The Balmer spectrum and tensor telescope conjecture for noetherian path algebras

2025/11/25 by Sabatini, Enrico
Mathematics · #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #Commutative Algebra and Its Applications

paper · doi:10.48550/arxiv.2511.20204

Abstract

Given a commutative noetherian ring R and a finite acyclic quiver Q, we study the tensor triangulated category D(RQ) endowed with the vertexwise tensor product. We find a description of the internal hom functor and show that the category is not rigid. We compute its Balmer spectrum and, despite the non-rigidity, we get a classification of all the thick tensor-ideals and a stratification result. After introducing the notion of tensor-t-structure, we give a classification of the compactly generated ones and prove the tensor telescope conjecture.

Citations

• Telescope conjecture for t-structures over noetherian path algebras
• The Balmer spectrum and telescope conjecture for infinite groups
• Gluing of Fourier-Mukai partners in a triangular spectrum and birational geometry
• Cosupport in tensor triangular geometry
• Rigid integral representations of quivers over arbitrary commutative rings
• Lifting and restricting t-structures
• Compactly generated t-structures in the derived category of a\n commutative ring
• A tour of support theory for triangulated categories through tensor triangular geometry
• The prime spectra of relative stable module categories
• Spectra of tensor triangulated categories over category algebras
• The direct limit closure of perfect complexes
• Weight structures and simple dg modules for positive dg algebras
• The spectrum of prime ideals in tensor triangulated categories
• Construction of t-structures and equivalences of derived categories

Related

• Connectedness of the Balmer spectra of right bounded derived categories
• Telescope conjecture for t-structures over noetherian path algebras
• Re-framing the classification of ideals in noncommutative tensor-triangular geometry
• Descent in tensor triangular geometry
• Functors on triangulated tensor categories