I’ve been making an effort to understand rigid analytic geometry and outgrowths of it, like the theory of adic spaces, and how they’re utilized, for instance, in the theory of perfectoid spaces. Here, I’ll attempt to give an exposition of the Gross-Hopkins period map, and later describe how this is useful in homotopy theory.

$\newcommand{\FF}{\mathbf{F}}$ Let $R$ be a complete local ring with maximal ideal $\mathfrak{m}$, such that $R\xrightarrow{q} R/\mathfrak{m} \simeq_{f} \FF_{p^n}$. $\newcommand{\GG}{\mathbf{G}}$ Let $\GG$ be a (one-dimensional) formal group of height $n$ over $R$. We can consider its sheaf of invariant differentials $\omega_\GG$; once we choose a coordinate $t$, this is exactly $(t)/(t^2)$.

The Dieudonne module is defined via a weaker notion: instead of looking at invariant differentials, we can look at the cohomologically invariant differentials sitting inside $H^1_{dR}(\GG)$. This can be defined precisely as follows. Let $\mu:\GG\times\GG\to \GG$ denote the multiplication on $\GG$, and let $p_1$ and $p_2$ denote the two projection maps. The invariant differentials $\omega_\GG$ arise as the kernel of $\mu^\ast - (p_1^\ast+p_2^\ast):C^1_{dR}(\GG)\to C^1_{dR}(\GG\times\GG)$.

$\newcommand{\DD}{D}$ If $R = W(k)$ and $\GG$ arises as the lift of some height $n$ formal group $\GG_0$ over $k$, the Dieudonne module $\DD(\GG_0)$ is defined to be the kernel of $\mu^\ast - (p_1^\ast+p_2^\ast):H^1_{dR}(\GG)\to H^1_{dR}(\GG\times\GG)$. To make sure that this is well-defined, we need to ensure that this is independent of the lift $\GG$ to $W(k)$. The following theorem of Katz tells us precisely this.

Theorem (Katz): Let $R$ be a $p$-local torsion-free ring. If $f(x),g(x)\in xR[[x]]$ agree modulo $p$, then for any $\omega\in R[[x]]dx$, the difference $f^\ast\omega - g^\ast\omega$ is exact.

If two maps between formal varieties that agree mod $p$, this result tells us that they induce the same map on $H^1_{dR}$. It follows that $\DD(\GG_0)$ is independent of the lift of $\GG_0$ to $\GG/W(k)$.

The Dieudonne module $\DD(\GG_0)$ is a $W(k)$-module with a Frobenius action (coming from the map $x\mapsto x^p$ on $\GG_0$) that is $\phi$-semilinear, where $\phi$ is a lift of Frobenius to $W(k)$. It also has a Verschiebung operation, satisfying the relation $FV=VF=p$. We can simply define the Verschiebung via

Theorem (Grothendieck, Demazure, …): Let $C$ denote the Cartier-Dieudonne ring $W(k)[F,V]/(FV = VF = p, Fx = \phi(x)F, V\phi(x) = xV)$. Then there’s an equivalence of categories between formal groups over $k$ of finite height and free $C$-modules of finite rank given by taking a formal group to its Dieudonne module.

The height of the formal group is the rank of its Dieudonne module. For instance, $\DD(\GG_m) \simeq W(k)$ on one generator $x$, such that $F(x) = px$ and $V(x) = x$. If $\GG_0$ is the Honda formal group of height $n$ over $k = \FF_{p^n}$, then

We can use this description to provide an explicit presentation for the Morava stabilizer group $\GG_n$. The Verschiebung is $F^{n-1}$. By the equivalence of categories described above, we see that

We’re now in a position to describe the Gross-Hopkins map. I’ll describe it as I learnt it from Eric Peterson. I also learnt a lot by talking to David Zureick-Brown and Jackson Morrow.

