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-mod, 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 Mod-R, R = C (P, P).

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