A narrow over-ring adjustment functor

Abstract

such that Xi is finitely generated, XL is finitely cogenerated, 9 is surjective, and the A-homomorphism Cp: XL -+ Hom,( aN,, Xg) adjoint to rp is injective. The category of adjusted right R-modules will be denoted by adj$R). Note that if R is an Artin algebra then X is in adj$R) if and only if X is finitely generated, sot X is a B-module, and top X is an A-module via the natural epimorphisms A t R -+ B. In this case adj i( R) is one of the classes of modules considered by Auslander and Smal@ [l] and therefore adj$( has the Auslander-Reiten sequences [ 1, Proposition 6.51. The category adjg(R) was introduced in [25,26] in order to give a module-theoretical interpretation of arbitrary bimodule matrix problems in the sense of Drozd and Roiter. It was shown in [26] that if N: KoP x k --+ mod(k) is an additive k-functor, k is a field, K and L are additive Krull-Schmidt k-categories, and Mat( JV,) is the category of &,-matrices then there are a faithfully right-B-traced ring R and a full dense functor 0: Mat( wL,) -+ adj$(R) preserving the representation type and such that 0 vanishes only on a small set of trivial idecomposable matrices (see also [14]). This allows to show in [14, 261 that the category Mat(,N,) has Auslander-Reiten sequences (see also [30)). Moreover, a Coxeter map is defined in terms of 0 which provides us a useful computa-