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The problem of whether the cohomological support map of a finite dimensionalHopf algebra has the tensor product property has attracted a lot of attentionfollowing the earlier developments on representations of finite group schemes.Many authors have focussed on concrete situations where positive and negativeresults have been obtained by direct arguments.  In this paper we demonstrate that it is natural to study questions involvingthe tensor product property in the broader setting of a monoidal triangulatedcategory. We give an intrinsic characterization by proving that the tensorproduct property for the universal support datum is equivalent to completeprimeness of the categorical spectrum. From these results one obtainsinformation for other support data, including the cohomological one. Twotheorems are proved giving compete primeness and non-complete primeness incertain general settings.  As an illustration of the methods, we give a proof of a recent conjecture ofNegron and Pevtsova on the tensor product property for the cohomologicalsupport maps for the small quantum Borel algebras for all complex simple Liealgebras.

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