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arXiv · 1610.03561

Local study of stable module categories via tensor triangulated geometry

Nicolas Ricka
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Abstract

We investigate the particular properties of the stable category of modules over a finite dimensional cocommutative graded connected Hopf algebra $A$, via tensor-triangulated geometry. This study requires some mild conditions on the Hopf algebra $A$ under consideration (satisfied for example by all finite sub-Hopf-algebras of the modulo $2$ Steenrod algebra). In particular, we study some particular covers of its spectrum of prime ideals $\mathrm{Spc}(A)$, which are related to Margolis' Work. We then exploit the existence of Margolis' Postnikov towers in this situation to show that the localization at an open subset $U$ of $\mathrm{Spc}(A)$, for various $U$, assembles in an $\infty$-stack. Finally, we turn to applications in the study of Picard groups of Hopf algebras and localizations in the stable categories of modules.

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Nicolas Ricka. 2016-10-11. Local study of stable module categories via tensor triangulated geometry. https://arxiv.org/abs/1610.03561

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