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Dualities and Finitely Presented Functors
[Thesis]. Manchester, UK: The University of Manchester; 2017.
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Abstract
We investigate various relationships between categories of functors. The major examples are given by extending some duality to a larger structure, such as an adjunction or a recollement of abelian categories. We prove a theorem which provides a method of constructing recollements which uses 0-th derived functors. We will show that the hypotheses of this theorem are very commonly satisï¬ed by giving many examples. In our most important example we show that the well-known Auslander-Gruson-Jensen equivalence extends to a recollement. We show that two recollements, both arising from diï¬erent characterisations of purity, are strongly related to each other via a commutative diagram. This provides a structural explanation for the equivalence between two functorial characterisations of purity for modules. We show that the Auslander-Reiten formulas are a consequence of this commutative diagram. We deï¬ne and characterise the contravariant functors which arise from a pp-pair. When working over an artin algebra, this provides a contravariant analogue of the well-known relationship between pp-pairs and covariant functors. We show that some of these results can be generalised to studying contravariant functors on locally ï¬nitely presented categories whose category of ï¬nitely presented objects is a dualising variety.
Keyword(s)
Abelian category; Additive category; Auslander-Gruson-Jensen duality; Auslander-Reiten formulas; Contravariant functor; Finitely presented functor; Hilton-Rees embedding; Localisation; Locally finitely presented category; Recollement of abelian categories; pp-pair