[Paper Review] Grothendieck-Serre in the quasi-split unramified case

Kęstutis Česnavičius|arXiv (Cornell University)|Sep 11, 2020

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper proves the Grothendieck–Serre conjecture for quasi-split reductive group schemes over unramified regular local rings, establishing that every generically trivial torsor is trivial. The proof uses a novel version of Noether normalization over discrete valuation rings and a presentation lemma for smooth relative curves in mixed characteristic, enabling reduction to the affine line via excision and patching techniques.

ABSTRACT

The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group scheme $G$ over a regular local ring $R$ is trivial. We settle it in the case when $G$ is quasi-split and $R$ is unramified. Some of the techniques that allow us to overcome obstacles that have so far kept the mixed characteristic case out of reach include a version of Noether normalization over discrete valuation rings, as well as a suitable presentation lemma for smooth relative curves in mixed characteristic that facilitates passage to the relative affine line via excision and patching.

Motivation & Objective

• To resolve the mixed characteristic case of the Grothendieck–Serre conjecture under the conditions that the base ring is unramified and the group scheme is quasi-split.
• To extend the conjecture to semilocal rings flat and geometrically regular over a Dedekind domain, broadening applicability beyond local rings.
• To establish a product formula for quasi-split reductive groups over unramified regular local rings, linking adelic and global structures.
• To prove that a reductive group scheme over a regular semilocal ring is split (resp. quasi-split) if and only if its generic fiber is split (resp. quasi-split), strengthening the conjecture’s implications.
• To provide a new cohomological toolset for torsors in mixed characteristic by leveraging patching and relative curve techniques.

Proposed method

• Develops a version of Noether normalization adapted to discrete valuation rings to handle the mixed characteristic setting.
• Introduces a presentation lemma for smooth relative curves in mixed characteristic, allowing reduction to the relative affine line via excision and patching.
• Applies a Lindel-type trick in the context of Cohen–Macaulay relative curves to lift torsors from the generic fiber to the total space.
• Uses patching techniques (e.g., Lemma 7.1) to descend torsors from a smooth relative curve to the affine line over the base ring.
• Employs nonabelian cohomology and twisting techniques to reduce the problem to torsors under tori, leveraging purity results from [CTS79] and [CTS87].
• Utilizes the structure of Borel subgroups and their unipotent radicals to reduce the problem to torus torsors, where the conjecture is already known.

Experimental results

Research questions

• RQ1 Does the Grothendieck–Serre conjecture hold for quasi-split reductive group schemes over unramified regular local rings in mixed characteristic?
• RQ2 Can the conjecture be extended to semilocal rings that are flat and geometrically regular over a Dedekind domain?
• RQ3 Is the triviality of a torsor over the fraction field of a regular ring sufficient to guarantee triviality over the ring itself when the group is quasi-split and the ring is unramified?
• RQ4 Under what conditions does the generic fiber of a reductive group scheme determine the splitting or quasi-splitting status of the group scheme over the base ring?
• RQ5 Can cohomological obstructions to trivializing torsors be eliminated via geometric reduction techniques in mixed characteristic?

Key findings

• The Grothendieck–Serre conjecture is proven for quasi-split reductive group schemes over unramified regular local rings, with the kernel of the map $ H^1(R, G) o H^1( ext{Frac}(R), G) $ being trivial.
• The conjecture is extended to semilocal rings flat and geometrically regular over a Dedekind ring, with the same triviality result holding in this broader setting.
• A product formula is established: for an unramified regular local ring $ R $, an element $ r eq 0 $, and a quasi-split reductive $ R $-group $ G $, it holds that $ G(reve{R}[1/r]) = G(reve{R})G(R[1/r]) $, where $ reve{R} $ is the $ r $-adic completion of $ R $.
• A reductive $ R $-group scheme $ H $ is split if and only if its generic fiber $ H_{ ext{Frac}(R)} $ is split, for $ R $ an unramified regular local ring.
• In the equicharacteristic case, a reductive $ R $-group scheme $ H $ is quasi-split if and only if its generic fiber is quasi-split, for $ R $ a regular semilocal ring.
• The injectivity of $ H^1(R, ext{SO}_n) o H^1( ext{Frac}(R), ext{SO}_n) $ and $ H^1(R, ext{O}_n) o H^1( ext{Frac}(R), ext{O}_n) $ is proven for regular semilocal rings with $ 2 otin R^ imes $, implying that non-isomorphic nondegenerate quadratic forms over $ R $ remain non-isomorphic over $ ext{Frac}(R) $.