  The asymptotic stability of several homological invariants of the graded
pieces of a graded module has attracted quite a lot of attention over the last
decades. We provide in this text several stability results together with
estimates of the degree from which it stabilizes. Before we establish these
regularity results, we prove several facts about depth and cohomological
dimension with respect to a finitely generated ideal and about
Castelnuovo-Mumford regularity of a graded module.
