Tuesday, June 17, 16:30, 7.527

Simone Virili (Barcelona), A study of Sylvester rank functions via functor categories.

Abstract:
Given a ring R, a Sylvester rank function on the category of finitely presented right R-modules is an isomorphism-invariant function which is additive on coproducts, subadditive on right-exact sequences, monotone on quotients, and taking the value 1 on R. In this talk I will start by observing that any Sylvester rank function can be uniquely extended to a so-called length function on the category of functors from finitely presented left R-modules to Abelian groups.
This enlargement of the setting comes with many advantages, in fact, the richer structure of the functor category makes it possible to use tools like torsion-theoretic localizations, rings of definable scalars and the Ziegler spectrum in the study of rank functions. To illustrate this, I will provide examples of results about rank functions whose initial proofs are technically demanding, yet can be derived almost effortlessly within the expanded framework.