Duality in tensor-triangular geometry via proxy-smallness

TL;DR

This paper develops a new framework in tensor-triangular geometry using proxy-smallness to analyze duality phenomena, enabling classification of objects and dualities in various mathematical contexts without restrictive assumptions.

Contribution

It introduces proxy-small geometric functors, classifies rigid objects in torsion categories, and generalizes Gorenstein duality in tensor-triangular geometry.

Findings

Classified rigid objects in torsion categories using proxy-smallness.

Proved Grothendieck duality for proxy-small geometric functors.

Unified duality phenomena across algebra, topology, and invariant theory.

Abstract

We make a systematic study of duality phenomena in tensor-triangular geometry, generalising and complementing previous results of Balmer--Dell'Ambrogio--Sanders and Dwyer--Greenlees--Iyengar. A key feature of our approach is the use of proxy-smallness to remove assumptions on functors preserving compact objects, and to this end we introduce proxy-small geometric functors and establish their key properties. Given such a functor, we classify the rigid objects in its associated torsion category, giving a new perspective on results of Benson--Iyengar--Krause--Pevtsova. As a consequence, we show that any proxy-small geometric functor satisfies Grothendieck duality on a canonical subcategory of objects, irrespective of whether its right adjoint preserves compact objects. We use this as a tool to classify Matlis dualising objects and to provide a suitable generalisation of the Gorenstein ring spectra of Dwyer--Greenlees--Iyengar in tensor-triangular geometry. We illustrate the framework developed with various examples and applications, showing that it captures Matlis duality and Gorenstein duality in commutative algebra, duality phenomena in chromatic and equivariant stable homotopy theory, and Watanabe's theorem in polynomial invariant theory.