Annihilators of local cohomology in characteristic zero

Abstract

This paper discusses the problem of whether it is possible to annihilate elements of local cohomology modules by elements of arbitrarily small order under a fixed valuation. We first discuss the general problem and its relationship to the Direct Summand Conjecture, and next present two concrete examples where annihilators with small order are shown to exist. We then prove a more general theorem, where the existence of such annihilators is established in some cases using results on abelian varieties and the Albanese map. 1. Almost vanishing of local cohomology The concept of almost vanishing that we use here comes out of recent work on Almost Ring Theory by Gabber and Ramero [4]. This theory was developed to give a firm foundation to the results of Faltings on Almost etale extensions [3], and these ideas have their origins in a classic work of Tate on p-divisible groups [21]. The use of the general theory, for our purposes, is comparatively straightforward, but it illustrates the main questions in looking at certain homological conjectures, as discussed later in the section. The approach is heavily influenced by Heitmann’s proof of the Direct Summand Conjecture for rings of dimension three [8]. Let A be an integral domain, and let v be a valuation on A with values in the abelian group of rational numbers; more precisely, v is a function from A to Q ∪ {∞} such that (1) v(a) =∞ if and only if a = 0, (2) v(ab) = v(a) + v(b) for all a, b ∈ A, and (3) v(a+ b) > min{v(a), v(b)} for all a, b ∈ A. Received November 6, 2006; received in final form April 4, 2007. 2000 Mathematics Subject Classification. Primary 13D22. Secondary 13D45, 14K05. P.R. and A.K.S. were supported in part by grants from the National Science Foundation. V.S. was supported by a Swarnajayanthi Fellowship of the DST. c ©2007 University of Illinois