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Balmer spectra and Drinfeld centers

Kent B. Vashaw

Abstract

The Balmer spectrum of a monoidal triangulated category is an important geometric construction which is closely related to the problem of classifying thick tensor ideals. We prove that the forgetful functor from the Drinfeld center of a finite tensor category ${\mathbf{C}}$ to ${\mathbf{C}}$ extends to a monoidal triangulated functor between their corresponding stable categories, and induces a continuous map between their Balmer spectra. We give conditions under which it is injective, surjective, or a homeomorphism. We apply this general theory to prove that Balmer spectra associated to finite-dimensional cosemisimple quasitriangular Hopf algebras (in particular, group algebras in characteristic dividing the order of the group) coincide with the Balmer spectra associated to their Drinfeld doubles, and that the thick ideals of both categories are in bijection. An analogous theorem is proven for certain Benson--Witherspoon smash coproduct Hopf algebras, which are not quasitriangular in general.

Balmer spectra and Drinfeld centers

Abstract

The Balmer spectrum of a monoidal triangulated category is an important geometric construction which is closely related to the problem of classifying thick tensor ideals. We prove that the forgetful functor from the Drinfeld center of a finite tensor category to extends to a monoidal triangulated functor between their corresponding stable categories, and induces a continuous map between their Balmer spectra. We give conditions under which it is injective, surjective, or a homeomorphism. We apply this general theory to prove that Balmer spectra associated to finite-dimensional cosemisimple quasitriangular Hopf algebras (in particular, group algebras in characteristic dividing the order of the group) coincide with the Balmer spectra associated to their Drinfeld doubles, and that the thick ideals of both categories are in bijection. An analogous theorem is proven for certain Benson--Witherspoon smash coproduct Hopf algebras, which are not quasitriangular in general.

Paper Structure

This paper contains 16 sections, 30 theorems, 89 equations.

Table of Contents

  1. Preliminaries
  2. Tensor triangular geometry
  3. Compactly-generated triangulated categories
  4. Stable categories and finite tensor categories
  5. Support data
  6. The Drinfeld center
  7. Drinfeld Centers and Balmer Spectra
  8. Construction of a continuous map between Balmer spectra
  9. A support data interpretation
  10. Recovering ideals from their supports
  11. The image of prime ideal contraction
  12. Conditions under which $f$ is injective, surjective, or a homeomorphism
  13. Applications
  14. Group algebras and dual group algebras
  15. Cosemisimple Hopf algebras
  16. ...and 1 more sections

Key Result

Theorem A

(See Proposition pfunct-drinf-cent-st and Proposition pfbar-induces-spc). Let $\mathbf C$ be a finite tensor category. There exists a monoidal triangulated functor $\overline{F}: \mathop{\mathrm{\sf st}}\nolimits(\mathop{\mathrm{\sf Z}}\nolimits(\mathbf C)) \to \mathop{\mathrm{\sf st}}\nolimits(\mat for $\mathbf P \in \mathop{\mathrm{Spc}}\nolimits \mathop{\mathrm{\sf st}}\nolimits(\mathbf C)$.

Theorems & Definitions (58)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Proposition \oldthetheorem
  • ...and 48 more