Ring stacks conjecturally related to the stacks $BT_n^{G,μ}$

TL;DR

The paper defines and develops ring stacks ${}^s\mathscr{R}}_n$ and ${}^s\mathscr{R}}_n^{\oplus}$ by encoding sheared Witt-vector data into cone constructions over ring-spaces ${}^sW$ and $Q=W/\hat{W}$. It builds several economical and self-dual models (via DG-rings, the Lau equivalence, and quotient/cone presentations) to realize these stacks, aiming to relate them to the moduli of truncations of Barsotti–Tate groups ${\mathrm{BT}}_n^{G,\mu}$. The work integrates prismatization, sheared prismatization, and delta-ring structures, and provides explicit descriptions in mixed and equal characteristic, including $p$-adic and semiperfect cases. A key theme is establishing duality and tractable presentations (cone/quasi-ideal) to enable comparisons with ${\mathrm{BT}}_n^{G,\mu}$ and to facilitate potential computational approaches. The paper also sketches autoduality conjectures and outlines how these ring stacks interpolate between Witt-theoretic data and the geometry of Barsotti–Tate stacks, with broad implications for the prismatization program and moduli of $p$-divisible groups.

Abstract

Using the ring space of sheared Witt vectors, we define certain ring stacks. We suggest several models for the ring stacks. Motivation: there is a conjectural description of the stack of n-truncated Barsotti-Tate groups and its Shimurian analogs in terms of the new ring stacks.

Theorems & Definitions (71)

• Proposition 2.3.4
• proof : Proof of Proposition \ref{['p:Q/hat V (Q)']}
• Corollary 2.3.5
• proof
• Corollary 2.3.6
• proof
• Proposition 3.1.6
• proof
• Lemma 3.2.5
• proof