In the early 70's, M. Auslander initiated a novel approach to the study of modules over artinian rings: rather than dealing directly with the modules, one works in the functor category (mod-R,Ab) of additive functors from the category of finitely presented modules to the category of abelian groups. In other words, one studies modules over the category of finitely presented modules. It might seem that this is piling complication upon complication, but Auslander's approach has been remarkably successful.

One main point of this chapter is to reconsider some of Auslander's results, especially the functorial characterisation of rings of finite representation type, in terms of pp formulas and pp-types. The other main purpose is to set down the material on pp-types and functors which should be useful in the classifications of infinite-dimensional indecomposable pure-injectives over particular (classes of) algebras.

Let U be the forgetful functor from mod-R to Ab – the functor which simply forgets that an R-module is anything more than an abelian group. If φ is a pp formula in one free variable, then the assignment M↦φ(M), with the induced action on morphisms, is a functor, Fφ, from mod-R to Ab: indeed, it is a subfunctor of U. We see in §1 that every subfunctor of U is a (possibly infinite) sum of such functors induced from pp formulas. It is also shown that the Fφ are finitely presented functors and that, if one allows pp formulas in more than one free variable, the Fφ are generating. Therefore, one may use these “pp-functors” in place of the more usual representable functors (M,−).