# Seminario: On a conjecture of M. Auslander

## Martedì 16 Giugno 2015, ore 11:30 - Aula 1BC50 - Alex Martsinkovsky

ARGOMENTI: Seminari

Martedì 16 Giugno 2015 alle ore 11:30 in Aula 1BC50, Alex Martsinkovsky (Northeastern University - Boston) terrà un seminario dal titolo "On a conjecture of M. Auslander".

Abstract
In his foundational La Jolla paper on coherent functors, M. Auslander described injective objects in certain functor categories as direct summands of the covariant Ext-functors, and conjectured that they are again Ext-functors. He established that result in the case the fixed argument was of finite projective dimension. In the same volume, P. Freyd gave a positive answer in the case the underlying abelian category has denumerable sums. Later, Auslander gave a unifying proof of these results, but also showed that the conjecture is not true in general.
In this talk, I will give a positive answer to the conjecture in the seemingly overlooked case when the contravariant argument is an object of finite length. Moreover, our result is established for any additive bifunctor whose endomorphisms lift to endomorphisms of the fixed argument. That this is the case for the Ext-functor is a consequence of the Hilton-Rees theorem, for which we give a short proof. Other immediate applications include Hom modulo an ideal, and, when the fixed argument is finitely presented, the Tor-functor.
No prior experience with coherent functors is needed for this expository talk. All concepts will be defined and explained during the talk.