ApolloniusSun

Outline

$K(n)$-local sphere is essentially the basic object of chromatic homotopy theory. However people even failed to completely understand its homotopy group so far. In 2024, Barthel, Schlank, Stapleton and Weinstein use input from $p$-adic geometry to fully understand the rational part of this homotopy group.

The key point is the fact that $L_{K(n)}\mathbb{S} \to E_n$ exhibts the latter as a pro-Galois extension of the former ring spectra with pro-Galois group $\mathbb G_n$ (the Morava stablizer group). In other words: $L_{K(n)}\mathbb {S}\simeq E_n^{h\mathbb G_n}$

The homotopy fixed point spectral sequence computing $L_{K(n)}\mathbb S\otimes \mathbb Q$ turns out to collapse immediately, hence there's no extra indeterminancy in this Galois descent process. In fact the only nontrivial terms in this spectral sequence is $H^*(\mathbb G_n, \pi_0 E_n)$.

The subtlety is that the action of the Morava stablizer group is in general hard to describe, this is where the p-adic geometry comes in.

Theorem.(Scholze-Weinstein) There exists a perfectoid space $\mathcal X$:

$LT_K\stackrel{GL_n(\Z_p)}{\longleftarrow} \mathcal X\stackrel{\mathcal O_D^\times}{\longrightarrow} \mathcal H_K$

where both arrows are pro-etale torsor, $LT_K$ denotes the Lubin-Tate space; $\mathcal H_K$ denotes (the base change of) the Drinfeld symmetric space.
Moreover, these two pro-etale torsor are dual to each other in the following sense: $\mathcal X$ carries an action of $GL_n(\Z_p)\times \mathcal O_D^\times$ action such that $GL_n(\Z_p)$-pro-etale torsor is $\mathcal O_D^\times$-equivariant and vice versa.

This duality phenomenon can be explained in the view point of "shtukas" and modification of vector bundles on Fargues-Fontaine curve. We'll provide a detailed proof of this theorem in terms of the language introduced by Scholze.

The previous result somehow translate the action of $\mathbb G_n$ (or $\mathcal O_D^\times$) on $\pi_0 E_n$ (i.e. the Lubin-Tate space) to the action of $GL_n(\Z_p)$ on the Drinfeld symmetric space $\mathcal H_K$. Considering these two torsors, intuitively one can translate the $\mathcal O_D^\times$-fixed point of the cohomology of $LT_K$ to the $GL_n(\Z_p)$-fixed point of the cohomology of $\mathcal H_K$.

Thus the remaining problems are the following two:

1. Precisely which cohomology theory can fit into the previous framework?
2. These cohomology might differ from the actual (global section of )structure sheaf, how to compare them?

The answer of the first question is that pro-etale cohomology and condensed solid group cohomology (fixed point) fits in well; the second question is the main computational part of the B-S-S-W's work.

To be precise, by constructing a comparison theorem one can show that the pro-etale cohomology $R\Gamma_{proet}$ and $\Gamma[\varepsilon]$ differs by bounded p-torsion, hence equivalence after rationalization. This comparison theorem is almost purely formal after a explicit computation for the case of a single point, i.e. $\mathrm{Spec}(K)$, whereas in this case the theorem turns out to be the integral refined version of the Ax-Sen-Tate theorem.

Notes

Rational K(n)-local sphere (incompleted)