The Balmer spectrum and tensor telescope conjecture for noetherian path algebras
Enrico Sabatini
TL;DR
The paper analyzes the derived category $\mathcal{D}(RQ)$ of representations of a finite acyclic quiver $Q$ over a commutative noetherian ring $R$, equipped with a vertexwise tensor product. It computes the Balmer spectrum of the compact objects, showing $\mathrm{Spc}(\mathcal{D}^c(RQ)) \cong \mathrm{Spec}(R) \times Q_0$ and proves a thick tensor-ideals classification via specialization-closed subsets, despite the lack of rigidity. It then develops tensor-t-structures on $\mathcal{D}(RQ)$, classifies compactly generated ones in terms of filtrations of Balmer-spectrum subsets, and proves the tensor telescope conjecture in this non-rigid (yet stratifiable) setting, including generalizations to filtration systems with Dynkin support. The results extend known rigid-case classifications (e.g., for field coefficients) to a broad Noetherian base and expand the toolkit for stratification in tensor-triangular geometry, offering new insights into representations of finite acyclic quivers over rings.
Abstract
Given a commutative noetherian ring $R$ and a finite acyclic quiver $Q$, we study the tensor triangulated category $\mathcal{D}(RQ)$ 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.