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Grothendieck–Neeman duality and the Wirthmüller isomorphism

Part of: Abelian categories Applied homological algebra and category theory (Co)homology theory

Published online by Cambridge University Press:  23 May 2016

Paul Balmer ,
Ivo Dell’Ambrogio  and
Beren Sanders
Paul Balmer
Affiliation:
Mathematics Department, UCLA, Los Angeles, CA 90095-1555, USA email balmer@math.ucla.edu
Ivo Dell’Ambrogio
Affiliation:
Laboratoire de Mathématiques Paul Painlevé, Université de Lille 1, Cité Scientifique – Bât. M2, 59665 Villeneuve-d’Ascq Cedex, France email ivo.dellambrogio@math.univ-lille1.fr
Beren Sanders
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark email sanders@math.ku.dk
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Abstract

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We clarify the relationship between Grothendieck duality à la Neeman and the Wirthmüller isomorphism à la Fausk–Hu–May. We exhibit an interesting pattern of symmetry in the existence of adjoint functors between compactly generated tensor-triangulated categories, which leads to a surprising trichotomy: there exist either exactly three adjoints, exactly five, or infinitely many. We highlight the importance of so-called 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 Pontryagin–Matlis duality à la Dwyer–Greenless–Iyengar in the theory of ring spectra, and of Brown–Comenetz duality à la Neeman in stable homotopy theory.

Keywords

Grothendieck dualityur-Wirthmüllerdualizing objectadjointscompactly generated triangulated categorySerre functor

MSC classification

Primary: 18E30: Derived categories, triangulated categories
Secondary: 14F05: Sheaves, derived categories of sheaves and related constructions 55U35: Abstract and axiomatic homotopy theory
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 8 , August 2016 , pp. 1740 - 1776
Copyright
© The Authors 2016 

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