Grothendieck–Neeman duality and the Wirthmüller isomorphism

By Paul Balmer, Ivo Dell’Ambrogio, Beren Sanders. Compositio Mathematica, 2015

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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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bd20d7e23548ded153d2f6c6f64ede0f3ae02909

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The spectrum of the equivariant stable homotopy category of a finite group (2015, background): “the tom Dieck splitting theorem) and therefore inflation does not have a left adjoint (by [9]).”