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Some background:

Let us fix a non-archimedean local field $E$ with residue characteristic $p$, and let $G$ be some connected reductive group over $E$. In [FS, §VIII.1.1] the authors define a moduli sheaf $Z^1(W_E,\widehat{G})$ of condensed $1$-cocycles for $\widehat{G}$ on $\mathbb{Z}_\ell$-schemes for $\ell\ne p$ (also defined by other authors). They show that this moduli space is representable with affine connected components.

More precisely, they first observe that

$$Z^1(W_E,\widehat{G})=\varinjlim_P Z^1(W_E/P,\widehat{G})$$

where here $P$ travels over open (normal) subgroups of the wild inertia group $P_E$ for $W_E$, and the transition maps are clopen. They then observe that, for a fixed $P$, one has that $Z^1(W_E/P,\widehat{G})= Z^1(W,\widehat{G})$ where $W\subseteq W_E/P$ is the discrete subgroup generated by the image of $P_E$ and choices of generators for tame inertia and a Frobenius. They then show that $Z^1(W,\widehat{G})$ is represented by an affine $\mathbb{Z}_\ell$-scheme which is finite type, flat, and a relative complete intersection.

$\DeclareMathOperator\colim{colim}$Wishing to study the affine GIT quotient(s) $Z^1(W,\widehat{G})\mathbin{//}\widehat{G}$ Fargues and Scholze observe that there is a $\widehat{G}$-equivariant identification $$\mathcal{O}(Z^1(W,\widehat{G}))=\colim_{\mathcal{J}}\,\mathcal{O}(Z^1(F_n,\widehat{G}))\tag{1}\label{1}$$

(where here $\mathcal{O}$ just means global sections of the structure sheaf). Here $\mathcal{J}$ is the category of all pairs $(n,\iota)$ where $n\geqslant 0$ is an integer and $\iota\colon F_n\to W$ is a group map. This identification is nearly tautological. From this we immediately see that one has an identification $$\mathcal{O}(Z^1(W,\widehat{G}))^{\widehat{G}}=\left(\colim_\mathcal{J} \,\mathcal{O}(Z^1(F_n,\widehat{G}))\right)^\widehat{G}.$$

$\DeclareMathOperator\Exc{Exc}$They then consider the excursion algebra

$$\Exc(W,\widehat{G}):=\mathrm{colim}_\mathcal{J}\, \mathcal{O}(Z^1(F_n,\widehat{G}))^{\widehat{G}},$$

and observe that one tautologically has a map of $\mathbb{Z}_\ell$-algebras $\Exc(W,\widehat{G})\to \mathcal{O}(Z^1(W,\widehat{G}))^{\widehat{G}}$.

The question

Fargues and Scholze make the following claim:

Claim: The map $\Exc(W,\widehat{G})\to \mathcal{O}(Z^1(W,\widehat{G}))^{\widehat{G}}$ induces a universal homeomorphism (of discrete $\mathbb{Z}_\ell$-schemes) and induces an isomorphism over $\mathbb{Q}_\ell$.

Their justification for this is that it follows from [H], but I am not sure how. As they give no further explanation, I would assume the reason is simple, and I am missing it.

One (possibly overcomplicated) explanation

$\DeclareMathOperator\Spec{Spec}$If I have not made a mistake, there is one way to justify this using the material in [A]. Namely, in op. cit. the author defines a ring map $A\to B$ to be adequate (resp. universally adequate) if for every $b$ in $B$ there exists some $N\geqslant 1$ and some $a$ in $A$ such that $a\mapsto b^N$ (resp. is adequate after every base change). He also then proves the following fact.

Fact 1 ([A, Lemma 3.14 and Lemma 3.15]): Let $A\to B$ be an injective (universally) adequate morphism. Then, $\Spec(B)\to \Spec(A)$ is a (universal) homeomorphism, and is an isomorphism if $A$ is a $\mathbb{Q}$-algebra.

From this we see that to prove the above stated claim of Fargues and Scholze it suffices to show that $\Exc(W,\widehat{G})\to \mathcal{O}(Z^1(W,\widehat{G}))^{\widehat{G}}$ is an injective universally adequate map. Injectivity is clear (cf. [B, Theorem 1]) and so it suffices to prove the map is universally adequate.

To prove this, one can use the following result in [A].

Fact 2 (cf. [A, Lemma 9.2.5]): Suppose that $H\to\Spec(\mathbb{Z}_\ell)$ is a geometrically reductive group scheme. If $A\to B$ is a surjection of finite type $\mathbb{Z}_\ell[H]$-algebras, then the induced map $A^H\to B^H$ is universally adequate.

