Associated sheaf functors in tensor triangular geometry

Rowe, James Cameron (2023) Associated sheaf functors in tensor triangular geometry. PhD thesis, University of Glasgow.

Abstract

Given a tensor triangulated category we investigate the geometry of the Balmer spectrum as a locally ringed space. Specifically we construct functors assigning to every object in the category a corresponding sheaf and a notion of support based upon these sheaves. We compare this support to the usual support in tt-geometry and show that under reasonable conditions they agree on compact objects. We show that when tt-categories satisfy a scheme-like property then the sheaf associated to an object is quasi-coherent, and that in the presence of an appropriate t-structure and affine assumption, this sheaf is in fact the sheaf associated to the object’s zeroth cohomology. When the tensor triangulated structure is replaced with a monoidal triangulated structure we show that one can form localising bimodules and central idempotents given particular localisation sequences. Finally, we provide a computation of the spectrum for the enveloping algebra of the A₂ quiver and determine that spectrum consists of a single point.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
• Supervisor's Name: Stevenson, Dr. Gregory and Muthiah, Dr. Dinakar
• Date of Award: 2023
• Depositing User: Theses Team
• Unique ID: glathesis:2023-83497
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 22 Mar 2023 16:32
• Last Modified: 23 Mar 2023 11:20
• Thesis DOI: 10.5525/gla.thesis.83497
• URI: https://theses.gla.ac.uk/id/eprint/83497