Derived algebras on formal stacks and prismatic gauges

TL;DR

This work develops a comprehensive framework for derived algebras in quasi-coherent sheaves on formal stacks, connecting Bhatt–Mathew/Raksit’s non-connective derived algebra theory with formal derived geometry and prismatization. It introduces and analyzes $J$-complete and graded derived algebras, proving monadicity/ comonadicity results and establishing descent and base-change properties along quasi-affine morphisms. Through the Rees construction and filtered/graded correspondences, it provides geometric classification theorems for derived algebras on formal classifying stacks such as $B\mathbf{G}_m$ and on formal filtered stacks, including identifications with Nygaard-prismatic cohomology in orientable prisms. The results lay groundwork for applications to prismatic cohomology and forthcoming work on Nygaard prismatization, delta-schemes, and Tannakian formalisms in the formal/derived setting. Overall, the paper advances the interaction between non-connective derived algebraic geometry and formal geometric structures, enabling new computations and conceptual clarity in prismatization and filtered geometry.

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.

Theorems & Definitions (190)

• Example 1.1
• Definition 1.4
• Example 1.5: $p$-complete $\infty\text{-categories}$ arise as categories of quasi-coherent sheaves on stacks.
• Example 1.6
• Example 1.7
• Example 1.8
• Theorem 1.10
• Theorem 1.11
• Theorem 1.12
• Theorem 1.13