Conference abstracts

Session S04 - Operator Algebras

July 26, 15:00 ~ 15:50

## Nonarchimedean bornological algebras and their cyclic homology

### Universidad de Buenos Aires, Argentina   -   gcorti@dm.uba.ar

Let $V$ be a complete discrete valuation domain with maximal ideal $\pi V$, fraction field $K=V[\pi^{-1}]$, and residue field $k=V/\pi V$. We are interested in developing a bivariant cohomology theory for $k$-algebras which takes values in $K$-vector spaces and has all the good properties (homotopy invariance, Morita invariance, excision, agreement with the relevant variant of de Rham cohomology in the commutative case, etc.). We assume that $K$ has characteristic zero, but make no assumption on the characteristic of $k$; in fact the main case for us is $\mathrm{char}(k)=p>0$. The general idea is to associate to each $k$-algebra $A$ a (pro-) $K$-algebra $T(A)$ and take (some variant of) the periodic cyclic homology of $T(A)$. Such a construction already exists for commutative $k$-algebras of finite type; it yields Bertherlot's rigid cohomology, which is the correct variant of de Rham cohomology in this setting. In this talk I will explain a result we have interpreting rigid cohomology (made $2$-periodic) of a commutative $k$-algebra $A$ of finite type as the periodic cyclic homology of a certain pro-complete bornological $K$-algebra $T(A)$. Along the way I will discuss bornological $V$ and $K$-algebras,

Joint work with Joachim Cuntz (Universit\"at M\"unster) and Ralf Meyer (Universit\"at G\"ottingen).