I won't define what geometrically reductive means, but the results of Haboush and Seshadri (I assume that's what the reference to [H] was about, but I thought it would be more relevant to cite [S]) imply that as $\widehat{G}\to \Spec(\mathbb{Z}_\ell)$ is connected and reductive it is geometrically reductive.

Now, as $\mathcal{O}(Z^1(W,\widehat{G}))$ is finite type over $\mathbb{Z}_\ell$, it's easy to see from \eqref{1} that $\mathcal{O}(Z^1(F_n,\widehat{G}))\to \mathcal{O}(Z^1(W,\widehat{G}))$ is surjective for some $(n,\iota)$. We may then apply Fact 2 to see that $\mathcal{O}(Z^1(F_n,\widehat{G}))^{\widehat{G}}\to \mathcal{O}(Z^1(W,\widehat{G}))^{\widehat{G}}$ is universally adequate, and from this it follows fairly quickly that $\Exc(W,\widehat{G})\to \mathcal{O}(Z^1(W,\widehat{G}))^{\widehat{G}}$ is universally adequate, and so again we're done by Fact 1.


Is my explanation of this correct? Is there a simpler way to understand the proof of the claim?

I apologize if I have overcomplicated things.

References:

[A] Alper, J., 2014. Adequate moduli spaces and geometrically reductive group schemes.

[B] Bergman, G.M., 2005. Direct limits and fixed point sets. Journal of Algebra, 292(2), pp.592-614.

[FS] Fargues, L. and Scholze, P., 2021. Geometrization of the local Langlands correspondence. arXiv preprint arXiv:2102.13459.

[H] Haboush, W.J., 1975. Reductive groups are geometrically reductive. Annals of Mathematics, 102(1), pp.67-83.

[S] Seshadri, C.S., 1977. Geometric reductivity over arbitrary base. Advances in Mathematics, 26(3), pp.225-274.

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    $\begingroup$ It seems to me that perhaps Scholze-Fargues are alluding to the fact that $\widehat{G}$ being geometrically reductive (by Haboush's theorem) implies that $\text{Exc}(W, \widehat{G})$ and $\mathcal{O}(Z^1(W, \widehat{G}))^{\widehat{G}}$ are finite type $\mathbb{Z}_\ell$-algebras? If you have this result, then I believe you can get things like the map in your question being finite and surjective easily. I don't quite see why it's radicial yet (without reasoning as you did above). $\endgroup$ May 22, 2022 at 12:01
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    $\begingroup$ Sorry for being cryptic here! After inverting $\ell$, taking $\hat{G}$-invariants is exact, so the result is clear. Also, taking $\hat{G}$-invariants preserves finite type $\mathbb Z_\ell$-algebras; I gather I missed the good reference for that. Then to prove that it's a universal homeomorphism, it suffices to identify $\overline{\mathbb F}_\ell$-points. These correspond, by Haboush, to closed $\hat{G}$-orbits of the algebras base changed to $\overline{\mathbb F}_\ell$. But these closed $\hat{G}$-orbits can be identified easily. $\endgroup$ May 22, 2022 at 17:29
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    $\begingroup$ Let $A=\mathcal O(Z^1(W,\hat{G}))$ and $B=\mathcal O(Z^1(F_n,\hat{G}))$ for some surjection $F_n\to W$. Then the closed $\hat{G}$-orbits in $\mathrm{Spec}(A)$ inject into the closed $\hat{G}$-orbits in $\mathrm{Spec}(B)$. Conversely, a closed $\hat{G}$-orbit in $\mathrm{Spec}(B)$ which extends to a compatible system of closed $\hat{G}$-orbits in all other $B'$'s, necessarily lives in $\mathrm{Spec}(A)$ (as all its closed points do, as $A$ is the colimit of the $B'$'s). $\endgroup$ May 23, 2022 at 6:05
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    $\begingroup$ About Haboush: I think this is the key theorem to justify this correspondence to closed $\hat{G}$-orbits. (It guarantees that there are enough $\hat{G}$-invariant functions to distinguish between any two distinct closed $\hat{G}$-orbits.) $\endgroup$ May 23, 2022 at 6:06
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    $\begingroup$ You may find my expository paper Reductivity properties over an affine base, Indagationes Mathematicae (2020), doi.org/10.1016/j.indag.2020.09.009 helpful. $\endgroup$ May 23, 2022 at 6:57

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