p-divisible moduli

This note reviews some constructions of Lurie-Davis

Idea ($M_n$: moduli of height n p-divisible groups) Let $A\in Adi{c}_{\mathbb{S}}$ be an adic ring in anima. We construct a stack on connective affines:

$M_n^A\in Fun(Ring(S),S):=P(Af{f}_{cn})$ The game is to consider formal formal geometry above $M_n^A$. We consider the $\infty$-category of formal stacks

${\widehat{Stk}}_{/M_n^A}\subset P(Af{f}_{cn}{)}_{/M_n^A},$

Construction of moduli. Construction of $M_n^A$ is designed to capture the definition that a $p$-divisible group over the the adic ring $A$ is data:

$$\mathbb{G}\in Fun(CRing(S{)}_{R/},Mo{d}_{H\mathbb{Z}}(Sp))\simeq Ab(Fun(CRing(S{)}_R,S))$$

here $Ab$ denotes strict abelian group objects, often modelled in the Lawvere way.

The maps $H\mathbb{Z}\to H{F}_p,H\mathbb{Z}\to \mathbb{Q}$ induce functors out of these categories of strict abelian group stacks, that keep track of torsion and torsion-free data respectively.

• The condition of being $p$-divisible is stated in these terms: $\mathbb{G}(B)[1/p]=0$ for any input $B$. That is, being p-divisible means p-locally, there is only p-torsion data. (p=Torsion-free = p-rational = p-indivible). Another way to say this is that $\mathbb{G}(B)$ is a $p^{\infty }$-torsion abelian group.

• Then there’s the condition of being “affine”: $Ma{p}_{Mo{d}_{H\mathbb{Z}}}(H\mathbb{Z},\mathbb{G}(B))\simeq Ma{p}_{CRin{g}_R}({\mathcal{O}}_{\mathbb{G}},B)$ It says the prestack of strict elements of $\mathbb{G}$ is given by $Spec({\mathcal{O}}_{\mathbb{G}})$. Furthermore we ask for a ${\mathcal{O}}_{\mathbb{G}}$ to be finite flat.

More generally, there are stacks of $p$-torsion elements $\mathbb{G}[p^n]:=Ma{p}_{Mo{d}_{H\mathbb{Z}}}(H{C}_{p^n},\mathbb{G}(-))$

(these assemble into the Tate-module of $\mathbb{G}$).

• Finally, Flatness is about how the scheme interacts with the $p$-map. (does this have something to do with the Frobenius?). But actually this is the property that’s said to be p-divisible?

we ask that the map $p:\mathbb{G}\to \mathbb{G}$ is surjective with respect to finite flat topology. This means that finite-flat-locally, everything is divisible by p: an element in $a\in \mathbb{G}(B)$ can be written as $p\cdot b$ for some $b$.

Heights of p-divisible groups Define a (possibly infinite) numerical meaure

$ht(\mathbb{G}):=rank(\mathbb{G}[p^1])$ The multiplicative group example of Lurie is instructive: it’s an explicit way to work with $v_1$ periodicity.

Remark A map $Y\to M_n^A$ is said to be a (n $A$-family of) height n p-divisible groups on $Y$.

Example on the p-local sphere (as an adic ring), $M_n^{\mathbb{S}}$ is the moduli stack of spectral $p$-divisble groups.

Construction: Landweber Stack at height n

For $A$ as before, construct $F_n:{\widehat{Stk}}_{/M_n^A}^{op}\to CAlg(Sp)$ Remark $F_n$ sends a formal affine $\mathbb{G}:Spf(R)\to M_n^A$ to an $E_{\infty }$ ring $F(Spf(R)/M_n^A):=E_{\mathbb{G}}$.

We’d like to view $F_n$ as a geometric object: as a formal ring stack, or as something like a spectral DM stack.

Remark This is supposed to resemble how the classical Landweber theorem is a constrution sending a formal group $Spf(R)\to (\text{moduli of d=1 algebraic groups})$ to a complex-oriented ring spectrum $E$

Basic properties of this correspondence

$E$ can be described by data of the $R$-family of height $n$ p-divisible groups $\mathbb{G}$

• ${\pi }_{0}E\simeq {\pi }_{0}R$
• ${\pi }_{odd}E=0$
• ${\pi }_{2k}E\simeq {\omega }^{\otimes k}$

where $\omega \in Pic({\pi }_{0}R)$ is the dualizing line of ${\pi }_0(\mathbb{G})$.

Example (Absolute case)

Take the initial case $A={\mathbb{S}}_p$ as an adic ring in anima.