Noncommutative tensor triangular geometry and the tensor product property for support maps

TL;DR

This paper explores the tensor product property of support maps in monoidal triangulated categories, providing intrinsic characterizations and applying these to quantum Borel algebras, advancing understanding in representation theory.

Contribution

It introduces an intrinsic characterization linking the tensor product property to complete primeness of the categorical spectrum, extending the analysis to broader settings.

Findings

Tensor product property is equivalent to complete primeness of the spectrum.

Provides criteria for when support data satisfy the tensor product property.

Proves the conjecture for small quantum Borel algebras across all complex simple Lie algebras.

Abstract

The problem of whether the cohomological support map of a finite dimensional Hopf algebra has the tensor product property has attracted a lot of attention following the earlier developments on representations of finite group schemes. Many authors have focussed on concrete situations where positive and negative results have been obtained by direct arguments. In this paper we demonstrate that it is natural to study questions involving the tensor product property in the broader setting of a monoidal triangulated category. We give an intrinsic characterization by proving that the tensor product property for the universal support datum is equivalent to complete primeness of the categorical spectrum. From these results one obtains information for other support data, including the cohomological one. Two theorems are proved giving compete primeness and non-complete primeness in certain general settings. As an illustration of the methods, we give a proof of a recent conjecture of Negron and Pevtsova on the tensor product property for the cohomological support maps for the small quantum Borel algebras for all complex simple Lie algebras.