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,-).

The category of coherent functors enjoys the property of being the universal abelian category over the one-object category R. The Serre subcategories of fp(mod-R, Ab) play a role analogous to that of the two-sided ideals of R in the classical theory of rings. If M is a right R-module the Serre subcategory S(M) of coherent functors that vanish at M corresponds to the annihilator ideal in the classical theory. In the third and fourth talks, we describe the localization of fp(mod-R, Ab)  at S(M) in two important cases:

a. M = E, the minimal injective cogenerator over a right coherent ring; and

b. M = k[x,y], the affine plane, considered as module over the Lie algebra sl(2,k), where k is a field of characteristic 0.