$\newcommand{\frakm}{\mathfrak{m}}$ A warning before we start: a lot of the results cited from Lurie’s Spectral Algebraic Geometry and Higher Algebra (referred to here as SAG and HA, respectively) are stated for connective ring spectra, but I believe that the results extend to nonconnective spectra as well. In particular, we’ve defined things so that the proofs of the desired results pass through.

This means that it might be true that none of these results are correct, but it’s still a good idea to work through this, because it’s a cute application of DAG.

# Generalities on Picard groups




• $f^\ast \ff$ is a projective $R$-module such that $\pi_0 f^\ast \ff$ is a finitely generated $\pi_0 R$-module
• $\pi_0(k\otimes_R f^\ast \ff)$ is a $k$-vector space of dimension $1$ where $k$ is a field with a map of $\Eoo$-rings $R\to k$.

$\newcommand{\QCoh}{\mathrm{QCoh}}$ $\pic(\dX)$ is the space of suspensions of line bundles on $\dX$, topologized as a subspace of $\QCoh(\dX)^\simeq$.

Let $R$ be an even periodic $\Eoo$-ring, and let $M$ be a line bundle over $R$. Then $\pi_\ast M$ is a projective $\pi_0 R$-module. Indeed, Proposition 7.2.2.18 of Higher Algebra, suitably adapted to the nonconnective scenario, shows that $\pi_0 M$ is a projective $\pi_0 R$-module. The result then follows from $R$ being even periodic, the fact that projective modules are flat, and the fact that $\pi_n M \simeq \pi_n R \otimes_{\pi_0 R} \pi_0 M$.

Let $E$ denote Morava E-theory at height $n$ (as constructed in the previous post), and $M$ be a line bundle over $E$. Then $\pi_\ast M$ is a free $E_\ast$-module of rank $1$ with the action of $\GG_n$. This follows from the claim in the previous paragraph, the fact that $E_\ast$ is even periodic, the definition of flatness, the fact that a projective module over a local ring is free, and the naturality of the action of $\GG_n$.

# The $K(n)$-local Picard group

Recall that we define $E^{hU}$ for $U\leq \GG_n$ a closed subgroup (of finite index) by the limit

As we’ll be using descent techniques, the following theorem is important.

$\newcommand{\Lk}{L_{K(n)}}$ Theorem (Devinatz-Hopkins): There is an equivalence $E^{h\GG_n} \simeq \Lk S$.

I don’t know if the adic topology on $\Lk S$ is the same as the adic topology on the homotopy limit used to construct $E^{h\GG_n}$. More generally, I don’t know if the $\infty$-category of adic $\Eoo$-rings is complete.

Let $\dX$ be the stack $\spf E/\GG_n$.

$\newcommand{\Sp}{\mathrm{Sp}}$ Theorem 1: There are symmetric monoidal equivalences

Proof. For the first statement, Corollary 8.2.4.15 of SAG tells us that it suffices to prove that an $E$-module is $\frakm$-complete iff it is $K(n)$-local. Let $M$ be a $p$-local complex oriented $E$-module. Then there is an equivalence:

The proof of the claim is similar to Proposition 4.1 of Chapter 6 of the TMF book. Since $M$ is an $E$-module, $\beta^{-1} M/\frakm$ is a $K(n)$-module spectrum, hence $K(n)$-local. We can obtain $\beta^{-1} M/\frakm^I$ (where $I = (i_0,i_1,\cdots,i_{n-1})$) from $\beta^{-1} M/\frakm$ via a finite number of cofibration sequences. This shows that $\beta^{-1} M/\frakm^I$ is a $K(n)$-local $E$-module. Taking the direct limit, we find that $\beta^{-1} M/\frakm^\infty$ is $K(n)$-local. It suffices to prove that $K(n)_\ast \beta^{-1} M/\frakm^\infty \simeq K(n)_\ast M$. For this we may use the same argument in Proposition 4.1 of Chapter 6 of the TMF book, again. The statement that $K(n)$-local is equivalent to $I$-complete now follows immediately from Corollary 7.3.3.3 of SAG.

