Group actions on monoidal triangulated categories and Balmer spectra

2023/11/30 by Hongdi Huang, Huang, Hongdi, Kent B. Vashaw +1 · 3 citations
Mathematics · #16T05 (Secondary) #16W22 #18G80 #18M05 (Primary) 14L30 #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Category Theory (math.CT) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2311.18638

openalex publication_date 2023/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a group acting on a left or right rigid monoidal triangulated category \mathbf K which has a Noetherian Balmer spectrum. We prove that the Balmer spectrum of the crossed product category of \mathbf K by G is homeomorphic to the space of G-prime ideals of \mathbf K, give a concrete description of this space, and classify the G-invariant thick ideals of \mathbf K. Under some additional technical conditions, we prove that the Balmer spectrum of the equivariantization of \mathbf K by G is also homeomorphic to the space of G-prime ideals. Examples of stable categories of finite tensor categories and perfect derived categories of coherent sheaves on Noetherian schemes are used to illustrate the theory.

Cited by

• A semisimple subcategory of Khovanov's Heisenberg category
• Noncommutative tensor triangular geometry: modules, bimodules, and unipotent Hopf algebras
• Re-framing the classification of ideals in noncommutative tensor-triangular geometry

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• The Balmer spectrum of certain Deligne–Mumford stacks
• The Balmer spectrum of certain Deligne-Mumford stacks
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