Modul: MAT770 Oberseminar: Algebraische Geometrie

## Representability of analytic K-theory within a rigid analytic motivic homotopy category

Talk by Dr. Christian Dahlhausen

Speaker invited by: Prof. Dr. Joseph Ayoub

**Date:** 17.10.23 **Time:** 13.15 - 14.45 **Room:** Y27H25

Algebraic K-theory of smooth schemes (over a regular noetherian base scheme) is representable within Morel-Voevodsky's motivic homotopy category, wherein the affine line is contractible. For rigid analytic spaces, Ayoub developed an analogous theory wherein the closed unit ball B^1 is contractible. Within Ayoub's category, Morrow's continuous K-theory and Kerz-Saito-Tamme's analytic K-theory are not representable for two reasons: First, they are not B^1-invariant and, secondly, the mapping objects are not pro-spaces. In this talk, I will sketch the construction of a rigid analytic motivic homotopy category with coefficients in condensed spectra. By design, the rigid affine line is contractible in this category and it is canonically enriched over the category of condensed spectra. I will explain how this yields that -- after passing from pro-spaces to condensed spaces -- continuous K-theory and analytic K-theory shall be representable. Furthermore, we can identify the representing object with the image of the representing object of algebraic K-theory under a canonical analytification functor. In future work, I intend to employ this representability in order to study Adams operations, similarly to Riou's work on Adams operations on higher algebraic K-theory. In the long run, this might be useful for studying the problem of lifting algebraic cycles as studied by Bloch-Esnault-Kerz who linked this problem to the Hodge conjecture for abelian varietes.