$\newcommand{\Eoo}{\mathbf{E}_\infty}$ It’s been a rather long pause since the last post; I apologize for that. In this post, I’d like to describe some work which I recently completed that resolves a question posed in this old post. That post contains a proof that the ring parametrizing Drinfel’d level $p^k$-structures at height $1$ does not lift to a $K(1)$-local $\Eoo$-ring, owing to it having a primitive $p$th root of unity in its homotopy. Here, I’ll describe a generalization of this result, and apply it to show that the ring parametrizing Drinfel’d level $p^k$-structures at any height and any prime (there’s a slight modification — as always — for $p=2$) does not lift to a $K(n)$-local $\Eoo$-ring. This is the subject of my paper.

# Roots of unity

I’d asked Tyler Lawson a few weeks ago whether he knew a proof for the nonexistence of derived lifts of Drinfel’d level structures in homotopy theory. He responded that he didn’t, but he gave an interesting argument for a related result.

Proposition (Lawson): There is no $K(1)$-local $\Eoo$-ring $R$ such that $\pi_0 R$ contains a primitive $2^k$th root of unity for some $k\geq 2$.

The proof is rather simple: suppose otherwise; then such an $\Eoo$-ring $R$ would necessarily have a primitive fourth root of unity $i$ in $\pi_0 R$. Since

if $x=y=i$, then $\theta(-1) = -1$ would be $-2(\theta(i) - \theta(i)^2)$. This, however, implies that $2$ is invertible, which is impossible.

Naturally, Tyler asked if this result could be generalized to all heights and primes. The answer, unsurprisingly, is yes:

Theorem: Let $p$ be an odd prime. There is no $K(n)$-local $\Eoo$-ring (in fact, one can replace $\Eoo$ with $H_\infty$ here) $R$ such that $\pi_0 R$ contains a primitive $p^k$th root of unity for some $k\geq 1$.

The proof critically hinges on the following recent result of Jeremy Hahn’s:

Theorem (Hahn): Any $K(n)$-acyclic $H_\infty$-ring spectrum is $K(n+1)$-acyclic.

Using Jeremy’s result, we can reduce to the height $1$ case: suppose $n\geq 2$, and let $R$ be a $K(n)$-local $\Eoo$-ring such that $\pi_0 R$ contains a primitive $p^k$th root of unity $\zeta_{p^k}$ for some $k\geq 1$. Then $\pi_0 L_{K(1)} R$ also contains a primitive $p^k$th root of unity, simply because $\pi_0 R\xrightarrow{f} \pi_0 L_{K(1)} R$ is a ring homomorphism, so $f(\zeta_{p^k})$ is another primitive $p^k$th root of unity. Assuming the result is true at height $1$, we find that $L_{K(1)} R\simeq 0$, i.e., $R$ is $K(1)$-acyclic. But Jeremy’s theorem tells us that $R$ is $K(h)$-acyclic for every $h\geq 2$, so $R\simeq L_{K(n)} R\simeq 0$, which is a problem.

We worked to reduce this to height $1$ because any $K(1)$-local $\Eoo$-ring $R$ admits power operations $\psi^p,\theta:\pi_0 R\to \pi_0 R$ with $\psi^p$ a ring homomorphism. This is necessary to prove the main result. Here’s an outline of the proof (at height $1$, of course).

• Prove (easily) the following formula for the $\theta$-operation:
• If $\pi_0 R$ contains a primitive $p^k$th root of unity, it also contains a primitive $p$th root of unity $\zeta$.
• Since the sum of all $p$th roots of unity is zero, the left hand side evaluates to zero when $x_k=\zeta_p^k$. One can show that the ugly sum on the right hand side evaluates to $-1$.
• Utilize the multiplicativity of $\psi^p$ to observe that $\theta(x^n) = ((x^p+p\theta(x))^n-x^{np})/p$. Plugging this into the first sum on the right hand side gives $p\cdot\text{something nonzero}$.
• Conclude that $p$ is invertible.
• Realize that that’s absurd.

# Drinfel’d level structures

We can now finally apply these results to proving that the Lubin-Tate tower does not lift to a tower of $\Eoo$-rings over Morava $E$-theory. We’ll use notions from the prequel, but we’ll slightly change notation: what’s denoted by $R_k$ at height $n$ in that post will be denoted by $A_k^n$ in this one. $\newcommand{\M}{\mathcal{M}}$

Corollary: Let $\M_k^h$ denote the Lubin-Tate space at level $p^k$ and height $h$, and let $E$ denote Morava $E$-theory at height $n$ and the prime $p$. Then $\M_k^h$ does not admit a lift to a derived stack over $E$ for any $k>0$. $\newcommand{\Z}{\mathbf{Z}}$

To prove this, we’ll need the “determinant” map $\M_k^h \to \M_k^1$ that’s constructed in Section 2.5 of Weinstein’s Semistable models for modular curves of arbitrary level. Both formal schemes are affine, i.e., $\M_k^h\simeq \text{Spf }A_k^h$ for some $A_k^h$. For instance, $A_k^1 \simeq \Z_p[\zeta_{p^k}]$ (see, e.g., the post linked to above). The image of $\zeta_{p^k}$ under the induced map $\Z_p[\zeta_{p^k}] \to A_k^h$ is a primitive $p^k$th root of unity in $A_k^h$. Applying the main result of this post, we conclude that $A_k^h$ does not lift to a $K(h)$-local $\Eoo$-ring; in particular, there is no $K(h)$-local $E$-algebra whose homotopy is isomorphic to $A_k^h$, as desired! (Note that this argument shows that $A_k^h$ doesn’t even lift to a $K(h)$-local $H_\infty$-ring.)

Note that at infinite level, we can provide a somewhat different proof of the nonexistence of a lift: there is an isomorphism $\M_\infty^h\simeq \text{Spf }A_\infty^h$, where $A_\infty^h$ is a $W(\mathbf{F}_{p^h})[[u_1,\cdots,u_{h-1}]]$-algebra. By Theorem 2.8.1 of the paper of Weinstein’s (linked to above), we have the following presentation of $A_\infty^h$:

It follows that $A_\infty^h$ contains a primitive $p$th root of unity, and hence does not lift by the main result of this post.

Before I end: as is evident above, this post sprouted from conversations with Tyler Lawson, who suggested this project. I also had some useful conversations with Marc Hoyois, Matt Ando, Catherine Ray, Alex Mennen, and Paul VanKoughnett on both the topic of this post and related stuff.