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Grothendieck–Neeman duality and the Wirthmüller isomorphism

Published online by Cambridge University Press:  23 May 2016

Paul Balmer
Affiliation:
Mathematics Department, UCLA, Los Angeles, CA 90095-1555, USA email [email protected]
Ivo Dell’Ambrogio
Affiliation:
Laboratoire de Mathématiques Paul Painlevé, Université de Lille 1, Cité Scientifique – Bât. M2, 59665 Villeneuve-d’Ascq Cedex, France email [email protected]
Beren Sanders
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark email [email protected]
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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.

Type
Research Article
Copyright
© The Authors 2016 

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