Bounded regularity

Let k be a field and S the polynomial ring k[x1,…,xn]. For a non-trivial finitely generated homogeneous S-module M with grading in Z, an integer D and some homogeneous polynomial f in S, it is defined what it means that f is regular on M up to degree D. Following the usual definition of regularity, a generalization to finite sequences of polynomials in S is given.Different criteria for a finite sequence of polynomials in S to be regular up to a particular degree are given: first a characterization with Hilbert series, then a characterization with first syzygies, and finally, for M=S, characterizations with Betti numbers as well as with the Koszul complex and free resolutions.