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 -structures at height does not lift to a -local -ring, owing to it having a primitive 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 -structures at any height and any prime (there’s a slight modification — as always — for ) does not lift to a -local -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 -local -ring such that contains a primitive th root of unity for some .
The proof is rather simple: suppose otherwise; then such an -ring would necessarily have a primitive fourth root of unity in . Since
if , then would be . This, however, implies that 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 be an odd prime. There is no -local -ring (in fact, one can replace with here) such that contains a primitive th root of unity for some .
The proof critically hinges on the following recent result of Jeremy Hahn’s:
Theorem (Hahn): Any -acyclic -ring spectrum is -acyclic.
Using Jeremy’s result, we can reduce to the height case: suppose , and let be a -local -ring such that contains a primitive th root of unity for some . Then also contains a primitive th root of unity, simply because is a ring homomorphism, so is another primitive th root of unity. Assuming the result is true at height , we find that , i.e., is -acyclic. But Jeremy’s theorem tells us that is -acyclic for every , so , which is a problem.
We worked to reduce this to height because any -local -ring admits power operations with a ring homomorphism. This is necessary to prove the main result. Here’s an outline of the proof (at height , of course).
- Prove (easily) the following formula for the -operation:
- If contains a primitive th root of unity, it also contains a primitive th root of unity .
- Since the sum of all th roots of unity is zero, the left hand side evaluates to zero when . One can show that the ugly sum on the right hand side evaluates to .
- Utilize the multiplicativity of to observe that . Plugging this into the first sum on the right hand side gives .
- Conclude that 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 -rings over Morava -theory. We’ll use notions from the prequel, but we’ll slightly change notation: what’s denoted by at height in that post will be denoted by in this one.
Corollary: Let denote the Lubin-Tate space at level and height , and let denote Morava -theory at height and the prime . Then does not admit a lift to a derived stack over for any .
To prove this, we’ll need the “determinant” map 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., for some . For instance, (see, e.g., the post linked to above). The image of under the induced map is a primitive th root of unity in . Applying the main result of this post, we conclude that does not lift to a -local -ring; in particular, there is no -local -algebra whose homotopy is isomorphic to , as desired! (Note that this argument shows that doesn’t even lift to a -local -ring.)
Note that at infinite level, we can provide a somewhat different proof of the nonexistence of a lift: there is an isomorphism , where is a -algebra. By Theorem 2.8.1 of the paper of Weinstein’s (linked to above), we have the following presentation of :
It follows that contains a primitive 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.