Picard groups in triangular geometry and applications to modular representation theory

For a tensor triangulated $\mathbb{Z}/p$-category $\mathscr{K}$, with spectrum  $\operatorname{Spc}(\mathscr{K})$, we construct an injective group homomorphism $\check{\operatorname{H}}^1 (\operatorname{Spc}(\mathscr{K}),\mathbb{G}_{\opera... ...bb{Z}[1/p]\hookrightarrow\operatorname{Pic} (\mathscr{K})\otimes\mathbb{Z}[1/p]$, where $\operatorname{Pic}(\mathscr{K})$ is the group of $\otimes$-invertible objects of  $\mathscr{K}$. In modular representation theory, we prove that this homomorphism induces a rational isomorphism between the Picard group of the projective support variety and the group of endotrivial representations.