FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2001, VOLUME 7, NUMBER 4, PAGES 983-992

## Grothendieck categories as quotient categories of $\left(R-mod, Ab\right)$

G. A. Garkusha
A. I. Generalov

Abstract

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A Grothendieck category can be presented as a quotient category of the category $\left(R-mod, Ab\right)$ 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 $\mathcal C / \mathcal S^P$, $\mathcal S^P = \\left\{C \in \mathcal C \mid \left\{\right\}_C \left(P, C\right) = 0\\right\}$ is equivalent to the module category $Mod-R$, $R =$C (P, P).

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