Powers of Ideals : Primary Decompositions , Artin -

Let R be a Noetherian ring and I an ideal in R. Then there exists an integer k such that for all n 1 there exists a primary decomposition I n = q 1 \ \ q s such that for all i, p q i nk q i. Also, for each homogeneous ideal I in a polynomial ring over a eld there exists an integer k such that the Castelnuovo-Mumford regularity of I n is bounded above by kn. The regularity part follows from the primary decompositions part, so the heart of this paper is the analysis of the primary decompositions. In S], this was proved for the primary components of height at most one over the ideal. This paper proves the existence of such a k but does not provide a formula for it. In the paper SS], Karen E. Smith and myself nd explicit k for ordinary and Frobenius powers of monomial ideals in polynomial rings over elds modulo a monomial ideal and also for Frobenius powers of a special ideal rst studied by Katzman. Explicit k for the Castelnuovo-Mumford regularity for special ideals is given in the papers by Chandler C] and Geramita, Gimigliano and Pitteloud GGP]. Another method for proving the existence of k for primary decompositions of powers of an ideal in Noetherian rings which are locally formally equidimensional and analytically unramiied is given in the paper by Heinzer and Swanson HS]. The primary decomposition result is not valid for all primary decompositions. Here is an example: let I be the ideal (X 2 ; XY) in the polynomial ring kX; Y ] in two variables