Regularity functions for homogeneous algebras

In the paper regularity functions are introduced satisfying the formal properties of the ordinary Castelnuovo-Mumford regularity. The most significant examples of such functions are the regularity functions arising from Koszul algebras. A comparison of regularity functions is given. This generalizes results of Avramov-Eisenbud and of Roemer. It is also shown that the residue class ring of a Koszul algebra module a componentwise linear ideal has a rational Poincare series, and that the regularity of the powers I^n of a graded ideal I in a Koszul algebra is bounded by a linear function for large n.