Types of Serre subcategories of Grothendieck categories

Every Serre subcategory S of an abelian category A is assigned a unique type (m,−n), where m (resp. n) counts how many times one can form left (resp. right) adjoints starting from i and Q, where i:S→A is the inclusion and Q is the quotient functor. The main result gives a complete list of all the types of Serre subcategories of Grothendieck categories:(0,0),(0,−1),(1,−1),(0,−2),(1,−2),(2,−1),(+∞,−∞). Two observations are technically crucial in proving the main result: the exactness of all the functors in a recollement of abelian categories forces the recollement to split; and any left recollement of a Grothendieck category can be extended to a recollement.