Powers of Ideals: Primary Decompositions, Artin-Rees Lemma and Regularity

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 find explicit k for ordinary and Frobenius powers of monomial ideals in polynomial rings over fields modulo a monomial ideal and also for Frobenius powers of a special ideal first 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 unramified 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, XY ) in the polynomial ring k[X,Y ] in two variables X and Y over a field k. For each positive integer m, I = (X) ∩ (X, XY, Y ) is an irredundant primary decomposition of I. However, for each integer k there exists an integer m, say m = k + 1, such that (X,Y ) 6⊆ (X, XY, Y ). Hence the result can only