We develop a general noncommutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (M∆Cs). Insight from noncommutative ring theory is used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an M∆C, K, and then to associate to K a topological space-the Balmer spectrum Spc K. We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that Spc K is a universal terminal object for the first two notions (support and weak support). The first two types of support data are then used in a theorem that gives a method for the explicit classification of the thick (two-sided) ideals and the Balmer spectrum of an M∆C. The third type (quasi support) is used in another theorem that provides a method for the explicit classification of the thick right ideals of K, which in turn can be applied to classify the thick two-sided ideals and Spc K.As a special case, our approach can be applied to the stable module categories of arbitrary finite dimensional Hopf algebras that are not necessarily cocommutative (or quasitriangular). We illustrate the general theorems with classifications of the Balmer spectra and thick two-sided/right ideals for the stable module categories of all small quantum groups for Borel subalgebras, and classifications of the Balmer spectra and thick two-sided ideals of Hopf algebras studied by Benson and Witherspoon.

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.

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 focused 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.

In this paper the authors prove fundamental decomposition theorems pertaining to the internal structure of monoidal triangulated categories (M∆Cs). The tensor structure of an M∆C enables one to view these categories like (noncommutative) rings and to attempt to extend the key results for the latter to the categorical setting. The main theorem is an analogue of the Chinese Remainder Theorem involving the Verdier quotients for coprime thick ideals. This result is used to obtain orthogonal decompositions of the extended endomorphism rings of idempotent algebra objects of M∆Cs. The authors also provide topological characterizations on when an M∆C contains a pair of coprime proper thick ideals, and additionally, when the latter are complementary in the sense that their intersection is contained in the prime radical of the category.As an application of the aforementioned results, the authors establish for arbitrary M∆Cs a general version of Carlson's theorem on the connnectedness of supports for indecomposable objects. Examples of our results are given at the end of the paper for the derived category of schemes and for the stable module categories for finite group schemes.

Finite tensor categories (FTCs) T are important generalizations of the categories of finite dimensional modules of finite dimensional Hopf algebras, which play a key role in many areas of mathematics and mathematical physics. There are two fundamentally different support theories for them: a cohomological one and a universal one based on the noncommutative Balmer spectra of their stable (triangulated) categories T.In this paper we introduce the new notion of the categorical center C • T of the cohomology ring R • T of an FTC, T. This enables us to put forward a complete and detailed program for determining the exact relationship between the two support theories, based on C • T of the cohomology ring R • T of an FTC, T. More specifically, we construct a continuous map from the noncommutative Balmer spectrum of an FTC, T, to the Proj of the categorical center C • T , and prove that this map is surjective under a weaker finite generation assumption for T than the one conjectured by Etingof-Ostrik. Under stronger assumptions, we prove that (i) the map is homeomorphism and (ii) the two-sided thick ideals of T are classified by the specialization closed subsets of Proj C • T . We conjecture that both results hold for all FTCs. Many examples are presented that demonstrate how in important cases C • T arises as a fixed point subring of R • T and how the two-sided thick ideals of T are determined in a uniform fashion (while previous methods dealt on a case-by-case basis with case specific methods). The majority of our results are proved in the greater generality of monoidal triangulated categories.