Writing

Abstract: We use the newly developed “stacky” prismatic technology of Drinfeld and Bhatt-Lurie to give a uniform, group-theoretic construction of smooth (derived) stacks $\textup{BT}_n^{\mathcal{G},\mu}$ associated to a smooth affine group scheme $\mathcal{G}$ over $\mathbb{Z}_p$ and 1-bounded cocharacter $\mu$, verifying a recent conjecture of Drinfeld. This can be viewed as a refinement of results of Bultel-Pappas, who gave a related construction using $(\mathcal{G},\mu)$-displays defined via $p$-typical Witt vectors. When $\mathcal{G}=\textup{GL}_h$ and $\mu$ is minuscule, we show that $\textup{BT}_n^{\mathcal{G},\mu}$ is isomorphic to the stack of $n$-truncated $p$-divisible groups of height $h$ and dimension $d$ (with the latter determined by $\mu$). This generalizes results of Anschutz-Le Bras, yielding a linear algebraic classification of $p$-divisible groups over very general $p$-adic bases, and verifying another conjecture of Drinfeld.

The proofs use deformation techniques from derived algebraic geometry, combined with an animated variant of Lau’s theory of higher frames and displays. Since we are interested in applications to the study of local and global Shimura varieties, we actually prove representability results for a wide range of stacks whose tangent complex is 1-bounded in a suitable sense. As an immediate consequence, we prove algebraicity for the stack of perfect $F$-gauges of level $n$ with Hodge-Tate weights contained in $[0,1]$.

Abstract: Let $\mathcal{O}$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ with finite residue field. Using the associated $\mathcal{O}$-typical Witt vectors $W_{\mathcal{O}}(-)$, K. Ito and S. Marks define the notion of $\mathcal{O}$-prisms and show that they behave much like usual ($p$-typical) prisms. Building off of this, we first transport the relevant parts of the literature on absolute prismatic cohomology, prismatization, and syntomification to the $\mathcal{O}$-typical setting. We then replicate the results of prior work with K. Madapusi in the $\mathcal{O}$-typical setting and study new phenomena that do not arise in the $p$-typical setting. As an important technical step, we relate our results to the literature on $\mathcal{O}$-typical Barsotti-Tate groups and establish associated covariant and contravariant forms of $\mathcal{O}$-prismatic Dieudonne theory.

Abstract: Smith ideals provide a homotopy coherent generalization of the classical notion of ideal pairs, with quasi-ideals (as studied, e.g., by Drinfeld) already providing interesting examples in the discrete setting. Using the Smith ideal formalism, we conduct a thorough study of derived ideal pairs in the setting of nonconnective derived geometry, providing new perspective on Rees algebras, derived deformation to the normal bundle, and derived scheme-theoretic images. Working with iterated derived deformation spaces, we define products and sums of derived ideal pairs and analyze how they behave in comparison to their classical counterparts, ultimately showing that these operations naturally upgrade the $\infty$-category of derived ideal pairs over a fixed derived stack to an $\mathbb{E}_{\infty}$-algebra object in $\textup{Cat}_{\infty}$. We then conclude with an in-depth look at various notions of completeness and completion and what they tell us about derived formal geometry.

Abstract: We construct prismatic Rapoport-Zink spaces associated to general linear groups.