Séminaire de Mathématique
# On the Category of Localizing Motives

→
Europe/Paris

Amphithéâtre Léon Motchane (IHES)
### Amphithéâtre Léon Motchane

#### IHES

Le Bois Marie
35, route de Chartres
91440 Bures-sur-Yvette

Description

I will explain recent new results about the category of localizing motives -- the target of the universal localizing invariant of stable *k*-linear infinity-categories (over some base *k*), commuting with filtered colimits. In particular, I will explain the most striking property of this category: it is rigid as a large symmetric monoidal category (in the sense of Gaitsgory and Rozenblyum).

I will also explain how to compute morphisms in this category, obtaining an effective description of the algebraic version of *K*-homology and more generaly of Kasparov's *KK*-theory. As a special case, we will deduce the corepresentability of *TR* (by the reduced motive of the affine line) and of the topological cyclic homology (by the unit object of the kernel of *A*^{1}-localization), when restricted to the motives of connective *E*_{1}-rings. Another special case is the comparison theorem of two approaches to *K*-theory of formal schemes: the classical continuous *K*-theory is equivalent to the K-theory of the category of nuclear modules, which was defined by Clausen and Scholze.

If time permits, I will explain an application to the p-adic analogue of the lattice conjecture. Namely, we construct a symmetric monoidal functor from smooth and proper dg categories over *C _{p}* to perfect modules over the

========

Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: "subscribe seminaire_mathematique PRENOM NOM"

(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Organized by

Dustin Clausen

Contact