TTF Triples in Functor Categories

Hügel, Lidia Angeleri; Bazzoni, Silvana

doi:10.1007/s10485-009-9188-1

Cited by 4 publications

(1 citation statement)

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“…11.1.11], that applies to locally finitely generated Grothendieck categories; it appears without proof as Proposition 11.1.11 in [Pr] (the proof follows the same lines of analogous results in [G, Po]). Furthermore, Theorem A is in the same spirit of [AB,Prop. 2.4 and 3.6], where related characterizations of hereditary torsion classes are given in the more general setting of Grothendieck categories with a projective generator.…”

Section: Introductionmentioning

confidence: 95%

Torsion Pairs in Categories of Modules over a Preadditive Category

Parra

Saorı́n

Virili

2020

Bull. Iran. Math. Soc.

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It is a result of Gabriel that hereditary torsion pairs in categories of modules are in bijection with certain filters of ideals of the base ring, called Gabriel filters or Gabriel topologies. A result of Jans shows that this bijection restricts to a correspondence between (Gabriel filters that are uniquely determined by) idempotent ideals and TTF triples. Over the years, these classical results have been extended in several different directions. In this paper we present a detailed and self-contained exposition of an extension of the above bijective correspondences to additive functor categories over small preadditive categories. In this context, we also show how to deduce parametrizations of hereditary torsion theories of finite type, Abelian recollements by functor categories, and centrally splitting TTFs.

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Section: Introductionmentioning

confidence: 95%

Torsion Pairs in Categories of Modules over a Preadditive Category

Parra

Saorı́n

Virili

2020

Bull. Iran. Math. Soc.

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Recollements of Module Categories

Psaroudakis

Vitória

2013

Appl Categor Struct

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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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