GrothendieckNeeman duality and the Wirthmüller isomorphism
Abstract
We clarify the relationship between Grothendieck duality à la Neeman and the Wirthmüller isomorphism à la FauskHuMay. We exhibit an interesting pattern of symmetry in the existence of adjoint functors between compactly generated tensortriangulated categories, which leads to a surprising trichotomy: There exist either exactly three adjoints, exactly five, or infinitely many. We highlight the importance of socalled relative dualizing objects and explain how they give rise to dualities on canonical subcategories. This yields a duality theory rich enough to capture the main features of Grothendieck duality in algebraic geometry, of generalized PontryaginMatlis duality à la DwyerGreenleesIyengar in the theory of ring spectra, and of BrownComenetz duality à la Neeman in stable homotopy theory.
 Publication:

arXiv eprints
 Pub Date:
 January 2015
 arXiv:
 arXiv:1501.01999
 Bibcode:
 2015arXiv150101999B
 Keywords:

 Mathematics  Category Theory;
 Mathematics  Algebraic Geometry;
 Mathematics  Algebraic Topology;
 Mathematics  KTheory and Homology;
 Mathematics  Representation Theory
 EPrint:
 36 pages. Minor revision due to referee's comments. Added Examples 3.27, 4.8 &