Local Cohomology of Segre Product Type Rings

The aim of this paper is to investigate properties of the local cohomology of rings of mixed characteristic that are analogous to Segre products of rings de ned over a eld. The main question is whether the local cohomology can be almost killed in a nite extension (we de ne what this means below). There are two reasons for considering this type of ring. First, there are special properties of these rings that make it possible to answer this question. Second, and perhaps more important, Segre products are a large source of normal non-Cohen-Macaulay domains; in fact, many examples of such rings that are de ned by other means turn out to be Segre products. A theorem of Goto and Watanabe [1] gives a formula for the local cohomology of the Segre product in terms of the local cohomology of the factors; this both gives a method for constructing examples of normal rings with local cohomology in given degrees and provides a method for analyzing the questions we are considering. The general question behind this research is whether the absolute integral closure R + of a normal domain R that is either complete local or of nite type over a eld is almost Cohen-Macaulay. The ring R + is known to be Cohen-Macaulay in positive characteristic by results of Hochster and Huneke [4] and Huneke and Lyubeznik [5]. For rings of mixed characteristic, which is the most interesting case since many of the homological conjectures are open in that case (see for example Hochster [3]), R. Heitmann [2] showed that R + is almost Cohen-Macaulay in dimension three. The higher dimensional case, as well as the question of whether R + is Cohen-Macaulay in dimension 3, are still open. In the equicharacteristic case, there are some examples where elements of local cohomology can be almost killed in Roberts, Singh and Srinivas [10], but the general situation is not known. In this paper we describe some examples of non-Cohen-Macaulay rings that arise from Segre products and show that the elements of local cohomology that make the ring non-Cohen-Macaulay can be almost killed in R + .