Minimal free resolutions and asymptotic behavior of multigraded regularity

Let S be a standard N^k-graded polynomial ring over a field. Let I be a multigraded homogeneous ideal in S and let M be a finitely generated Z^k-graded S-module. We prove that the resolution regularity, a multigraded variant of Castelnuovo-Mumford regularity, of I^nM is asymptotically a linear function. This shows that the well known Z-graded phenomenon carries to multigraded situation.Comment: Final version to appear in J. Algebra; 18 page

Castelnuovo -Mumford regularity, postulation numbers, and reduction numbers

Suppose G is a standard graded ring over an infinite field, with positively graded piece G+. From the minimal graded free resolution of G, it is possible to derive several invariants, among them the multiplicity, the Castelnuovo-Mumford regularity, the Hilbert series, and the postulation number. We discuss a sharp lower bound for the regularity of G in terms of the postulation number, the depth, and the dimension of G. We present a class of examples in dimension 1 where the postulation number is 0 and the regularity of G can take on any value between 1 and the embedding codimension of G. This family of examples demonstrates that the regularity is not determined by the Hilbert function. Suppose G = grm (R) is the associated graded ring of a Cohen-Macaulay local ring (R, m). It is well known that R being Cohen-Macaulay does not imply G is Cohen-Macaulay. For 1 ≤ n ≤ 4, we present an example of a local complete intersection (R, m) of dimension n and embedding dimension 2 n + 1 such that grade G+ = 0. We compute the regularity, the reduction number and the postulation number of G for these examples, and consider the relationship among these invariants for G. In the case where dim G - grade G+ ≤ 1, a precise description is known as to how these integers are related. We consider the case where dim G - grade G+ = 2, and prove that if dim G - grade G+ = 2, then reg G = max{p + dim G - 1, r( m)}, where p is the postulation number of G and r(m) is the reduction number of m