Endomorphism rings and category equivalences
If \({\mathcal C}\) is a locally finitely generated Grothendieck category and \(M\in Ob {\mathcal C}\), \(S=End_{{\mathcal C}}(M)\), the class of the M- distinguished objects of \({\mathcal C}\) is the torsion free class of a torsion theory (T,F) of \({\mathcal C}\) and the functor \(Hom_{{\mathcal C}}(M\),- ): \({\mathcal C}\to S\)-mod induces an equivalence between the quotient cateogory of \({\mathcal C} modulo T\) and a certain quotient category (S,\({\mathcal F})\)-mod of S-mod. This quotient category consists of all the S-modules iff M is a finitely generated quasiprojective object of \({\mathcal C}\) which is CQF-3. The properties of this construction are studied. If M is codivisible, then three different full subcategories of \({\mathcal C}\) exist and they are equivalent to (S,\({\mathcal F})\)-mod. These results are applied to characterize the modules M such that the endomorphism ring S of M is a left semihereditary ring, a left CS-ring or a left continuous ring.
locally finitely generated Grothendieck category
scientific article; zbMATH DE number 4122169
scientific article