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Computational approach to the Artinian conjecture

Le : 08/04/2013 14h00
Par : Phillip Linke (Bielefeld, Allemagne)
Lieu : I 001
Lien web :
Résumé : What is generic representation theory? When looking at the category of functors Func(mod F_q, Mod F_q) for a finite field F_q, we obtain that a functor F generically gives rise to representations of GL(V ) for all spaces V. By the Yoneda lemma, for each V, F [Hom(V, -)] is projective; such a projective is called a standard projective. It turns out that these standard projectives generate the whole category. In the 1980s Lionel Schwartz conjectured that all the standard projectives are noetherian. If true this would imply that every finitely generated functor admits a projective resolution by finitely-generated projectives. There are partial results that back up this conjecture but no solution so far. In the talk we will not reach quite as far. The aim is to give an idea why the category of functors is at least coherent. That means that every finitely presented functor admits a resolution by finitely generated projectives. To get to this goal we will use certain combinatorial properties of the dimension function n ->dim F(F^n) for a functor F.