Overview
This research investigates duality phenomena within the framework of tensor-triangular geometry. It introduces an approach centered on 'proxy-smallness' to address and generalize existing results concerning duality. The developed framework aims to provide a systematic study of these phenomena.
Research Context
The study builds upon and extends previous work in tensor-triangular geometry, specifically referencing results by Balmer–Dell'Ambrogio–Sanders and Dwyer–Greenlees–Iyengar. A reported limitation in prior approaches is the reliance on assumptions that functors preserve compact objects. This research introduces proxy-smallness as a mechanism to remove these assumptions. In relation to classifications of rigid objects, the work offers a new perspective on findings by Benson–Iyengar–Krause–Pevtsova.
Approach
The core of the approach involves the introduction and characterization of 'proxy-small geometric functors'. These functors are critical for removing assumptions about the preservation of compact objects by other functors. The methodology focuses on establishing the key properties of these proxy-small geometric functors. The framework then uses these properties to classify rigid objects within the torsion categories associated with these functors. Furthermore, it examines the satisfaction of Grothendieck duality on specific subcategories of objects.
Findings
- The research introduces and establishes the key properties of proxy-small geometric functors.
- It provides a classification of rigid objects within the torsion category associated with a given proxy-small geometric functor. This classification offers a new perspective on prior results.
- Any proxy-small geometric functor is shown to satisfy Grothendieck duality on a canonical subcategory of objects. This holds irrespective of whether its right adjoint preserves compact objects.
- The developed framework serves as a tool for classifying Matlis dualizing objects.
- It provides a suitable generalization of the Gorenstein ring spectra as formulated by Dwyer–Greenlees–Iyengar within tensor-triangular geometry.
Potential Applications
The developed framework has been illustrated with various examples and applications. These applications demonstrate its capacity to capture:
- Matlis duality in commutative algebra.
- Gorenstein duality in commutative algebra.
- Duality phenomena observed in chromatic stable homotopy theory.
- Duality phenomena observed in equivariant stable homotopy theory.
- Watanabe's theorem in polynomial invariant theory.