  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.
