[Paper Review] On Grothendieck-Serre conjecture concerning principal G-bundles over regular semi-local domains containing a finite field: I

Ivan Panin|arXiv (Cornell University)|Jun 2, 2014

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper proves a key case of the Grothendieck-Serre conjecture for principal $G$-bundles over regular semi-local domains containing a finite field, establishing that triviality over the function field implies triviality over the base ring. The core result, Theorem 1.1, constructs a deformation of a trivial $G$-bundle over a polynomial ring using elementary fibrations and Bertini-type theorems, enabling reduction to the case of simply connected simple groups.

ABSTRACT

In three preprints [Pan2],[Pan3] and the present one we prove Grothendieck-Serre's conjecture concerning principal G-bundles over regular semi-local domains R containing a finite field (here G is a reductive group scheme). The present preprint contains main geometric presentation theorems which are necessary for that. The preprint [Pan2] contains reduction of the Grothendieck-Serre's conjecture to the case of a simple simply-connected group scheme. The preprint [Pan3] contains a proof of Grothendieck-Serre's conjecture for regular semi-local domains R containing a finite field. One of the main result of the present preprint is Theorem 1.1. The Grothendieck--Serre conjecture for the case of regular semi-local domains containing an infinite field is proven in joint work due to R.Fedorov and I.Panin (see [FP]). Thus the conjecture holds for regular semi-local domains containing a field.

Motivation & Objective

• To establish the Grothendieck-Serre conjecture for regular semi-local domains containing a finite field by proving the key deformation result in Theorem 1.1.
• To provide geometric tools—specifically elementary fibrations and nice triples—for reducing the general reductive group scheme case to the simply connected simple case.
• To complete the proof of the conjecture in the finite field case by combining results from this preprint, [Pan2], and [Pan3].
• To extend the known result for infinite fields (from [FP]) to the finite field case, completing the conjecture for all regular semi-local domains over a field.

Proposed method

• Uses elementary fibrations to construct a deformation of a principal $G$-bundle over $\mathcal{O}[t]$ from a trivial bundle over the function field.
• Applies Bertini-type theorems from [Poo] and [ChPoo] to ensure the existence of suitable hypersurfaces and étale neighborhoods.
• Employs the notion of a 'nice triple' to control the geometry of the base scheme and ensure compatibility with the group scheme action.
• Constructs a principal $G$-bundle $\mathcal{G}_t$ over $\mathbf{A}^1 \times U$ that is trivial over a localization $h^{-1}$ and restricts to the original bundle at $t=0$, using a section $\Delta'$ and an isomorphism $\Phi$.
• Relies on D. Popescu's approximation theorem to reduce the general case to the case of regular semi-local rings with finite residue fields.
• Uses the structure of the semi-local ring $\mathcal{O}$ as the ring of functions at finitely many points on a smooth $k$-variety to apply cohomological and geometric techniques.

Experimental results

Research questions

• RQ1 Does the Grothendieck-Serre conjecture hold for regular semi-local domains containing a finite field, specifically for principal $G$-bundles with $G$ reductive?
• RQ2 Can one construct a deformation of a trivial $G$-bundle over a polynomial ring that recovers the original bundle at $t=0$ and remains trivial over a localization?
• RQ3 How can one ensure the existence of a suitable elementary fibration and étale neighborhood that preserve the $G$-bundle structure over a finite field?
• RQ4 To what extent do Bertini-type theorems over finite fields allow the construction of hypersurfaces with desired geometric and cohomological properties?
• RQ5 Can the reduction to the simply connected simple case be achieved via geometric and cohomological techniques in the finite field setting?

Key findings

• Theorem 1.1 establishes that for a finite field $k$, a semi-local ring $\mathcal{O}$ of finitely many points on a smooth $k$-variety, and a simple simply connected group scheme $G$ over $\mathcal{O}$, any $G$-bundle trivial over the function field $K$ extends to a $G$-bundle over $\mathcal{O}[t]$ that is trivial over a localization $\mathcal{O}[t]_h$ and recovers the original bundle at $t=0$.
• The construction of the bundle $\mathcal{G}_t$ and the polynomial $h$ satisfies conditions (i) and (ii) of Theorem 1.1, proving the existence of such a deformation.
• The proof relies on the existence of a nice triple and an étale morphism $\sigma_g$ that allows lifting the trivialization from the function field to a neighborhood in $\mathcal{O}[t]$, using the isomorphism $\Phi$ to identify the group structures.
• The section $s = \sigma_g \circ \Delta'$ is shown to be the zero section, ensuring that the restriction of $\mathcal{G}_t$ to $\{0\} \times U$ is isomorphic to the original bundle $P_U$, which is the key step in the deformation argument.
• The result confirms that the kernel of the map $H^1_{\text{ét}}(\mathcal{O}, G) \to H^1_{\text{ét}}(K, G)$ is trivial for $G$ simply connected and simple, completing a major step in the full conjecture.
• The work completes the proof of the Grothendieck-Serre conjecture for regular semi-local domains over a field, as the case over infinite fields was already settled in [FP].