REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: Auslander's study of finitely presented functors has had enormous impact in the field of representation theory.  According to Auslander himself, many of his results were discovered by looking at representation theory from the functorial point of view.

In this talk I will recall the defect functor studied by Auslander and discuss some major applications of the defect functor to representation theory. In particular I will show how to obtain almost split sequences from the defect, how the defect regulates the dual of a finitely presented functor, and how the defect approximates finitely presented functors by representable functors.

I will finish with a discussion of the Auslander-Gruson-Jensen duality. This duality, originally discovered by Auslander and independently discovered by Gruson and Jensen, has also been recovered using model-theoretic techniques by Prest, Herzog, and Burke, and later -- using functorial techniques -- by Hartshorne.  I will include a proof that the category of finitely presented functors from an abelian category with enough projectives always has enough injectives. These injectives provide yet another way of recovering the Auslander-Gruson-Jensen duality. This duality is used to extend the notion of horizontal linkage from algebraic varieties over a commutative noetherian ring to finitely presented functors over an arbitrary coherent ring.