On the $\infty$-stack of complexes over a scheme

Abstract

We study fppf descent for enhanced derived categories. We revisit the work of [HS] and [TV08] in a lax context. More precisely, we construct a Cartesian and coCartesian fibration ${}^{\mathrm{op}}{\mathcal{D}}_S^{+}\to N({\mathrm{Sch}}_S)$ whose fibre over an $S$-scheme $T$ is the opposite ${\mathcal{D}}^{+}(T{)}^{\mathrm{op}}$ of the quasicategory of bounded below complexes of ${\mathcal{O}}_T$-modules. We show that this fibration satisfies fppf-descent for schemes.

37 pages. Minor modification to conventions