information on the graded Betti numbers from the knowledge of regions
where the local cohomology with support in a given graded ideal vanishes. This
is one of our main results. Conversely, vanishing of local cohomology with
support in any graded ideal is deduced from the shifts in a free resolution and
the local cohomology of the polynomial ring. Furthermore, the flexibility of
treating local cohomology with respect to any $B$ open new possibilities for
passing information.
  We provide new persistence results for the vanishing of local cohomology that
extend the fact that weakly regular implies regular in the classical case, and
we give sharp estimates for the regularity of a truncation of a module.
  In the last part, we present a result on Hilbert functions for multigraded
polynomial rings, that in particular provides a simple proof of the
Grothendieck-Serre formula.
