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

TL;DR

This paper constructs an $ ext{infinity}$-stack of bounded below complexes over schemes, demonstrating that it satisfies fppf descent, thereby extending the understanding of derived categories in algebraic geometry.

Contribution

It introduces a new $ ext{infinity}$-stack framework for complexes over schemes and proves its descent properties, building on and generalizing previous work in the field.

Findings

The $ ext{infinity}$-stack satisfies fppf descent for schemes.

The construction uses a Cartesian and coCartesian fibration approach.

Revisits and extends prior results by [HS] and [TV08].

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 ${}^{\mathrm{op}}\mathscr D^+_S\rightarrow N(\mathop{\mathrm{Sch}}_S)$ whose fibre over an $S$-scheme $T$ is the opposite $\mathscr D^+(T)^{\mathrm{op}}$ of the quasicategory of bounded below complexes of $\mathscr O_T$-modules. We show that this fibration satisfies fppf-descent for schemes.