**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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