Universität Göttingen, Deutschland
I will survey my joint work with Guillermo Cortiñas and other coauthors about analytic cyclic homology theories for algebras over nonarchimedean local fields. The point of this work is to define a homology theory for noncommutative algebras in characteristic p by lifting them to suitably completed algebras over the p-adic integers and then using cyclic homology. The relevant completion is understood very nicely in the setting of bornological algebras. Ideally, the theory should generalise rigid cohomology for commutative algebras, which in turn generalises Monsky-Washnitzer cohomology. At the moment, however, this goal has not yet been reached. On the one hand, the periodic cyclic homology of the appropriate liftings is known to agree with rigid cohomology in the commutative case. On the other hand, the analytic cyclic homology for suitable dagger algebras is known to depend only on their reduction mod p.