[Paper Review] R'esolutions flasques des groupes lin'eaires connexes

Jean-Louis Colliot-Thélène|ArXiv.org|Sep 19, 2006

Autoimmune and Inflammatory Disorders17 citations

TL;DR

This paper introduces flasque resolutions for connected reductive groups over a field, presenting them as quotients $ G = H/S $, where $ H $ is an extension of a quasi-trivial torus by a simply connected semisimple group, and $ S $ is a flasque torus central in $ H $. The key contribution is a functorial, choice-free approach to Galois cohomology and related invariants—such as the Brauer group of smooth compactifications and $ R $-equivalence classes—using these resolutions, which simplifies and unifies known results for number fields and local fields.

ABSTRACT

A connected reductive group G over a field k may be written as a quotient H/S, where the k-group H is an extension of a quasitrivial torus by a simply connected semisimple group, and S is a flasque k-torus, central in H (a flasque torus is a torus whose cocharacter group is an H^1-trivial Galois lattice). The flasque torus S is well-defined up to multiplication by a quasitrivial torus. Such presentations G=H/S lead to a simplified approach of the Galois cohomology of G and of related objects, such as the Brauer group of a smooth compactification of G. When k is a number field, one also recovers known formulas, in terms of S, for the quotient of the group of rational points by R-equivalence, and for the abelian groups which measure the lack of weak approximation and the failure of the Hasse principle for principal homogeneous spaces.

Motivation & Objective

• To develop a new, systematic method for studying the Galois cohomology of connected reductive algebraic groups over a field.
• To replace the reliance on maximal tori and complex torus resolutions (as in Borovoi, Sansuc, Kottwitz) with a more flexible, intrinsic construction based on flasque tori.
• To provide a unified framework for computing invariants such as the Brauer group of smooth compactifications and the quotient $ G(k)/R $, particularly over number fields.
• To establish a new definition of the algebraic fundamental group $ \pi_1(G) $ via flasque resolutions, independent of earlier constructions.
• To show that flasque resolutions yield functorial and canonical results in Galois cohomology, especially in the context of weak approximation and the Hasse principle.

Proposed method

• Construct a resolution $ 1 \to S \to H \to G \to 1 $, where $ H $ is an extension of a quasi-trivial torus by the universal cover of $ G^{\mathrm{der}} $, and $ S $ is a flasque $ k $-torus central in $ H $.
• Use the flasque property of $ S $—that its cocharacter module is $ H^1 $-trivial—to simplify Galois cohomology computations.
• Define the algebraic fundamental group $ \pi_1(G) $ as the cocharacter module of the flasque torus $ S $ in a flasque resolution, independent of maximal torus choices.
• Relate the Brauer group of a smooth compactification of $ G $ to the flasque torus $ S $, providing two equivalent formulas: one in terms of $ S $, one in terms of $ \pi_1(G) $.
• Apply the resolution to compute $ G(k)/R $ and $ H^1(k, G) $ over global and local fields, using known results on simply connected groups (Kneser, Harder, Chernousov).
• Establish quasi-isomorphism between Borovoi’s complex of tori and the complex derived from flasque resolutions, proving equivalence of the approaches without maximal torus dependence.

Experimental results

Research questions

• RQ1 Can a canonical, choice-free resolution of a connected reductive group $ G $ be constructed using flasque tori, independent of maximal torus choices?
• RQ2 How do flasque resolutions simplify the computation of Galois cohomology, especially $ H^1(k, G) $ and the Brauer group of smooth compactifications?
• RQ3 What is the relationship between the flasque resolution and the algebraic fundamental group $ \pi_1(G) $, and how does this definition compare to Borovoi’s or Merkur’ev’s?
• RQ4 To what extent do flasque resolutions recover known results on $ R $-equivalence, weak approximation, and the Hasse principle for principal homogeneous spaces over number fields?
• RQ5 Can the cohomological invariants of $ G $—such as the Brauer group and $ G(k)/R $—be expressed functorially in terms of the flasque torus $ S $ in a resolution?

Key findings

• For any connected reductive group $ G $ over a field $ k $, a flasque resolution $ 1 \to S \to H \to G \to 1 $ exists, with $ H $ an extension of a quasi-trivial torus by the simply connected cover of $ G^{\mathrm{der}} $, and $ S $ a flasque $ k $-torus central in $ H $.
• The flasque torus $ S $ is uniquely determined up to multiplication by a quasi-trivial torus, making it a canonical invariant of $ G $.
• The Brauer group of a smooth compactification of $ G $ is isomorphic to $ H^1(k, S) $, and also to $ H^1(k, \pi_1(G)) $, providing two equivalent formulas.
• Over a number field, the quotient $ G(k)/R $ of rational points modulo $ R $-equivalence is isomorphic to $ H^1(k, S) $, recovering known formulas via this new method.
• The cohomology $ H^1(k, G) $ classifying principal homogeneous spaces is computed via the resolution, and the results align with those of Sansuc, Kottwitz, Borovoi, and Gille.
• The flasque resolution approach is equivalent to Borovoi’s complex of tori, but avoids maximal torus choices and complex resolutions, offering a more direct and functorial framework.