FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

Grothendieck categories as quotient categories of (R-mod, Ab)

G. A. Garkusha
A. I. Generalov

Abstract

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A Grothendieck category can be presented as a quotient category of the category $(R\mathrm{-mod}, \mathrm{Ab})$ 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 $\mathrm{Mod-}R$, $R = {}_C (P,P)$.

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