§2 · Agent 2 · Integral models · 2026-10-18T03:18:30Z

Lemma 2.4 · Smooth stabilizers; orbit immersions $B/B_x\hookrightarrow X$ over $\mathbb Z[1/2]$

proved audit: correct ★ key step

Second audit: correct. Re-derived step by step: B∩θ_nB=T⋉U_{Φ⁺∩θ_nΦ⁺} and BB″=U_{Φ⁺∩Φ″⁻}×T×U_{Φ″⁺} is closed in the open cell Ω″ (Conrad Thm 5.1.13 and Prop 5.1.16, stated for a split reductive group over any non-empty base, checked in the local text); B_y and F are fixed loci of Z/2≅μ₂ on smooth Z[1/2]-schemes (CGP A.8.10(2), Edixhoven Prop 3.4: not available locally, standard), and dim B_{y,k}=16+c_x on every fibre since the +1-eigenlattice of an involution has the same rank mod every p≠2. dμ₁=1−θ_n is onto T₁F=(b+b″)^{−θ_n} because Z=(1−θ_n)(b₁−b₂)/2, so on fibres B_k/B_{y,k}→F_k is an étale monomorphism, an open immersion (EGA IV 17.9.1), and μ is flat by the fibrewise criterion (EGA IV 11.3.10; B flat of finite presentation over Z[1/2]). τ_n(by)=bθ_n(b)⁻¹ is right; Y_n=X×_G O_n→O_n is a monomorphism (base change of τ_n) and an fppf epimorphism (μ factors through it), hence an isomorphism; lifts nht of vC_W(w_L) give points of T(Z[1/2])·y, so Y_n depends only on x, and everything commutes with base change to O.

For $n\in N_G(T)(\mathbb Z[1/2])$, $\operatorname{Stab}_B(nx_0)$ is smooth with fibre dimension independent of the fibre, and the orbit map identifies $B/\operatorname{Stab}_B(nx_0)$ with a smooth locally closed subscheme of $X$: through $\tau$ the orbit is open in a fixed locus inside the locally closed $B\theta_n(B)\subseteq G$.

Lemma 2.4.

Let $n\in N_G(T)(\mathbb Z[1/2])$ lift $v$, $y=nx_0$, $\theta_n=\operatorname{Int}(n)\theta\operatorname{Int}(n)^{-1}$, $x=vw_Lv^{-1}$, $B''=\theta_n(B)$. Over $\mathbb Z[1/2]$, hence over every $\mathcal O\ni\frac12$:

$B_y=B\cap G^{\theta_n}=(T\ltimes U_{\Phi^+\cap\theta_n\Phi^+})^{\theta_n}$ is smooth with fibres of dimension $16+c_x$.
$BB''\subseteq G$ is smooth locally closed; with $\mathcal F$ the fixed locus of $g\mapsto\theta_n(g)^{-1}$ on it, $\mu(b)=b\theta_n(b)^{-1}$ is flat onto an open $\mathcal O_n\subseteq\mathcal F$ and $B/B_y\simeq\mathcal O_n$.
With $\tau_n(z)=\tau(z)\tau(y)^{-1}$, $Y_n:=\tau_n^{-1}(\mathcal O_n)$ is locally closed in $X$ and $b\mapsto by$ induces $B/B_y\simeq Y_n$: smooth of relative dimension $26-c_x$, fibres the orbits $B_ky$.
$Y_n$ depends only on $x$ (write $Y_x\ni y_x$).

Root groups (any base). For closed $\Psi$ with $\Psi\cap-\Psi=\emptyset$, $\prod_{\gamma\in\Psi}U_\gamma\simeq U_\Psi$, a smooth closed subgroup (Conrad, Prop. 5.1.16); $U_{-\Phi'^+}\times T\times U_{\Phi'^+}\to G$ is an open immersion for every positive system (Conrad, Thm. 5.1.13).
$BB''$. With $\Phi''^+=\theta_n\Phi^+$, inside the open cell $\Omega''=U_{\Phi''^-}\times T\times U_{\Phi''^+}$:

$$B=U_{\Phi^+\cap\Phi''^-}\times T\times U_{\Phi^+\cap\Phi''^+},\quad B''=1\times T\times U_{\Phi''^+},\quad BB''=U_{\Phi^+\cap\Phi''^-}\times T\times U_{\Phi''^+}.$$

So $BB''$ is closed in $\Omega''$, smooth affine, $T_1(BB'')=\mathfrak b+\mathfrak b''$, and $B\cap B''=T\ltimes U_{\Phi^+\cap\Phi''^+}$.

