Picard groups in triangular geometry and applications to modular representation theory

Abstract:

For a tensor triangulated $\mathbb {Z}/p$-category $\mathscr {K}$, with spectrum $\operatorname {Spc}(\mathscr {K})$, we construct an injective group homomorphism $\check {\operatorname {H}}^1 (\operatorname {Spc}(\mathscr {K}),\mathbb {G}_{\operatorname {m}} )\otimes \mathbb {Z}[1/p]\hookrightarrow \operatorname {Pic} (\mathscr {K})\otimes \mathbb {Z}[1/p]$, 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.