Cohomological Study of Bruhat-Tits Subgroups

• The paper introduces an explicit type morphism that defines exact cohomological sequences connecting parahoric subgroup actions to combinatorial automorphism orbits.
• It computes precise obstruction groups for quasi-split adjoint groups using 2-torsion counting methods and detailed combinatorial analysis.
• It applies Grothendieck–Serre injection results to bridge local and global descent in the rationality of Bruhat–Tits subgroup schemes.

The cohomological paper of Bruhat–Tits subgroups centers on the analysis of parahoric subgroups and their integral models within the framework of reductive groups over discretely valued local or global fields. This investigation draws on Bruhat–Tits buildings, the structure of parahoric group schemes, and Galois cohomology to quantify and classify obstructions to rationality, provide precise subgroup decomposition theorems, and compute exact obstruction groups in specific cases, especially for quasi-split adjoint groups and simply connected cases. The approach refines and supplements the foundational 1987 work of Bruhat and Tits by incorporating explicit exact sequences built from the automorphism group of the affine Dynkin diagram, and by relating the local and global classifying data for torsors and parahoric stabilizers.

1. Type Morphism, Exact Sequences, and Cohomological Framework

A principal tool in the refined cohomological analysis is the "type" (or extended type) morphism, which encodes the action of the group $G(K)$ on the combinatorial facets of the associated Bruhat–Tits building. For a reductive $K$-group $G$, with $K$ the fraction field of a henselian discrete valuation ring $R$, and Bruhat–Tits building $\mathcal{B}(G, K)$, there is a canonical surjective morphism

$\xi: G(K) \longrightarrow \Xi,$

where $\Xi$ is a finite group, typically derived from the automorphism group of the affine Dynkin diagram associated to $G$. This yields an exact sequence: $$1 \to G(K)^c \to G(K) \stackrel{\xi}{\to} \Xi \to 1,$$ where $G(K)^c$ (alternatively $G(K)^b$) denotes the subgroup acting trivially on the types of facets. Such exact sequences allow for passage to Galois cohomology in the context of the Galois group $\Gamma = \operatorname{Gal}(K^\mathrm{nr}/K)$, leading to the cohomological exact sequence: $$1 \to \widetilde{\xi}(H) \to (\Xi_H)^{\Gamma} \to H^1(\Gamma, H^c) \to H^1(\Gamma, H) \to H^1(\Gamma, \Xi_H),$$ where $H \subset G(K)$ is typically the stabilizer of a facet or a parahoric subgroup, and $\Xi_H$ denotes the image of $H$ under $\xi$.

2. Obstruction Groups and Their Combinatorial Computation

The kernel of the natural map

$$H^1(\Gamma, G(K)_{\widetilde{F}}) \rightarrow H^1(\Gamma, G(K))$$

(where $G(K)_{\widetilde{F}}$ is the stabilizer of a given $n.r.$-invariant facet $\widetilde{F}$ in the extended building) is central to the descent problem for torsors and the rational structure of subgroup embeddings. The refined results identify this kernel as being governed by the $\Gamma$-fixed orbits of the facet's type under the action of the group $\Xi_H$, yielding

$$\ker\bigl( H^1(\Gamma, G(K)_{\widetilde{F}}) \to H^1(\Gamma, G(K)) \bigr) \cong \frac{\bigl(\mathrm{Orb}(\widetilde{F})_{G}\bigr)^{\Gamma}}{G(K)},$$

and, when phrased in terms of "types," as

$$\ker\left(H^1(\Gamma, G(K)_{\widetilde{F}}) \to H^1(\Gamma, G(K))\right) \cong \frac{\{\omega \cdot T_{\max}\, |\, \omega \in \Xi_H \}^\Gamma}{\Xi_H},$$

with $T_{\max}$ the type associated to the $n.r.$-chamber.

This combinatorial description directly expresses the cohomological obstruction in terms of automorphism orbits of the Dynkin diagram, thus reducing the computation of the kernel to finite group theory and group action on types.

