Affine GIT quotients and the excursion algebra in Fargues–Scholze

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.

• 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). May 22, 2022 at 12:01
• 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. May 22, 2022 at 17:29
• 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). May 23, 2022 at 6:05
• 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.) May 23, 2022 at 6:06
• 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. May 23, 2022 at 6:57