Abstract
We provide an axiomatic approach for studying support varieties of objects in a triangulated category via the action of a tensor triangulated category, where the tensor product is not necessarily symmetric. This is illustrated by examples, taken in particular from the representation theory of finite dimensional algebras.
Highlights
The main purpose of this paper is to present a common framework where most of the existing occurrences of support varieties fit in
An inspiration for this work have been the notes on axiomatic stable homotopy theory by Hovey et al [24], where tensor triangulated categories play a central role
The purpose of this paper is to point out (1) that one often misses a vital underlying structure, namely a tensor triangulated category acting on the category where the theory of support is constructed and (2) that one obtains a central ring action from the graded endomorphism ring of the tensor identity of the acting tensor triangulated category
Summary
The main purpose of this paper is to present a common framework where most of the existing occurrences of support varieties fit in. The purpose of this paper is to point out (1) that one often misses a vital underlying structure, namely a tensor triangulated category acting on the category where the theory of support is constructed and (2) that one obtains a central ring action from the graded endomorphism ring of the tensor identity of the acting tensor triangulated category. This point of view has been taken successfully by Stevenson in [36,37], but there the tensor triangulated category acting has a symmetric tensor product.
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