Stratification, costratification, relative tensor-triangular geometry and singularity categories

Verasdanis, Charalampos (2024) Stratification, costratification, relative tensor-triangular geometry and singularity categories. PhD thesis, University of Glasgow.

Abstract

We develop the theory of stratification for a rigidly-compactly generated tensor-triangulated category using the smashing spectrum and the small smashing support. Our first result, outside the stratified context, is that the Hochster dual of the Balmer spectrum is the Kolmogorov quotient of the smashing spectrum equipped with a certain topology. Within the stratified context, we prove that it suffices to check stratification on certain smashing localizations and we investigate connections between big prime ideals, objectwise-prime ideals and homological primes. We give a characterization of the Telescope Conjecture in terms of the homological spectrum and the homological support. Moreover, we study induced maps between smashing spectra and prove a descent theorem for stratification.

We develop the theory of costratification in the setting of relative tensor-triangular geometry and prove that costratification is equivalent to the colocal-to-global principle and cominimality. We also introduce and study prime localizing submodules and prime colocalizing hom-submodules, in the first case, generalizing objectwise-prime localizing tensor-ideals. Further, we prove that it suffices to check costratification on certain localizations with respect to smashing submodules and certain covers of the associated space of supports/cosupports. We apply our results to show that the derived category of quasi-coherent sheaves over a noetherian separated scheme is costratified, generalizing a result of Neeman for noetherian rings.

Finally, we classify the colocalizing subcategories of the singularity category of a locally hypersurface ring and then we generalize this result to schemes with hypersurface singularities.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Additional Information: Supported by funding from the Engineering and Physical Sciences Research Council (EPSRC).
• Subjects: H Social Sciences > HA Statistics; Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
• Funder's Name: Engineering and Physical Sciences Research Council (EPSRC)
• Supervisor's Name: Whittaker, Professor Mike
• Date of Award: 2024
• Depositing User: Theses Team
• Unique ID: glathesis:2024-84703
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 15 Nov 2024 14:19
• Last Modified: 15 Nov 2024 14:19
• Thesis DOI: 10.5525/gla.thesis.84703
• URI: https://theses.gla.ac.uk/id/eprint/84703

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