Castelnuovo-mumford Regularity of Products of Ideals

This is not the case in general. There are examples already with M = I such that reg(I) > 2 reg(I), see Sturmfels [15] and Terai [16]. On the other hand, Chandler [5] and Geramita, Gimigliano and Pitteloud [11] have shown that reg(I) ≤ k reg(I) holds for ideals with dimR/I ≤ 1. In general one has that reg(I) is asymptotically a linear function of k, see [14, 8]. If one takes I = m and M any graded R-module, then reg(mM) ≤ reg(M) + 1 holds. So it is natural to ask whether (1) holds whenever I is generated by a regular R-sequence or at least by a sequence of linear forms. Unfortunately this is also not the case, even when M is a monomial ideal with a linear resolution and I is generated by a subset of the variables, see Example 2.1. The purpose of this note is to describe some cases where (1) is nonetheless valid. In Section 1 we recall some generalities about regularity and show in Section 2 that (1) is valid for ideals generated by sequences which are almost regular with respect to M and regular with respect to R, see 2.3. For example, any generic sequence of homogeneous forms of length ≤ dimR has these properties. We also show the validity of (1) when the dimension of I is ≤ 1. The argument is similar as in the corresponding result of Chandler. More surprising is the fact, proved in Section 3 (Theorem 3.1), that any product of ideals of linear forms has a linear resolution. This is obtained as a consequence of a description of a primary decomposition of such an ideal, see 3.2. In Section 4 we consider ideals with linear quotients, that is, ideals which can be generated by a minimal system of generators whose successive colon ideals are generated by linear forms. Examples of such ideals are stable, and squarefree stable ideals in the sense of Eliahou-Kervaire [10] and Aramova-Herzog-Hibi [1], as well as polymatroidal ideals, as noted in [13]. Again it turns out that the property of having linear quotients is not preserved under taking products or powers. However we show in Section 5 that products of polymatroidal ideals are again polymatroidal,