On the tensor product of Grothendieck categories (Julia Ramos González)Séminaire « Topologie »
M3 - Salle des Séminaires
Orateur : Julia Ramos González
Lieu : salle des Séminaires M3 Résumé : Grothendieck abelian categories are used, in a categorical approach to noncommutative algebraic geometry, as models for noncommutative schemes. In the first part of the talk we define a tensor product of Grothendieck categories. Our main tool is the Gabriel-Popescu theorem, and more concretely its generalization by Lowen, which allows to present Grothendieck categories as localizations of module categories. In particular, we define compatible tensor products in the three different equivalent approaches to the localization theory of module categories (linear Grothendieck topologies, localizing Serre subcategories and strict localizations) which give rise to a well-defined tensor product of Grothendieck categories. In the second part of the talk, we relate the tensor product of Grothendieck categories to the Bird-Kelly tensor product of locally presentable categories. This allows us to use Brandenburg’s work on categorical algebraic geometry to show that the tensor product of Grothendieck categories generalizes the product of quasi-compact quasi-separated schemes and thus can be seen as the correct notion of product of noncommutative schemes. This is joint work with Wendy Lowen and Boris Shoykhet.