Localizations of endomorphism rings and fixpoints

Abstract

Let U E ‘8 be an object of an abeiian category with exact direct limits. Let A = [U, U] be the endomorphism ring of U and let Modn be the category of right A-modules. For each X E 2l the ring A acts on the group [U, X] of all morphisms U X by means of composition and the functor [U, -I: % + Mod,r , X [U, A] has a left adjoint @)A U: Mod, + %. Let FI the motivation being the theorem of Gabriel and Popescu [4]. For this, consider the filter 5 of all right ideals I C .A which cover U in the sense that U = uvel im y, where im y denotes the image of y: U + U. Let (Mod& be the full subcategory of all modules YE ModA such that the restriction map [A, Y] [I, Y] is bijective for each 1~ 3. Then by [S, 8.6(b)] the inclusion (Mod& -+ Mod=,, has a left adjoint 3-10~: Mod,, -+ (Mod& (= g-localization). In general the functor @lot is not exact. Roughly speaking, the main result is a necessary and sufficient condition on C for the composite F,(S) (Mod& , X$‘$loc[tr, X] to be an equivalence and for I in particular when % is a category of modules over an h-local domain or a Dedekind domain. If U is a direct sum of finitely generated modules or if U is divisible or torsion or algebraically