The overconvergent site (Q2885313)

``Rigid cohomology is a cohomology theory for algebraic varieties over a field \(k\) of positive characteristic \(p\) with values in a vector space over a \(p\)-adic field \(K\) whose residue field is \(k\).''NEWLINENEWLINERigid cohomology is of `de Rham type' in that it is defined in terms of certain de Rham complexes on smooth formal liftings to characteristic \(0\). It is natural to ask if -- like usual de Rham cohomology of algebraic varieties over a field of characteristic \(0\), which is known to be equivalent with the cohomology of the infinitesimal site -- also rigid cohomology has an interpretation as the cohomology of an analogous site. As long as one is interested in smooth proper \(k\)-varieties only, Berthelot's crystalline site is the perfect solution to this problem (in fact this concerns crystalline cohomology rather then rigid cohomology), but for more general varieties this cannot work.NEWLINENEWLINEIn the present book, Bernard Le Stum defines such a site, the `overconvergent site'. By its very definition as given by Pierre Berthelot, rigid cohomology is defined by means of \textit{chosen} embeddings of the \(k\)-varieties in question (or just of sufficiently small open subsets of them) into smooth formal schemes over mixed characteristic valuation rings. It is then a theorem that the definitions do not depend on the choices made, that they glue as desired and make up a completely functorial theory. The point in introducing the overconvergent site is to \textit{intrinsically} build in functoriality into the theory.NEWLINENEWLINE``Of course, this article owes much to Berthelot's previous work on rigid cohomology\dots We only want to rewrite his theory with a slightly different approach.''NEWLINENEWLINEWhile rigid cohomology theory is classically formulated in terms of Tate's theory of rigid spaces, here Berkovich analytic spaces are used instead.NEWLINENEWLINEThe book consists of three chapters. In the first chapter the necessary geometric foundations are laid, culminating in the local section theorem. The overconvergent site is defined. In the second chapter it is shown that finitely generated modules (over structure sheaves in the overconvergent site) correspond to overconvergent isocrystals (as defined in the classical theory). In the third chapter it is shown the cohomology coincides, as desired.NEWLINENEWLINEThe first appendix section provides background material on sites and toposes, the second one on analytic varieties.