On a theorem of auslander and applications

Throughout t,his paper A will denote an artin algebra and modA the category of finitely generated right A-modules. Let M be an A-module and denote by addM the fill1 subcategory of modA consisting of the direct sums of direct summands of M . In [All M.Auslander considered the full subcategory Cy of modA consisting of the modules X having a presentation MI Mo -+ X -+ 0 wit,h Mt in addM, such that the induced sequence HomA(M, MI) -t HomA(M, Mo) -, HomA(hl, X) -+ 0 is exact, and proved that the restriction of the functor FM = HomA(hf, -) : modA -+ m o d E n d ~ ( M ) to Cp is full and faithful. T h w FM induces an equivalence between subcategories of modA and modEndA(M) respectively. More precisely, between Cy and h ( F M I c y ) , the image of the re~t~riction of the functor FM to C;f. When the module M is projective one getas the well known equivalence betweefl the subcategory of modA consisting of t,hc modules with a presentation in addM and modEndA(M). This paper is motivated by the above result. We start by defining for an A-module M and for every n > 0 full subcategories Cf of modA consisting of the modules X such that there is an exact sequence Mn -+ . . . -+ MI -, Mo