For the second statement, we use descent. We have an equivalence:

Since $\spf E \to \dX$ is a $\GG_n$-Galois etale cover and $\spf(E\widehat{\wedge} E) \simeq \spf E\times_{\dX} \spf E$, it follows that the cosimplicial diagram is the cobar construction for homotopy fixed points. So, we have $\QCoh(\dX) \simeq \QCoh(\spf E)^{h\GG_n}$. Thus $\QCoh(\dX) \simeq \Lk \Sp$. This concludes the proof.

A word of warning: $\dX$ is not $\spf L_{K(n)} S$! For instance, let $L_{K(1)} S$ denote the $K(1)$-local sphere with the $p$-adic topology. Then $\QCoh(\spf L_{K(1)} S)\not\simeq \QCoh(\dX)$. Indeed, by Corollary 8.2.4.15 of SAG, we know that $\QCoh(\spf L_{K(1)} S)\simeq \Mod(L_{K(1)} S)^{p\text{-complete}}$. Clearly this is not equivalent to $L_{K(1)} \Sp$.

Anyway, we can use descent in derived algebraic geometry to provide a slick proof of the following result of Hopkins-Mahowald-Sadofsky from the 1990s. (Their result said that these statements are equivalent to a further condition, but as I don’t know how to prove it using derived algebraic geometry, I’ve not included it in the statement below.)

Theorem 2 (Hopkins-Mahowald-Sadofsky): The following statements are equivalent.

• A $K(n)$-local spectrum $M$ is in the $K(n)$-local Picard space $\pic_n$;
• $E^\vee_\ast M$ is a free $E_\ast$-module of rank $1$.

Proof. The previous theorem shows that $\pic_n = \pic(\dX)$, and we know that $\pic(\dX) \simeq \pic(\spf E)^{h\GG_n}$. Note that here we are critically using the fact that $\pic$ is the Picard space, and not the Picard group! The Picard group doesn’t satisfy any kind of descent (because it’s $\pi_0\pic$, and $\pi_0$ doesn’t commute with limits or colimits).

Suppose $M$ is in $\pic_n$. Since $\spf E$ is connected, every invertible object of $\QCoh(\spf E)$ is of the form $\Sigma^k \mathcal{L}$ where $\mathcal{L}$ is a line bundle on $\spf E$ and $k\in\mathbf{Z}$; so we can assume that $f^\ast M$ is a line bundle. We know that $f^\ast M \simeq E \ \widehat{\wedge} \ M$, so since $\GG_n$ acts on the first factor, it follows that $E^\vee_\ast(M)$ is a free $E_\ast$-module of rank $1$.

Now, assume the converse. As a consequence of the main result of the main result of this paper of Baker-Richter, we know that $E \ \widehat{\wedge} \ M$ is in $\pic(\spf E)$. This has a $\GG_n$-linearization since (by Hopkins-Miller) $\GG_n$ acts (continuously) via the first factor of $E \ \widehat{\wedge} \ M$, and $E$ descends to the structure sheaf $\Lk S$ on $\dX$.

And we’re done!

Corollary 3: There is an equivalence $\pic(\spf E) \simeq \pic(E)$ that respects the $\GG_n$-action.

This follows from Theorem 1 and Theorem 8.5.0.3 of SAG. Now, Theorem 2 and Corollary 3 furnish a homotopy equivalence $\pic(E)^{h\GG_n} \simeq \pic_n$. This gives a spectral sequence for computing $\pic_n$:

If we want to understand $\pic_n$, we therefore can try to understand $\pic(E)$ and the $\GG_n$-action on it. Note that there is a homotopy equivalence $\pic(E) \simeq BGL_1(E) \simeq \Omega^\infty \Sigma gl_1 E$. This tells us that $\pi_0\pic(E) \simeq \mathbf{Z}/2$ (I don’t think I have a reference for this statement), and that $\pi_1 \pic(E) \simeq (W(\mathbf{F}_{p^n})[[u_1,\cdots,u_{n-1}]])^\times$. (In fact, there is a fiber sequence $\Sigma gl_1 E \to \mathfrak{pic} E \to H\mathbf{F}_2$.)