The eventual stability of depth, associated primes and cohomology of a   graded module
  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. 
Marc Chardin, Jean-Pierre Jouanolou, Ahad Rahimi
math.AC math.AG