$\newcommand{\LT}{\mathrm{LT}}$ Like above, let $\GG_0$ be the Honda formal group of height $n$ over $k = \FF_{p^n}$. The idea behind the Gross-Hopkins map is simple: according to Katz’s theorem above, the Dieudonne module $\DD(\GG_0)$ is independent of the lift of our formal group to $W(k)$. The cochain complex defining de Rham cohomology, however, is not. $\newcommand{\PP}{\mathbf{P}}$ Consequently, if $\GG$ is a lift to $W(k)$, the “strictly” invariant differentials $\omega_\GG$ select a one-dimensional subspace of $\DD(\GG_0)$. This defines the Gross-Hopkins period map $\LT_n(W(k)) \xrightarrow{\pi_{GH}} \PP(\DD(\GG_0))(W(k))$, given by

Note that the cokernel of $\omega_\GG\subseteq \DD(\GG_0)$ is the Lie algebra of the Cartier dual of $\GG_0$; this is supposed to be analogous to the Hodge–de-Rham decomposition. Gross and Hopkins prove that this map is equivariant for the action of $\GG_n = \mathrm{Aut}(\GG_0)$, identified above.

$\newcommand{\Sp}{\text{Sp }}$ The Gross-Hopkins period map is trying really hard to be a map at the rigid analytic level. Indeed, we can consider the rigid analytic fiber of $\LT_n$. Since $\LT_n$ is an affine formal scheme, the construction is not hard to describe.

$\newcommand{\spf}{\text{Spf }}$ The generic fiber of $\LT_n$ is supposed to be like the analytification (there’s even a notion of GAGA in this context!) of $\LT_n$. Recall that if $A$ is an affinoid $K$-algebra, $X = \Sp A$ denotes the space of maximal ideals of $A$. Let $f_0,\cdots,f_k\in A$ have no common zeros (i.e., $V(f_0,\cdots,f_k) = \emptyset$). A \emph{rational subdomain} of $\Sp A$ is a subset of the form $\{x\in \Sp A: |f_i(x)|\leq |f_0(x)|\text{ for all }1\leq i\leq k\}$. This is the same as $\Sp A\langle \frac{f_1}{f_0},\cdots,\frac{f_k}{f_0}\rangle$, where

This admits a presheaf of “affinoids” $\mathcal{O}_X$ defined by

Tate showed that this is indeed a sheaf, and that the higher sheaf cohomology of $X$ vanishes (just like for ordinary affine schemes). Now suppose $X = \spf R$ is a formal $A$-scheme that’s locally of finite type, where $A$ is the valuation ring of a complete non-Archimedian field $K$. Assume that $R$ has a finitely generated ideal $(u_0,\cdots,u_n)$ of definition and uniformizer $\pi$. One defines $X^\text{rig}$ to be $\Sp(R\otimes_A K)$. More generally, if $X = \spf R$ is a formal affine scheme, where $R$ has a finitely generated ideal $(u_1,\cdots,u_n)$ of definition and uniformizer $\pi$, define

Notice that $S_k/\pi S_k = R/(\pi, u_1,\cdots,u_k)[T_1,\cdots,T_n]$ is a $A/\pi$-algebra of finite type. We can therefore take the rigid analytic fiber $\spf (S_k)^\mathrm{rig}$. Define

It turns out that this is independent of the choice of generators $u_i$ — which is good, because the generators that we wrote down for $\LT_n$ are noncanonical.

For instance, suppose $X = \spf A[[x]]$ with ideal of definition $(\pi, x)$, so that $S_k\simeq A[[x]]\langle T\rangle/(\pi T - x^n)$. Thus, $\spf (S_k)^\mathrm{rig} = \Sp(S_k\otimes_{A} K) = \Sp(K\langle x,x^n/\pi\rangle)$ is the closed disk centered at zero and radius $|\pi|^{1/k}$. Consequently, $X^\mathrm{rig}$ is the open disk of radius one. This example generalizes in a straightforward way to unit open balls.

Theorem (Gross-Hopkins): Once we pass to the generic fiber, the Gross-Hopkins map $(\LT_n)^\mathrm{rig}\xrightarrow{\pi_{GH}} \PP^{n-1} = \PP(\DD(\GG_0))$ is $\GG_n$-equivariant and etale!

In the original Gross-Hopkins paper, the construction of this map is given as follows. Construct a line bundle $\mathcal{L}$ on $(\LT_n)^\mathrm{rig}$, and define a map $\DD(\GG_0)\to \Gamma((\LT_n)^\mathrm{rig},\mathcal{L})$. Then $\pi_{GH}$ sends a point to the hyperplane of sections of $\mathcal{L}$ supported at $x$. This is equivalent to our construction, since we can choose $\mathcal{L}$ to be the sheaf corresponding to the Lie algebra of the Cartier dual of $\GG_0$.