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
Morava -theory is -complete, where is the maximal ideal of . Thus it makes sense to consider the derived affine formal scheme . Let me imprecise here and claim that this has a continuous action of the profinite Morava stabilizer group , although I have not constructed this action yet (note that the action exists – proving that it is continuous on (and not is hard).
Let be an -ring. A projective -module is a retract of a free -module. Projective -modules are flat, since direct sums and retracts of flat modules are flat.
Definition: Let be a connected derived stack, i.e., a (possibly nonconnective) connected spectral Deligne-Mumford stack. A line bundle on is a quasicoherent sheaf such that for every etale , the pullback satisfies the following properties:
- is a projective -module such that is a finitely generated -module
- is a -vector space of dimension where is a field with a map of -rings .
is the space of suspensions of line bundles on , topologized as a subspace of .
Let be an even periodic -ring, and let be a line bundle over . Then is a projective -module. Indeed, Proposition 220.127.116.11 of Higher Algebra, suitably adapted to the nonconnective scenario, shows that is a projective -module. The result then follows from being even periodic, the fact that projective modules are flat, and the fact that .
Let denote Morava E-theory at height (as constructed in the previous post), and be a line bundle over . Then is a free -module of rank with the action of . This follows from the claim in the previous paragraph, the fact that 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 .
The -local Picard group
Recall that we define for a closed subgroup (of finite index) by the limit
As we’ll be using descent techniques, the following theorem is important.
Theorem (Devinatz-Hopkins): There is an equivalence .
I don’t know if the adic topology on is the same as the adic topology on the homotopy limit used to construct . More generally, I don’t know if the -category of adic -rings is complete.
Let be the stack .
Theorem 1: There are symmetric monoidal equivalences
Proof. For the first statement, Corollary 18.104.22.168 of SAG tells us that it suffices to prove that an -module is -complete iff it is -local. Let be a -local complex oriented -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 is an -module, is a -module spectrum, hence -local. We can obtain (where ) from via a finite number of cofibration sequences. This shows that is a -local -module. Taking the direct limit, we find that is -local. It suffices to prove that . For this we may use the same argument in Proposition 4.1 of Chapter 6 of the TMF book, again. The statement that -local is equivalent to -complete now follows immediately from Corollary 22.214.171.124 of SAG.
For the second statement, we use descent. We have an equivalence:
Since is a -Galois etale cover and , it follows that the cosimplicial diagram is the cobar construction for homotopy fixed points. So, we have . Thus . This concludes the proof.
A word of warning: is not ! For instance, let denote the -local sphere with the -adic topology. Then . Indeed, by Corollary 126.96.36.199 of SAG, we know that . Clearly this is not equivalent to .
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 -local spectrum is in the -local Picard space ;
- is a free -module of rank .
Proof. The previous theorem shows that , and we know that . Note that here we are critically using the fact that is the Picard space, and not the Picard group! The Picard group doesn’t satisfy any kind of descent (because it’s , and doesn’t commute with limits or colimits).
Suppose is in . Since is connected, every invertible object of is of the form where is a line bundle on and ; so we can assume that is a line bundle. We know that , so since acts on the first factor, it follows that is a free -module of rank .
Now, assume the converse. As a consequence of the main result of the main result of this paper of Baker-Richter, we know that is in . This has a -linearization since (by Hopkins-Miller) acts (continuously) via the first factor of , and descends to the structure sheaf on .
And we’re done!
Corollary 3: There is an equivalence that respects the -action.
This follows from Theorem 1 and Theorem 188.8.131.52 of SAG. Now, Theorem 2 and Corollary 3 furnish a homotopy equivalence . This gives a spectral sequence for computing :
If we want to understand , we therefore can try to understand and the -action on it. Note that there is a homotopy equivalence . This tells us that (I don’t think I have a reference for this statement), and that . (In fact, there is a fiber sequence .)