Formal Cohomology: I

Abstract

The aim of this, and the succeeding papers, is to construct a cohomology theory of DeRham type for non-singular varieties defined over a field k of finite characteristic, the coefficient field being a complete valued field K of characteristic 0 with residue class field k. The theory is strongly motivated by the work of Dwork, and indeed in certain cases the cohomology groups may be identified with spaces studied by Dwork. The theory, though still highly incomplete, has many pleasant features. In particular a form of the Lefschetz fixed point theorem, generalizing results of Dwork and Reich, may be proved, and this leads to a new proof, closely related to Dwork's, of the rationality of the zeta function. One of the indications that such a cohomology theory should exist is the fact that for varieties over the complexes the DeRham cohomology may be recovered algebraically. Specifically, let V be a non-singular affine complex variety, and let D(V) be the complex of algebraic differential forms on V. Then results of Atiyah, Hodge, and Grothendieck (see [7]) show that the homology of the complex D(V) is precisely the classical DeRham cohomology of V. Recently, Hartshorne, starting with this algebraic definition of DeRham cohomology, has succeeded in giving algebraic proofs of such results as finite dimensionality of the cohomology and Poincare duality; his techniques work over any ground field k of characteristic 0. When the characteristic of k is p, the algebraic definition above does not give reasonable cohomology groups. Suppose for example that V is the affine line over Z/(p). Then 1-forms on V such as X"P-'dX cannot be integrated, so the 1-dimensional algebraic DeRham cohomology of V is infinite dimensional. A natural attempt to remedy the above difficulty is to lift the coordinate ring A of V to some characteristic 0 ring A, and to use closed/exact differentials on A rather than on A. For in characteristic 0, one can introduce denominators, and integrate these troublesome forms. Now, there is a natural candidate for A. Namely, let (R, (p)) be a complete unramified discrete valuation ring of characteristic 0 with residue class field k and K = R 0z Q. Then, Grothendieck (see [3, Expose III, Cor. 6.10, p. 25]) has shown that there is a unique R algebra As complete in the p-adic topology, flat over R, and reduc-This content downloaded from 124.35.175.171 on Thu, 19 Jun 2025 09:31:41 UTC All use subject to https://about.jstor.org/terms