Compact derived VI-modules are classified by their support types, and the Balmer spectrum for families of bounded-rank abelian p-groups is computed explicitly.
Cohen's theorem in tensor triangular geometry
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A theorem of Cohen from 1950 states that a commutative ring is Noetherian if and only if every prime ideal is finitely generated. In this note, we establish analogues of this result in tensor triangular geometry. In particular, for an essentially small tensor triangulated category $\mathscr{K}$ with weakly Noetherian spectrum, we show that every prime ideal in $\mathscr{K}$ can be generated by finitely many objects if and only if the set of prime ideals of $\mathscr{K}$ is finite.
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The spectrum of global representations for families of bounded rank and VI-modules
Compact derived VI-modules are classified by their support types, and the Balmer spectrum for families of bounded-rank abelian p-groups is computed explicitly.