Item 1. $B_y$ is the fixed locus of $\theta_n$ on the smooth $B\cap\theta_nB$; $\mathbb Z/2\simeq\mu_2$ over $\mathbb Z[1/2]$, so it is smooth with Lie algebra the fixed vectors (CGP A.8.10(2); Edixhoven 3.4): $4+12+c_x$ by Lemma 2.2, 2–3, for every $p\ne2$.
$\mathcal F$. $\sigma(g)=\theta_n(g)^{-1}$ preserves $BB''$ (as $\theta_nB''=B$); $\mathcal F$ is smooth with $T_1\mathcal F=(\mathfrak b+\mathfrak b'')^{-\theta_n}$. $\mu$ is the orbit map at $1$ of $b*g=bg\theta_n(b)^{-1}$ on $\mathcal F$, with scheme-theoretic stabilizer $B_y$.
Fibres. $d\mu_1=1-\theta_n$ has kernel $\operatorname{Lie}B_y$ and image $T_1\mathcal F$ ($Z=b_1+\theta_nb_2$, $\theta_nZ=-Z$ gives $Z=(1-\theta_n)\frac{b_1-b_2}2$). So $B_k/B_{y,k}\to\mathcal F_k$ is an étale monomorphism, an open immersion (EGA IV 17.9.1).
Item 2. $\mu$ is flat by the fibrewise criterion (EGA IV 11.3.10), hence open; $B\to\mathcal O_n$ is fppf with $B\times_{\mathcal O_n}B\simeq B\times B_y$, so $\mathcal O_n\simeq B/B_y$.
Item 3. $\tau_n(by)=bn\theta(n)^{-1}\theta(b)^{-1}\theta(n)n^{-1}=b\theta_n(b)^{-1}$. $\mathcal O_n$ is locally closed in $G$, so $Y_n$ is locally closed; $\tau_n|_{Y_n}$ is a monomorphism whose composite with $B\to Y_n$ is the epimorphism $\mu$, hence an isomorphism.
Item 4. Lifts of $vC_W(w_L)$ are $nht$, $h\in N_H(T)$ (Lemma 2.2, 6), and $nhtx_0\in T\cdot y$. All steps commute with base change $\mathbb Z[1/2]\to\mathcal O$.

Builds on

Lemma 2.2 · The involutions $\theta_x$: compact imaginary roots, no real roots
Proposition 2.3 · Borel orbits of $E_6/F_4$ over an algebraically closed field of characteristic $\ne2$

Built on by

Proposition 2.5 · Rank-one quotients: a point (type $G$) or $\mathbb P^1_{\mathcal O}$ (type $U$)
Lemma 2.7 · Orbit closures over $\mathcal O$ have the expected special fibres
Theorem 2.8 · Theorem: Axiom 2.2.1 holds for $\mathcal M_\theta$ over every DVR with $2$ invertible
Audit 6.4 · Audit of smooth stabilizers and orbit immersions over $\mathbb Z[1/2]$ (claim A2-4)

Sources

conrad-reductive-group-schemes.pdf, Thm. 5.1.13, Prop. 5.1.16
CGP Prop. A.8.10; Edixhoven 1992 Prop. 3.4 (cited)
EGA IV 11.3.10, 17.9.1 (cited)

Table 1. Page production by session
	tokens	US $
The run, its in-run audits and the pages (from 01:50 UTC)01:50–05:20 UTC	349 M	$144
This conversation: the audit pass (eight reviewers, one salvage agent, re-checks) and the simplified VIb page05:21–07:33 UTC	267 M	$106
Removing the detail switch from the published trees05:34–05:51 UTC	15 M	$7
Total	631 M	$257

Lemma 2.4 · Smooth stabilizers; orbit immersions \(B/B_x\hookrightarrow X\) over \(\mathbb Z[1/2]\)

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