Nonarchimedean bornologies, cyclic homology and rigid cohomology

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.