Grothendieck categories as quotient categories of $(R\mathrm{\text{-}mod},\mathrm{Ab})$

G. A. Garkusha ; A. I. Generalov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 7 (2001) no. 4, pp. 983-992.

Résumé

A Grothendieck category can be presented as a quotient category of the category $(R\mathrm{\text{-}mod},\mathrm{Ab})$ of generalized modules. In turn, this fact is deduced from the following theorem: if $\mathcal C$ is a Grothendieck category and there exists a finitely generated projective object $P\in\mathcal C$, then the quotient category $\mathcal C/\mathcal S^P$, $\mathcal S^P=\{C\in\mathcal C \mid{}_C(P,C)=0\}$ is equivalent to the module category $\mathrm{Mod\text{-}}R$, $R={}_C(P,P)$.