# Seminario: “Decomposition and classification of length functions”

## Mercoledì 11 Luglio 2018, ore 16:45 - Aula 1BC50 - Dario Spirito

ARGOMENTI: Seminari

Mercoledì 11 Luglio 2018 alle ore 16:45 in Aula 1BC50, Dario Spirito (Università di Roma Tre) terrà un seminario dal titolo “Decomposition and classification of length functions”.

Abstract
A (generalized) length function on a ring $R$ is a map from the class of $R$-modules to the set of non-negative real numbers (plus infinity) that is additive on exact sequences and such that the length of a module is the supremum of the length of its finitely generated submodules. This concept was introduced by Northcott and Reufel as a generalization of the classical length of a module; they classified those function on valuation domains, while Vámos classified all length function on Noetherian rings.
In this talk, we present some results on ways to link a length function on an integral domain $D$ to length functions on $D$-algebras; in particular, we are interested in representing a length function on $D$ as a sum of length function coming from a set of length functions, each defined on an overring of $D$. We show that standard representations can be found when $D$ admits a Jaffard family, when $D$ is Noetherian and when $D$ is a Prüfer domains such that every ideal has only finitely many minimal primes. We also show that there is a natural bijection between singular length functions and localizing systems.