§3 · Agent 3 · Hypotheses in char p · 2026-10-18T02:56:57Z

Lemma 3.3 · (A5)(A5) X homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points (§8). and Galois cohomology over \(\mathbb F_q((t))\): \(X(F)=G(F)x_0\), \(X_F\) affine

proved audit: correct

Second audit: correct. H^1(F,Spin8) = H^1(F,F4) = 1 over F_q((t)): Bruhat-Tits III §4.7 (H^1 = 1 for simply connected semisimple groups over henselian discretely valued fields with perfect residue field of cd <= 1). That paper is not available locally; I confirmed the content and the number through the secondary source arXiv:2309.11769 (https://arxiv.org/abs/2309.11769), Prop. 3(2), which covers F_p((t)). H_F is split F4 because it contains the split T_8. Hilbert 90 (multiplicative and additive) handles the split tori and the split unipotent B_8 part; the exact sequences give X(F) = G(F)x0, L(F)x0 = T(F)x0 = A_X(F)x0, A_X(O) = im T(O) (Pic O = 0) and X-ring(F) = B(F)x0. X_F is affine by Richardson 1977 / Borel 1985, valid in every characteristic (Milne, Aside 9.46, confirms), with fpqc descent of affineness from F-bar to F. Only H^1 of smooth connected groups enters.

Replaces Kneser's theorem by Bruhat–Tits III for \(F=\mathbb F_q(\!(t)\!)\) and draws the consequences used later: \(X(F)\) is one \(G(F)\)-orbit, the torus orbit \(A_X(F)x_0\) equals \(T(F)x_0\), \(X_F\) is affine, and the open Borel orbit has a single \(B(F)\)-orbit of \(F\)-points, so (A5)(A5) X homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points (§8). holds.

Lemma 3.3.

Let \(F=\mathbb F_q(\!(t)\!)\), \(p\) odd (the same holds for \(p\)-adic \(F\), \(p\) odd).

  1. \(H^1(F,\mathrm{Spin}_8)=H^1(F,F_4)=1\) (Bruhat–Tits III, Thm 4.7: \(H^1(K,G)=1\) for \(G\) semisimple simply connected over a henselian discretely valued field \(K\) with perfect residue field of cohomological dimension \(\le1\)). \(H^1(F,T')=1\) for split tori and \(H^1(F,U')=1\) for split unipotent groups.
  2. \(X(F)=G(F)\,x_0\).
  3. \(A_X(F)x_0=L(F)x_0=T(F)x_0\), and \(A_X(\mathcal O)\) is the image of \(T(\mathcal O)\).
  4. \(X_F\) is affine.
  5. \(\mathring X(F)=B(F)\,x_0\).

Hence (A5)(A5) X homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points (§8). holds for \(\mathcal M_\theta\) over \(\mathbb F_q(\!(t)\!)\). Only \(H^1\) of smooth connected groups is used, for which étale and fppf cohomology agree; \(H^1(F,\mu_2)=F^\times/F^{\times2}\) has order 4; no \(\mu_p\), \(\alpha_p\) or Artin–Schreier classes occur.

  1. (1). Bruhat–Tits III (J. Fac. Sci. Univ. Tokyo 34 (1987)), Thm 4.7, cited (not in the local sources); its standing hypotheses cover every complete discretely valued field with perfect residue field of dimension \(\le1\), in particular \(\mathbb F_q(\!(t)\!)\) — it is the characteristic-free replacement of Kneser's theorem. \(\mathrm{Spin}_8\) and \(F_4\) are split simply connected. Split tori: Hilbert 90. Split unipotent: a central series with quotients \(\mathbb G_a\) and \(H^1(F,\mathbb G_a)=0\).
  2. (2). Exact sequence of pointed sets \(G(F)\to X(F)\to H^1(F,H)\) for the étale \(H\)-torsor \(G\to X\) (Proposition 3.2) and \(H^1(F,H)=H^1(F,F_4)=1\).
  3. (3). \(Lx_0\simeq L/L^\theta=L/\mathrm{Spin}_8=A_X\simeq T/T_8\) (Proposition 3.2); \(L(F)\to A_X(F)\) is onto since \(H^1(F,\mathrm{Spin}_8)=1\), \(T(F)\to A_X(F)\) is onto since \(H^1(F,T_8)=1\), and \(T(\mathcal O)\to A_X(\mathcal O)\) is onto since \(H^1(\mathcal O,T_8)=\mathrm{Pic}(\mathcal O)^4=1\).
  4. (4). \(H_F\) is reductive, so \(G_F/H_F\) is affine: Richardson, Bull. LMS 9 (1977), Thm A (\(G/H\) affine iff \(H\) reductive, every characteristic, via Haboush's theorem); equivalently Haboush plus Mumford's GIT, Thm 1.1 and Amplification 1.3 (the free right action of \(H\) on the affine \(G\) has closed orbits, so \(\mathrm{Spec}\,F[G]^H\) is a geometric quotient). Cited, not opened; this replaces Matsushima's theorem used in S4-4.
  5. (5). \(\mathring X=B/B_8\) (Proposition 3.2), \(B_8=T_8\ltimes U_8\) a split connected solvable group over \(\mathbb F_q\); \(H^1(F,B_8)\to H^1(F,T_8)=1\) has trivial fibre \(\mathrm{im}\,H^1(F,U_8)=1\), so \(H^1(F,B_8)=1\) and \(B(F)\to(B/B_8)(F)\) is onto.

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