Nonarchimedean bornologies, cyclic homology and rigid cohomology

Abstract

Let V be a complete discrete valuation ring with residue field k and with fraction field K of characteristic 0. We clarify the analysis behind the Monsky–Washnitzer completion of a commutative V -algebra using spectral radius estimates for bounded subsets in complete bornological V -algebras. This leads us to a functorial chain complex for commutative k-algebras that computes Berthelot’s rigid cohomology. This chain complex is related to the periodic cyclic homology of certain complete bornological V -algebras.