On the cohomology of nash sheaves

Abstract

NASH introduced in [4] a concept of real algebraic manifold, and in [I], Artin and Mazur made precise the appropriate category. For definitions and examples, see [I] and [3]. These structures, which will be called Nash manifolds in this paper, occupy an intermediate position between real algebraic varieties and differentiable manifolds; in particular they sometimes allow the use of algebraic techniques in differential topology. For a highly successful example, see [l]. It had been hoped that the techniques of sheaf cohomology, which have proved so powerful in algebraic geometry, could be applied. The object of this paper is to show that this is not the case. Indeed, the only reasonable known method of computing sheaf cohomology is the tech construction, and because of the direct limit involved, it is essential to have open coverings by cohomologically trivial subsets; Stein manifolds play this part in complex analytic geometry, and affine schemes in algebraic geometry. The obvious candidates in the Nash category would be open balls with their canonical structure. It is easy to see that if any cohomologically trivial subsets of an arbitrary Nash manifold exist, then these open sets must be among them. Unfortunately, they are not. We shall show that on an open interval there are tech cocycles which are not coboundaries. In fact, consider the open interval (1 E, f 1 + E) for some positive E. (( 1 E, + l), (1, + 1 + s)} is an open cover, and a I-cocycle for this cover is an algebraic functionfon