[Paper Review] Groupe de Brauer non ramifié algébrique des espaces homogènes (The unramified algebraic Brauer group of homogeneous spaces)

Giancarlo Lucchini Arteche|arXiv (Cornell University)|Dec 11, 2013

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper provides explicit formulas for computing the unramified algebraic Brauer group of homogeneous spaces $V = G\backslash G'$, where $G'$ is simply connected semisimple, over both finite fields and fields of characteristic zero. By reducing Galois cohomology of the stabilizer group $G$ to that of a finite subquotient, the authors show that the Brauer group is entirely determined by a finite subgroup constructed from the component group and the $m$-torsion of the maximal torus, generalizing earlier results to non-connected stabilizers.

ABSTRACT

Using a reduction of the Galois cohomology of a linear algebraic group $G$ to that of a certain finite subquotient, we give different formulas allowing the calculation of the unramified algebraic Brauer group of a homogeneous space $V=G\backslash G'$ with $G'$ semisimple and simply connected, both over a finite field and over an arbitrary field of characteristic $0$.

Motivation & Objective

• To extend existing formulas for the unramified algebraic Brauer group of homogeneous spaces to cases with arbitrary stabilizer groups, not just finite or commutative ones.
• To resolve the case of non-connected stabilizers by showing that the Brauer group depends only on a finite subquotient of the stabilizer.
• To establish a general method for computing $\mathrm{Br}_{\mathrm{nr,al}}V$ using Galois cohomology reduction to finite groups.
• To prove that the cohomological information needed for the Brauer group is captured by a finite subgroup constructed from the component group and torus torsion.
• To generalize results from [BDH13] and [LA14] to stabilizers that are extensions of finite groups by split tori, covering the general linear algebraic group case.

Proposed method

• Reduction of Galois cohomology of a linear algebraic group $G$ to that of a finite subquotient, leveraging the structure of $G$ as an extension of a finite group $F$ by a split torus $T$.
• Construction of a finite $k$-subgroup $H \subset G$ such that $H$ fits into a commutative diagram with $T^{nd} \to H \to F$ and $T \to G \to F$, where $n = |F|$, $d = [L:k]$ for a splitting field $L$ of $T$.
• Use of the multiplication-by-$m$ map $\phi_m: T \to T$ with $m = nd$, and defining $H$ as the subgroup generated by $H_0$ and $\phi_m^{-1}(H_0 \cap T)$, ensuring $H$ is finite.
• Proof that the natural map $H^1(k', H) \to H^1(k', G)$ is surjective for all field extensions $k' \supset k$, implying $H$ captures all relevant Galois cohomology data.
• Application of a result from [CGR08] (Proposition 3.1) to show that the Brauer group is determined by the cohomology of this finite subgroup.
• Reduction of the computation of $\mathrm{Br}_{\mathrm{nr,al}}V$ to the Galois cohomology of a finite group, enabling explicit calculation in both finite and characteristic zero fields.

Experimental results

Research questions

• RQ1 Can the unramified algebraic Brauer group of a homogeneous space $V = G\backslash G'$ be computed when the stabilizer $G$ is non-connected and not of type 'ssumult'?
• RQ2 Does the Brauer group depend only on a finite subquotient of the stabilizer $G$, even when $G$ is an extension of a finite group by a split torus?
• RQ3 Is it possible to reduce the computation of $\mathrm{Br}_{\mathrm{nr,al}}V$ to Galois cohomology of a finite group, even over fields of characteristic zero?
• RQ4 How can one construct such a finite subgroup $H \subset G$ that controls the Brauer group, given $G$ is an extension of a finite group $F$ by a split torus $T$?
• RQ5 To what extent do the results from [LA14] (on finite stabilizers) extend to more general stabilizers via cohomological reduction?

Key findings

• For a stabilizer $G$ that is an extension of a finite group $F$ of order $n$ (prime to the characteristic) by a split torus $T$ over a field $k$, the unramified algebraic Brauer group $\mathrm{Br}_{\mathrm{nr,al}}V$ is determined by the Galois cohomology of a finite $k$-subgroup $H \subset G$.
• The finite subgroup $H$ is constructed as the group generated by a lift $H_0 \subset G$ of $F$ and the $m$-torsion of $T$, where $m = nd$, with $d = [L:k]$ for a splitting field $L$ of $T$, and $H$ fits into an exact sequence $1 \to T^{nd} \to H \to F \to 1$.
• The surjectivity of the map $H^1(k', H) \to H^1(k', G)$ for all $k' \supset k$ implies that all Galois cohomological data relevant to the Brauer group is captured by $H$, so $\mathrm{Br}_{\mathrm{nr,al}}V$ depends only on $H$.
• The formula for $\mathrm{Br}_{\mathrm{nr,al}}V$ is given in terms of $H^1(k, H)$ and $H^2(k, H)$, reducing the computation to finite group cohomology.
• The result generalizes previous formulas from [BDH13] and [LA14] to the case of stabilizers that are extensions of finite groups by split tori, covering the general linear algebraic group case.
• The method applies uniformly over both finite fields and fields of characteristic zero, providing a uniform computational framework for the unramified algebraic Brauer group of such homogeneous spaces.