Formal Cohomology: III. Fixed Point Theorems

Abstract

Let (R, (wi)) be a complete discrete valuation ring of characteristic 0 with quotient field K and finite residue class field k = GF(q). In our previous papers, [1] and [2], (which we shall refer to as FC I and FC II), we constructed the formal cohomology groups, Hi(A; K), of certain smooth k algebras A and developed several of their basic properties. The Frobenius endomorphism F: X Xq of A operates on the groups H'(A; K). In this paper we shall prove a fixed point theorem for the action of F, which generalizes results of and Reich and leads via techniques of to a new proof of the rationality of the zeta function of a k-variety. Applying the same techniques to the study of Fod, where j is an automorphism of A of finite order, we shall obtain a new and rather simple proof of the rationality of the Artin L-functions. Let A be a w. c. f. g. algebra over (R, (wr)). (See FC I for terminology.) A Frobenius endomorphism of A is a lifting of F: A d A, where A = A/wA. Such liftings exist provided A is very smooth. Suppose then that A is very smooth, that A is pure n dimensional and that F is a fixed Frobenius endomorphism of A. By FC I, Th. 8.6 Cor. 1, p 216, F induces a bijection F* on H(A, K). In fact Corollary 2 of the same theorem constructs an operator A: D(A) D(A) such that A* = qn(Fh)on H(A; K). The main object of this paper is to show that qlfF~l behaves like a compact operator on H(A, K) and that the alternating sum of the traces of (qnfF;l)s on the Hi(A; K) is equal to the number of points of Spec A rational over GF(qs). (It is likely that the H'(A; K) are always finite dimensional but we have only been able to prove this in general for i < 1.) In outline the paper goes as follows. Section 1 introduces a class of linear operators on vector spaces over K, the operators. The axiomatics here are essentially the Riesz-Serre decomposition theorem for compact endomorphisms of a Banach space. In particular, we are able to define the trace and characteristic power series of a nuclear operator. Section 2 studies Dwork operators. If M is a finite A-module, a