Derived algebras on formal stacks and prismatic gauges

Abstract

This paper studies how the theory of derived algebras (in the sense of Bhatt-Mathew and Raksit) interacts with formal derived geometry, specifically the formal derived stacks which show up in the theory of prismatization. As an application we prove some classification theorems for derived algebras in quasi-coherent sheaves on a certain class of filtered \emph{formal} stacks, which includes those whose quasi-coherent sheaves are prismatic gauges over a perfectoid ring. Along the way, among other things, we study the behavior of derived algebras along schematic quasi-affine morphisms in derived geometry, and for example, classify derived algebras on the source as precisely those derived algebras on the target which receive a map from the pushforward of the structure sheaf of the source. We also indicate how to extend some of our results to (formal) classifying stacks of diagonalizable group schemes. As an aside, we also show some classification theorems even for quasi-coherent sheaves on formal stacks which (to our knowledge) weren't available in the literature on derived geometry previously. These results are motivated by forthcoming work of the author but hoped to be generally useful.