Grothendieck-Neeman duality and the Wirthmüller isomorphism

Abstract

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-Greenlees-Iyengar in the theory of ring spectra, and of Brown-Comenetz duality à la Neeman in stable homotopy theory.