Abstract
such that 01 is an epimorphism of rings and B is a finitely generated projective right A-module through a. (Properly speaking, OL is the localization.) Suppose B is a finite left localization of A and let Mod A, Mod B be the respective categories of finitely generated left modules. The following facts can be found in [6, Sect. 4.161. The functor T = B @A -: Mod A Mod B is exact, the forgetful functor S: Mod B -+ Mod A is also exact, and TS is naturally equivalent to the identity on Mod B. Further, let Tor(A, B) be the full subcategory of Mod A whose objects X satisfy TX = 0. Then Tor(A, B) is a localizing (Serre) subcategory of Mod A, and the quotient category Q
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