Etale Cohomology of Rigid Spaces

Abstract

Fundamental groups, étale topology and étale cohomology have been conceived in algebraic geometry as a theory which captures topological properties, well known for real and complex varieties. Especially for algebraic varieties over a field of positive characteristic, this theory produces surprising analogies with the algebraic topology of real or complex varieties. One of the early successes is of course the proof of the Weil conjectures. For rigid spaces, a first glimpse of étale cohomology appeared in the work of Drinfel’d on the Langlands conjectures for function fields in positive characteristic. For certain rigid spaces, e.g., the Drinfel’d symmetric spaces Ω(ⁿ) (see Examples 7.4.9) and more general symmetric spaces, étale cohomology forms the basis for the study of automorphic representations and Galois representations. There are many names attached to these “p-adic theories” (P. Schneider, U. Stuhler, M. Rapoport, P. Deligne, Th. Zink et al.) Etale cohomology for rigid spaces is developed by V.G. Berkovich, O. Gabber (unpublished), R. Huber, A.J. de Jong, M. van der Put, K. Fujiwara et al. Berkovich, in the paper [21], develops an étale cohomology for “analytic spaces”. This category of analytic spaces was introduced in [22] and extended in [21]. It is different from the category of rigid spaces. For this reason we will not borrow from his work. However, we have to mention that the approach taken here, in some sense, does not differ from his. Furthermore, using the equality of Berkovich cohomology with the one presented here in the case of paracompact varieties (see [143], Section 8.3), all the results presented here are in principle deducible from the references [21, 20, 19, 18]. R. Huber constructed an étale cohomology theory for his adic spaces. This theory specializes to a theory for rigid spaces, too.