| Résume | In work of Gaitsgory, the notion of quasi-coherent sheaf of categories is introduced. He shows that the assigment $Y\mapsto \mathrm{QCoh}(Y)$-Mod satisfies fppf descent and gives criteria for prestacks to be 1-affine. He restricts to field of characteristic 0. The goal of this talk is to present such a theory that also works in positive characteristic, and using ind-perfect complexes instead of quasi-coherent sheaves. This is complicated by the lack of Künneth formulas and general base change for perfect complexes, we will see how non-connective geometry can help with this issue. This will give a way to construct spectral actions without restrictions like $\ell\nmid|\pi_0(Z(G))|$. |