3. Explicit Results for Quasi-Split Adjoint Groups

For quasi-split adjoint groups, the obstruction group can be calculated precisely in terms of the combinatorics of the group and its Dynkin diagram. The main result is the identification of the kernel as a $2$-torsion group whose order reflects the number of exceptional factors in any Weil restriction decomposition: $$\# \ker\left(H^1(\Gamma, G(K)_{\widetilde{F}}) \rightarrow H^1(\Gamma, G(K))\right) = 2^k,$$ with $k$ bounded above by the number of factors of type ${}^2D_n$ ($n \geq 4$) or ${}^2A_{4n+3}$ that split over a non-ramified extension. Case-by-case analysis, depending on the size of $\Xi^{\mathrm{n.r.}}$ and orbit structure, shows the kernel is either trivial or of order $2$. This precise calculation replaces earlier partial vanishing results, providing a fully combinatorial obstruction description.

4. The Role of the Grothendieck–Serre Theorem

A key structural result used in the analysis is the Grothendieck–Serre theorem in the context of principal $G$-bundles over regular local rings. For Bruhat–Tits group schemes $\mathcal{G}_F$ associated to facets $F$,

$$H^1_{\mathrm{ét}}(R, \mathcal{G}_{F}) \rightarrow H^1(K, G)$$

is injective if $F$ is hyperspecial. In cohomological terms, this implies that

$$\ker\Bigl( H^1(\Gamma, G(K)^0_F) \rightarrow H^1(\Gamma, G(K)) \Bigr)=\{1\},$$

thus excluding any nontrivial obstruction when the subgroup is as "large" as it can be (corresponding to hyperspecial or maximal-volume facets).

5. Comparison with the 1987 Bruhat–Tits Analysis

While the classical 1987 work provided basic decomposition theorems, the description of parahorics, and partial cohomological vanishing results, the refined approach of this paper provides:

• An explicit combinatorial description of the Galois cohomology kernel in terms of type orbits;
• Application to both simply connected and adjoint quasi-split cases, thus extending prior results to broader classes of groups;
• An identification of the precise relationship between local stabilizer cohomology and that of the ambient group via the "type" map.

The previous inductive and geometric methods are thus complemented by new functorial exact sequences and symmetry-based methods.

6. Key Formulas Summarizing the Structure

The following formulas represent core results:

• Exact sequence for the type morphism: $$1 \to G(K)^c \to G(K) \stackrel{\xi}{\longrightarrow} \Xi \to 1,$$
• Associated Galois cohomological sequence: $$1 \to \widetilde{\xi}(H) \to (\Xi_H)^{\Gamma} \to H^1(\Gamma, H^c) \to H^1(\Gamma, H) \to H^1(\Gamma, \Xi_H),$$
• Description of the obstruction kernel: $$\ker\Bigl(H^1(\Gamma, G(K)_{\widetilde{F}}) \to H^1(\Gamma, G(K))\Bigr) \cong \frac{(\mathrm{Orb}(T_{\max})_\Xi)^{\Gamma}}{\Xi},$$
• Special case cardinality: $$\# \ker\left( H^1(\Gamma, G(K)_{\widetilde{F}}) \rightarrow H^1(\Gamma, G(K)) \right) = 2^k$$ as dictated by the quasi-split adjoint structure.

7. Implications and Applications

This refined cohomological analysis has broad implications for the rationality of principal bundles, the structure and descent of Bruhat–Tits group schemes, and the understanding of local-global principles in arithmetic geometry. The exact kernel computations specifically inform the rational classification of torsors, the potential for unique lifting from local to global objects, and the nature of obstructions in patching and descent problems. Furthermore, these results connect the combinatorics of affine Dynkin diagrams, the geometry of the building, and the arithmetic and cohomology of associated group schemes in a fully explicit manner.

This comprehensive approach sets the foundation for deeper results regarding Galois cohomology, torsors under reductive group schemes, and the broader arithmetic theory of algebraic and arithmetic groups over local and global fields (Zidani, 22 Sep 2025).