Types of Serre subcategories of Grothendieck categories

Abstract

Every Serre subcategory of an abelian category is assigned a unique type. The type of a Serre subcategory of a Grothendieck category is in the list: $$(0, 0), \ (0, -1), \ (1, -1), \ (0, -2), \ (1, -2), \ (2, -1), \ (+\infty, -\infty);$$ and for each $(m, -n)$ in this list, there exists a Serre subcategory such that its type is $(m, -n)$. This uses right (left) recollements of abelian categories, Tachikawa-Ohtake [TO] on strongly hereditary torsion pairs, and Geigle-Lenzing [GL] on localizing subcategories. If all the functors in a recollement of abelian categories are exact, then the recollement splits. Quite surprising, any left recollement of a Grothendieck category can be extended to a recollement; but this is not true for a right recollement. Thus, a colocalizing subcategory of a Grothendieck category is localizing; but the converse is not true. All these results do not hold in triangulated categories.