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


Corollary: Let $\omega$ denote the sheaf of invariant differentials on the universal deformation $\GG$ of $\GG_0$ to $\LT_n$, so that $\omega = z^\ast\Omega^1_{\GG/\LT_n}$ (where $z:\LT_n\to \GG$ is the zero section). Then there is a $\GG_n$-equivariant isomorphism of line bundles

Here’s a sketch of my attempt at a proof.

Proof Sketch. Gross and Hopkins prove (see Proposition 23.2 of their paper) that the canonical bundle of $\PP^{n-1}$ is, as a $\GG_n$-equivariant bundle, given by $\mathcal{O}_{\PP^{n-1}}(-n)[\det]$. Since $\pi_{GH}$ is etale on the generic fiber, we can pull this back to get the desired isomorphism. By Proposition 20.3 of their paper, we’re done.

$\newcommand{\QQ}{\mathbf{Q}}$ Here’s why homotopy theorists care about this result. Like in any closed symmetric monoidal category, we can try to take the dual of an object. In general, this object won’t necessarily behave as we expect a dual to behave (i.e., to have evaluation and coevaluation maps that play nicely), but if the object is finite (say, compact), the notion of dualizability works well. We can play this game of dualizing objects in the category of spectra. One needs to be a little careful, because spectra form an $\infty$-category, and one has to tiptoe their way through $\infty$-category–land.


$\newcommand{\Lk}{L_{K(n)}}$ Gross-Hopkins duality provides a relationship between these two kinds of duality, namely Spanier-Whitehead duality and Brown-Comenetz/Anderson duality.

$\newcommand{\sph}{S}$ Theorem (Gross-Hopkins duality): Let $\Lk\sph$ denote the $K(n)$-local sphere. Then $I_{\QQ/\Z}\Lk\sph=:I$ is $K(n)$-locally invertible. Moreover, if $E$ denotes Morava $E$-theory (at height $n$) and $E^\vee_\ast (X)$ denotes $\pi_\ast \Lk(E\wedge X)$, then there’s a $\GG_n$-equivariant isomorphism

This might seem like an unnecessarily complicated result to even be useful. It turns out that if $p\gg n$, the formula in the above equation determines $I$ uniquely. In fact, one can write

where $\sph[\det]$ is a spectrum that exists only after $K(n)$-localization, called the “determinantal sphere”. If $n=1$ and $p>2$, then $\sph[\det]$ is just the $2$-sphere, but in general, it’s a very exotic.

$\newcommand{\Hom}{\mathrm{Hom}}$ We can use this formula to write down a relationship between Spanier-Whitehead and Brown-Comenetz duality:

To deduce this, we needed the statement of the completed $E$-homology of $I$; this can be proven by using the Corollary from above: One uses the following facts to do so (a great, but technical, exposition is Strickland’s paper, titled Gross-Hopkins duality).

• There’s an equivalence $\pi_\ast E \simeq \pi_{n+\ast} E/\frakm^\infty$, whatever the latter symbols mean.
• There’s another equivalence $\pi_\ast (E/\frakm^\infty) \simeq \Omega^{n-1}_{\LT_n/W(\FF_{p^n})}\otimes_{\mathcal{O}_{\LT_n}} \pi_{-\ast} E$. If you’d like, take this to be the definition of $\pi_\ast (E/\frakm^\infty)$, as we won’t ever be using the spectrum $E/\frakm^\infty$.
• There’s yet another equivalence $\Lk DE \simeq \Sigma^{-n^2} E$. From these equivalences, we get:

By the corollary from above, it now suffices to show that $\omega^{\otimes k} \simeq \pi_{2k} E$. Recall that the reduced cohomology $\widetilde{E}^\ast(X)$ is defined as the kernel of $E^\ast(X)\to E^\ast(\ast)$, induced by the inclusion of the basepoint into $X$. If $E$ is complex-oriented, so that $E^\ast(\mathbf{CP}^\infty)\simeq E^\ast[[x]]$ (with $|x| = 2$), then $\widetilde{E}^\ast(\mathbf{CP}^\infty) \simeq x \cdot E^\ast[[x]]$. This is exactly the maximal ideal $\frakm$ of the formal group $\GG_E = \spf E^\ast[[x]]\simeq \spf E^\ast(\mathbf{CP}^\infty)$. Now, $\pi_2 E = [\sph^2,E] = [\mathbf{CP}^1,E] = \widetilde{E}^\ast(\mathbf{CP}^1)$ is exactly $(x)/(x^2) = \frakm/\frakm^2$ — precisely the relative cotangent space/module of invariant differentials $\omega$! Remembering that $E$ is complex-oriented, this tells us that $\pi_{2k} E = \omega^{\otimes k}$.

