In this chapter we will introduce Auslander's general theory by which he reached the idea of AR sequences. We do this by considering Auslander categories. The proof of Theorem (4.18) is one of our main purposes here. Theorem (4.22) is also a remarkable result due to Auslander. The theorems can be stated without using Auslander categories, but they are required in its proof.

We keep the notation of the previous chapter, so that R is a Henselian CM local ring with maximal ideal m and with residue field k. The dimension of R is denoted by d. Furthermore we always assume that R has the canonical module KR. We denote by ℭ(R), or more simply ℭ, the category of all CM modules over R and R-homomorphisms. We also denote by (Ab) the category of all Abelian groups.

The idea is to move our attention from ℭ into the category of functors on ℭ.

(4.1) DEFINITION. Denote by Mod(ℭ) the category of contravariant additive functors from ℭ to (Ab). Namely, objects in Mod(ℭ) are the contravariant functors F ℭ → (Ab) with F(M ⊕ N) = F(M) ⊕ F(N) for any M and N in ℭ and, morphisms from F to G are the natural transformations of functors from F to G.