Recollements of Module Categories

Abstract

We establish a correspondence between recollements of abelian categories up to equivalence and certain TTF-triples. For a module category we show, moreover, a correspondence with idempotent ideals, recovering a theorem of Jans. Furthermore, we show that a recollement whose terms are module categories is equivalent to one induced by an idempotent element, thus answering a question by Kuhn. IntroductionA recollement of abelian categories is an exact sequence of abelian categories where both the inclusion and the quotient functors admit left and right adjoints. They first appeared in the construction of the category of perverse sheaves on a singular space by Beilinson, Bernstein and Deligne ([5]), arising from recollements of triangulated categories with additional structures (compatible t-structures). Properties of recollements of abelian categories were more recently studied by Franjou and Pirashvilli in [12], motivated by the MacPherson-Vilonen construction for the category of perverse sheaves ([23]).Recollements of abelian categories were used by Cline, Parshall and Scott to study module categories of finite dimensional algebras over a field (see [28]). Later, Kuhn used them in the study of polynomial functors ([20]), which arise not only in representation theory but also in algebraic topology and algebraic K-theory. Recollements of triangulated categories have appeared in the work of Angeleri Hügel, Koenig and Liu in connection with tilting theory, homological conjectures and stratifications of derived categories of rings ([1], [2], [3]). In particular, Jordan-Hölder theorems for recollements of derived module categories were obtained for some classes of algebras ([2], [3]). Also, Chen and Xi have investigated recollements in relation with tilting theory ([7]) and algebraic K-theory ([8]). Homological properties of recollements of abelian and triangulated categories have also been studied in [29].Recollements and TTF-triples of triangulated categories are well-known to be in bijection ([5], [24], [25]). We will show that such a bijection holds for Mod-A (see Proposition 5.2), where Mod-A denotes the category of right A-modules, for a unitary ring A. Similar considerations in Mod-A were made for split TTF-triples in [26]. More generally, we show that recollements of an abelian category A (up to equivalence) are in bijection with bilocalising TTF-classes (see Theorem 4.3).Examples of recollements are easily constructed for the module category of triangular matrix rings (see [9], [17], [22]) or, more generally, using idempotent elements of a ring (see Example 2.9). In fact, we will see that there is a correspondence between idempotent ideals of A and recollements of Mod-A, recovering Jans' bijection ([19]) between TTF-triples in Mod-A and idempotent ideals. Moreover, Kuhn conjectured in [20] that if the categories of a recollement are equivalent to categories of modules over finite dimensional algebras over a field, then it is equivalent to one arising from an idempotent element. In our main result, we prove this ...

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