Moduli of prismatic $(\mathcal{G},\mu)$-apertures

We study the derived moduli stack $\textup{BT}_n^{\mathcal{G},\mu}$ of (level-$n$ truncated) prismatic $(\mathcal{G},\mu)$-apertures, where $\mathcal{G}$ is a smooth affine group scheme over $\mathbb{Z}_p$ and $\mu$ is a $1$-bounded cocharacter of $\mathcal{G}$ defined over an unramified extension of $\mathbb{Z}_p$. Inspired by ideas of Drinfeld and Lau, we show that $\textup{BT}_n^{\mathcal{G},\mu}$ serves as a powerful group-theoretic generalization of the moduli stack of ($n$-truncated) $p$-divisible groups, endowed with analogous smoothness, finiteness, and representability properties as well as natural analogues of Dieudonne theory and Grothendieck-Messing theory. This opens the door to an intrinsically group-theoretic study of Rapoport-Zink spaces and potentially suggests new applications of derived algebraic geometry to the study of $F$-zips. This is joint work with Keerthi Madapusi.