Duality in tensor-triangular geometry via proxy-smallness
Thomas Peirce, Jordan Williamson
TL;DR
This work develops a unified framework for duality phenomena in tensor-triangular geometry by introducing proxy-small geometric functors, enabling Grothendieck duality on canonical subcategories without requiring preservation of compact objects. It defines Matlis dualising objects and the Gorenstein property in tt-categories, and provides a Morita-theoretic classification of Matlis lifts along with ascent/descent results. The paper then applies these tools to obtain local duality and Gorenstein duality in diverse contexts, including commutative algebra, chromatic homotopy theory, and invariant theory, yielding new perspectives on classical dualities such as Matlis duality and Watanabe’s theorem. Overall, the approach bridges and extends duality theories from algebra to tt-geometry, offering versatile techniques for identifying and exploiting dualising phenomena in structured categories.
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.