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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:728052 |
| Date | January 2017 |
| Creators | Dean, Samuel |
| Contributors | Prest, Michael |
| Publisher | University of Manchester |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://www.research.manchester.ac.uk/portal/en/theses/dualities-and-finitely-presented-functors(8ff515ee-6579-410d-88c1-ea4236d704f9).html |
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