Module categories over affine supergroup schemes

Abstract

Let $k$ be an algebraically closed field of characteristic $0$ or $p>2$. Let $\mathcal{G}$ be an affine supergroup scheme over $k$. We classify the indecomposable exact module categories over the tensor category ${\rm sCoh}_{\rm f}(\mathcal{G})$ of (coherent sheaves of) finite dimensional $\mathcal{O}(\mathcal{G})$-supermodules in terms of $(\mathcal{H},\Psi)$-equivariant coherent sheaves on $\mathcal{G}$. We deduce from it the classification of indecomposable {\em geometrical} module categories over $\sRep(\mathcal{G})$. When $\mathcal{G}$ is finite, this yields the classification of {\em all} indecomposable exact module categories over the finite tensor category $\sRep(\mathcal{G})$. In particular, we obtain a classification of twists for the supergroup algebra $k\mathcal{G}$ of a finite supergroup scheme $\mathcal{G}$, and then combine it with \cite[Corollary 4.1]{EG3} to classify finite dimensional triangular Hopf algebras with the Chevalley property over $k$.