Finiteness of De Rham Cohomology

Abstract

When k = C these groups have topological significance. Grothendieck proved in [1] that HDRt(A) -H$(V;C) where V is the complex manifold attached to A. From this one can see that IDR (A) is finite dimensional when k C, and the result extends easily to all k of characteristic zero. But it is desirable (especially to a p-adic cohomologist) to give a purely algebraic proof of finite dimensionality. Such a proof has been given by Hartshorne (unpublished); it is global in nature and like Grothendieck's proof makes essential use of resolution of singularities. The object of this note is to give a purely local proof. In llartshorne's proof, the Gysin sequence is used to reduce a question concerning affine varieties to one on projective varieties. We also will use the Gysin sequence to reduce to the case when A is a localization of a polynomial ring. In this case HDR(A) turns out to be the homology of a Koszul complex of first order differential operators on a polynomial ring. Deformation techniques from [2] may then be used to handle this complex. Indeed our paper is nothing more than a simplification and reinterpretation of some results from [2] in terms of DeRham cohomology. That such translations may be made has been shown by Katz, and the map of Lemma 2. 1 relating the De Rham complex t.o a K?oszul complex on a polynomial ring comes from his thesis ( [3] ).