Notice the eerie similarity with Serre duality, which states that if $X$ is a projective variety of dimension $n$ over $k$ with dualizing sheaf $\omega_X$, there is a canonical isomorphism

where $\cf$ is a coherent sheaf on $X$. As far as I know, there’s no explanation for this similarity with Gross-Hopkins duality in homotopy theory. I think, though, that one can deduce the invertibility of $I$ from some general principles of Serre duality.


$\newcommand{\spec}{\text{Spec }}$ Dualizing sheaves are unique up to invertible objects (if $I_1$ and $I_2$ are two dualizing sheaves for $\dX$, then $I_1\simeq I_2\otimes \Hom(I_2,I_1)$). Thus, self-duality implies that $f^! I_\Z$ is invertible, where $f:\dX\to \spec S$ is the structure map.

$\newcommand{\Eoo}{\mathbf{E}_\infty}$ For instance, if $R$ is a $\Eoo$-ring such that $\pi_0 R$ is a regular Noetherian ring of finite Krull dimension, then $R$ is a dualizing complex for $\spec R$. This is Corollary 6.5.4.10 of Spectral Algebraic Geometry.

Let $R$ be an even periodic $\Eoo$-ring. Suppose $u\in\pi_0 R$ is a nonzero divisor. In general, the quotient $R/u$ is not an $\Eoo$-ring. Suppose, though, that $R/u$ is another $\Eoo$-ring. Then $f:\spec R/u\to \spec R$ is a closed immersion, and $f^! M\simeq \mathrm{Map}_R(R/u,M)$. This is exactly the fiber of $M\xrightarrow{u} M$, which is $\Sigma^{-1} M/u$. Inductively, it follows that if $I$ is an ideal generated by a regular sequence $u_1,\cdots,u_n$ such that $R/I$ is an $\Eoo$-ring, then $f:\spec R/I\to \spec R$ is a closed immersion, and $f^! M\simeq \Sigma^{-n} M/I$. Thus if $K$ is a dualizing sheaf for $\spec R$ and $I$ is an ideal satisfying the above conditions, then $\Sigma^{-n} K/I$ is a dualizing sheaf for $\spec R/I$.

It follows from this discussion that any $\Eoo$-quotient of a regular Noetherian $\Eoo$-ring of finite Krull dimension is also self-dual.

$\newcommand{\pic}{\mathrm{Pic}}$ We can deduce the invertibility of the Gross-Hopkins spectrum from this: it follows from the above discussion that $\dX = \spf E/\GG_n$ is self-dual. Indeed, the above discussion shows that $\spf E$ is self-dual. Since the map $\spf E\to \dX$ is etale and surjective, Remark 6.6.1.3 of Spectral Algebraic Geometry implies that $\dX$ is self-dual, which tells us that the Gross-Hopkins spectrum is ($K(n)$-locally) invertible.

$\newcommand{\QCoh}{\mathrm{QCoh}}$ A natural question to ask is the following. Can recent results by Barthel-Beaudry-Stojanoska on the Anderson self-duality of higher real K-theories be deduced from the discussion above? One step in this direction is the following.

$\newcommand{\Mod}{\mathrm{Mod}}$ If $G$ is any finite subgroup of $\GG_n$, the map $\spf E\to \spf E/G$ is an etale surjection. We know, by an argument similar to the proof of Theorem 1 in this post, that $\QCoh(\spf E/G)\simeq (\Lk\Mod(E))^{hG}$. Suppose $G$ is such that $(\Lk\Mod(E))^{hG}\simeq \Lk\Mod(E^{hG})$. Again utilizing Remark 6.6.1.3 of Spectral Algebraic Geometry, we learn that $\spf E/G$ is self-dual, so that $I_\Z E^{hG}$ is in $\pic \spf E/G \simeq \pic(E^{hG})\simeq \pic(E)^{hG}$. It’d follow that $I_\Z E^{hG}$ is self-dual if we knew that $\pi_0 \pic(E^{hG})$ is cyclic. This is known to be true in some cases; it’d be interesting to find necessary and sufficient conditions on $G$ that ensure that this group is cyclic.