# Seminario di Algebra: Length functions of Grothendieck categories with applications to infinite group representations I

## Mercoledì 15 Ottobre 2014, ore 16:15 - Aula 2AB45 - Simone Virili

ARGOMENTI: Seminari

Mercoledì 15 Ottobre 2014 alle ore 16:15 in aula 2AB45, Simone Virili, assegnista del nostro Dipartimento, terrà il seminario su argomenti della sua tesi di dottorato "Length functions of Grothendieck categories with applications to infinite group representations I"

Abstract
In any Grothendieck category (e.g. a category of modules), one can define the composition length of the objects. A length function is a numerical invariant on the objects of the category with properties analogous to that of the composition length.
In this first talk we introduce the concept of Gabriel dimension of a Grothendieck category and we use it to classify all the length functions of a Grothendieck category with Gabriel dimension. The formalism of torsion theories and localization is fundamental in this part.
Given a ring R, a group G and a fixed crossed product R*G (e.g. the usual group algebra R[G]) we introduce a suitable compatibility condition of a length function L of R-Mod with R*G. We can then formulate the problem we are interested in:
Main Question. Is it possible to extend a length function L of R-Mod, which is compatible with R*G, to a length function R*G-Mod?