Weak completions, bornologies and rigid cohomology

Abstract

Let V be a complete discrete valuation ring with residue field k of positive characteristic and with fraction field K of characteristic 0. We clarify the analysis behind the Monsky-Washnitzer completion of a commutative V -algebra using completions of bornological V -algebras. This leads us to a functorial chain complex for a finitely generated commutative algebra over the residue field k that computes its rigid cohomology in the sense of Berthelot.

Noncommutative Geometry, the spectral standpoint

“…This resurgence of cyclic homology in the area of number theory was totally unexpected and is a witness of the coherence of the general line of thoughts. In fact the recent work [96,97] of G. Cortinas, J. Cuntz, R. Meyer and G. Tamme relates rigid cohomology to cyclic homology.…”