Gersten's conjecture for arithmetic surfaces

Abstract

One of the more promising applications of higher algebraic K-theory is to the study of algebraic cycles in algebraic geometry. By verifying a certain conjecture of Gersten for regular schemes of finite type over a field, Quillen has established a connection between the Chow group and certain sheaf cohomology groups defined in terms of K-theory [3]; Grayson has recently extended this result [2]. Their work indicates that verification of Gersten’s Conjecture might eventually prove to be the key step in developing intersection theory for regular schemes of unequal characteristic. For A a regular local ring, let &'(A) denote the (abelian) category consisting of those finitely generated A-modules M such that the support of M is of codimension at least p in Spec A. Gersten’s Conjecture is that the inclusion of categories JUPt’(A) -+ JGIP(A) induces the zero map of K-groups K,,(JH~“(A)) -+ K,(JUP(A)) for all n 2 0, p Z 0. In the paper in which he first stated the Conjecture, Gersten took the first step for regular local rings of unequal characteristic by proving the Conjecture for discrete valuation rings with finite residue class field [ 11. In this paper we take the next step by proving the Conjecture for the local rings of Spec R[t], where R is any Dedekind domain whose maximal ideals have finite residue class field; in particular, this applies when R is the ring of integers of an algebraic number field. The main theorem of this paper was first proved in the author’s 1976 doctoral dissertation at Rice University. The proof given here is basically the same, although the presentation has been greatly simplified by using the techniques developed in [5].