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We show that in an essentially small rigid tensor triangulated category with connected Balmer spectrum there are no proper non‐zero thick tensor ideals admitting strong generators. This proves, for instance, that the category of perfect complexes over a commutative ring without non‐trivial idempotents has no proper non‐zero thick subcategories that are strongly generated. The main theorem is also applied to the finite stable homotopy category, and we show that there are no strongly generated thick subcategories except the trivial one.

Strong generators in tensor triangulated categories