Grothendieck-Neeman duality and the Wirthm\"uller isomorphism

TL;DR

This paper explores the deep connections and symmetries between various duality theories and adjoint functor patterns in tensor-triangulated categories, revealing a trichotomy in the number of adjoints and unifying multiple duality frameworks.

Contribution

It introduces a unifying duality framework based on relative dualizing objects, linking Grothendieck duality, Pontryagin-Matlis duality, and Brown-Comenetz duality through a symmetry pattern.

Findings

Existence of exactly three, five, or infinitely many adjoints in certain categories.

Introduction of relative dualizing objects that induce dualities on subcategories.

A unified duality theory capturing key features of multiple duality concepts across mathematics.

Abstract

We clarify the relationship between Grothendieck duality \`a la Neeman and the Wirthm\"uller isomorphism \`a 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 \`a la Dwyer-Greenlees-Iyengar in the theory of ring spectra, and of Brown-Comenetz duality \`a la Neeman in stable homotopy theory.