Tuesday, March 03, 15:15, 7.527

Simone Virili (Barcelona), On the space of Sylvester rank functions.

Abstract:
Given a ring R, let P(R) be the space of Sylvester rank functions on the category of finitely presented modules mod-R. Recall that P(R) can also be described as the space of normalized additive functions on the abelianization Ab(R), which is the category of finitely presented objects in the additive functor category [R-mod,Ab]. Compared to mod-R, the category Ab(R) has a much richer structure, which can be used for the study of P(R). In this talk we will discuss two such applications: for the first one we will describe the space P_ex(R) of exact rank functions, while for the second we will describe the irreducible discrete rank functions using the topology of the Ziegler spectrum. As a concrete application, we will give a complete description of the space P(R) when R is a Dedekind domain with a polynomial identity.