Audit 6.4 · Audit of smooth stabilizers and orbit immersions over \(\mathbb Z[1/2]\) (claim A2-4)

1. Products of root groups. Conrad Thm. 5.1.13 (local PDF): for any positive system \(\Phi^+\) of a split reductive group over a non-empty scheme \(S\), \(U_{-\Phi^+}\times T\times U_{\Phi^+}\to G\) is an open immersion. Prop. 5.1.16(1): for a closed \(\Psi\) with \(\Psi\cap-\Psi=\emptyset\), multiplication in any order is an isomorphism onto a smooth closed subgroup \(U_\Psi\). Ordering \(\Phi^+\) with \(\Phi^+\cap\Phi''^-\) first (both pieces closed) gives \(U_{\Phi^+}=U_{\Phi^+\cap\Phi''^-}\times U_{\Phi^+\cap\Phi''^+}\), which is the one step A2 leaves implicit; then \(B\), \(B''=\theta_n(B)\), \(BB''\) and \(B\cap B''=T\ltimes U_{\Phi^+\cap\Phi''^+}\) are as displayed in Lemma 2.4 inside \(\Omega''\). \(BB''\) is closed in the open \(\Omega''\), hence locally closed and smooth in \(G\).
2. Stabilizer. \(\operatorname{Stab}_G(nx_0)=nG^\theta n^{-1}=G^{\theta_n}\) as schemes, so \(B_y=B\cap G^{\theta_n}\); if \(b=\theta_n(b)\) then \(b\in\theta_n(B)\), so \(B_y=(B\cap\theta_nB)^{\theta_n}\). Over \(\mathbb Z[1/2]\) the constant group \(\mathbb Z/2\) is \(\mu_2\), of multiplicative type, so the fixed locus on the smooth affine \(B\cap\theta_nB\) is smooth (CGP, Prop. A.8.10(2); Edixhoven, Prop. 3.4) with Lie algebra the fixed vectors: \(4+|\Phi_x\cap\Phi^+|+c_x=16+c_x\) (fixed roots contribute a line each because \(\theta_n\) fixes their root groups pointwise, Audit 6.2 step 1; each \(\theta_n\)-pair in \(\Phi^+\cap\theta_n\Phi^+\) one line; no real roots).
3. \(\mathcal F\) and \(\mu\). \(\sigma(g)=\theta_n(g)^{-1}\) is an involution of \(BB''\) since \(\theta_n\) swaps \(B\) and \(B''\); \(\sigma(\mu(b))=\mu(b)\), so \(\mu\) lands in \(\mathcal F=(BB'')^\sigma\), smooth by the same citation, with \(T_1\mathcal F=(\mathfrak b+\mathfrak b'')^{-\theta_n}\). \(d\mu_1=1-\theta_n\) on \(\mathfrak b\) has kernel \(\mathfrak b^{\theta_n}=\operatorname{Lie}B_y\), and its image is all of \(T_1\mathcal F\): for \(Z=b_1+\theta_nb_2\) with \(\theta_nZ=-Z\), \(Z=(1-\theta_n)\tfrac{b_1-b_2}2\). On each geometric fibre \(B/B_y\to\mathcal F\) is therefore étale at the base point, étale everywhere by \(B\)-equivariance, and a monomorphism, hence an open immersion (EGA IV 17.9.1: a flat monomorphism locally of finite presentation is an open immersion).
4. Flatness and the quotient. \(B\) is flat over the base, \(\mathcal F\) is of finite presentation, and \(\mu\) is flat on every fibre, so \(\mu\) is flat (EGA IV 11.3.10); flat and locally of finite presentation, it is open. \(B\to\mathcal O_n:=\mu(B)\) is an fppf covering and \(\mu(b)=\mu(b')\iff b'^{-1}b=\theta_n(b'^{-1}b)\), so \(B\times_{\mathcal O_n}B\simeq B\times B_y\) and \(\mathcal O_n\) represents \(B/B_y\).
5. Immersion into \(X\). \(\tau(gH)=g\theta(g)^{-1}\) is a monomorphism because \(H\) is the whole fixed-point scheme \(G^\theta\) (not an open subgroup of it). A direct computation gives \(\tau_n(by)=b\theta_n(b)^{-1}=\mu(b)\), so \(b\mapsto by\) maps \(B\) into \(Y_n=\tau_n^{-1}(\mathcal O_n)\), which is locally closed because \(\mathcal O_n\) is locally closed in \(G\). \(Y_n\to\mathcal O_n\) is a monomorphism, and an fppf epimorphism since its composite with \(B\to Y_n\) is \(\mu\); so it is an isomorphism of sheaves, hence of schemes, and \(Y_n\simeq B/B_y\) is smooth of relative dimension \(42-(16+c_x)=26-c_x\).
6. Independence and base change. Lifts of \(vC_W(w_L)\) are \(nht\) with \(h\in N_H(T)\) (Audit 6.2 step 6), and \(nhtx_0\in T\cdot y\subseteq By\). Open cells, fixed-point schemes, stabilizers and images of flat maps commute with base change \(\mathbb Z[1/2]\to\mathcal O\).

Outcome: holds.