Dualities and Finitely Presented Functors

Student thesis: Phd


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 satisfied 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 different 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 define 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 finitely presented categories whose category of finitely presented objects is a dualising variety.
Date of Award31 Dec 2017
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorMichael Prest (Supervisor) & Nigel Ray (Supervisor)


  • Additive category
  • Abelian category
  • Contravariant functor
  • pp-pair
  • Localisation
  • Hilton-Rees embedding
  • Auslander-Gruson-Jensen duality
  • Auslander-Reiten formulas
  • Recollement of abelian categories
  • Finitely presented functor
  • Locally finitely presented category

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