In a series of four talks, we will give a systematical introduction of the theory of coherent functors, introduced by M. Auslander. Let R be an associative ring with identity 1. A coherent functor is defined to be a finitely presented additive functor F : mod-R -> Ab, where mod-R denotes the category of finitely presented right R-modules. The most important coherent functor is the forgetful functor (R,-) := Hom (R,-).

In the first talk, we will define the category fp(mod-R, Ab) of coherent functors and give other examples of a coherent functor. Then we will use Yoneda's Lemma to prove Auslander's result that the category of coherent functors is abelian. In the second talk, we describe the duality D: fp(mod-R, Ab) --> fp(R-mod, Ab) of Auslander and Gruson-Jensen. We will also use positive primitive formulae from the language of left R-modules to describe the lattice Sub (R,-) of subobjects of the forgetful functor.