Atomical Grothendieck Categories

Motivated by the study of Gabriel dimension of a Grothendieck category, we introduce the concept of atomical Grothendieck category, which has only two localizing subcategories, and we give a classification of this type of Grothendieck categories. 1. Introduction. Given a Grothendieck category Ꮽ, we can associate with it the lattice of all localizing categories of Ꮽ and denote it by Tors(Ꮽ). We will show (Theorem 3.3) that if Ꮽ has Gabriel dimension, then the lattice Tors(Ꮽ) is semi-Artinian. Moreover, the Gabriel dimension of Ꮽ is exactly the Loewy length of this lattice. Example 3.4 proves that the converse statement does not hold. (Therefore, the properties of the lattice Tors(Ꮽ) do not determine the properties of the category Ꮽ.) These facts suggest introducing the concept of atomical Grothendieck category. Precisely, Ꮽ will be called atomical if the lattice Tors(Ꮽ) has only two elements, that is, Ꮽ has only two localizing subcategories, namely, {0} and Ꮽ. The classification of atomical Grothendieck categories is given in Section 4.