[Paper Review] On the purity conjecture of Nisnevich for torsors under reductive group schemes

Roman Fedorov|arXiv (Cornell University)|Sep 21, 2021

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper proves the Nisnevich purity conjecture for torsors under reductive group schemes over regular semilocal integral domains containing an infinite field, under an isotropy condition, showing that rationally trivial torsors over $ R_f $ are trivial. Without isotropy, a counterexample is constructed using affine Grassmannians, and an analogue for higher étale cohomology is established for commutative group schemes of multiplicative type in characteristic zero.

ABSTRACT

Let $R$ be a regular semilocal integral domain containing an infinite field $k$. Let $f\in R$ be an element such that for all maximal ideals $\mathfrak m$ of $R$ we have $f otin\mathfrak m^2$. Let $\mathbf G$ be a reductive group scheme over $R$. Under an isotropy assumption on $\mathbf G$ we show that a $\mathbf G$-torsor over the localization $R_f$ is trivial, provided it is rationally trivial. We show that it is not true without the isotropy assumption. Finally, if $\mathbf G$ is a commutative group scheme of multiplicative type and the regular semilocal ring contains a field of characteristic zero, we prove an analogue of Nisnevich purity conjecture for higher étale cohomology groups. The first statement is derived from its abstract version concerning presheaves of pointed sets satisfying some properties. The counterexample is constructed by providing a torsor over a local family of affine lines that cannot be extended to the family of projective lines. The latter is accomplished using the technique of affine Grassmannians.

Motivation & Objective

• To establish the Nisnevich purity conjecture for torsors under reductive group schemes over regular semilocal rings containing an infinite field.
• To demonstrate that the conjecture fails without an isotropy assumption on the group scheme.
• To extend the purity phenomenon to higher étale cohomology groups for commutative group schemes of multiplicative type in characteristic zero.
• To provide a general framework via abstract properties of pointed presheaves for proving the main result.

Proposed method

• The main result is derived from an abstract criterion for purity in terms of pointed presheaves satisfying specific exactness and extension properties.
• A counterexample is constructed by showing that a ${\mathbf{G}}$-torsor over a local family of affine lines cannot be extended to the corresponding projective lines.
• The construction relies on the geometry of affine Grassmannians, using a closed embedding $ \mathbb{P}^1_k \hookrightarrow \mathrm{Gr}_G $ for a split semisimple group $ G $.
• The non-extendability of morphisms from $ \mathbb{A}^1_{B'} $ to $ \mathrm{Gr}_G $ is used to obstruct extension of torsors.
• The proof uses the fact that the orbit of a nontrivial cocharacter in $ \mathrm{Gr}_G $ contains a $ \mathbb{P}^1 $, via the flag variety $ G/P_\lambda $.
• The cohomological analogue is established using the same framework, applying the abstract purity result to the multiplicative group scheme.

Experimental results

Research questions

• RQ1 Under what conditions is a rationally trivial $ \mathbf{G} $-torsor over $ R_f $ trivial, for $ R $ a regular semilocal domain and $ f \notin \mathfrak{m}^2 $ for all maximal ideals $ \mathfrak{m} $?
• RQ2 Can the Nisnevich purity conjecture fail for non-isotropic reductive group schemes over regular semilocal rings?
• RQ3 Is there a cohomological analogue of Nisnevich purity for higher étale cohomology groups when $ \mathbf{G} $ is commutative of multiplicative type and $ R $ contains a field of characteristic zero?
• RQ4 How do affine Grassmannians facilitate the construction of non-extendable torsors in the absence of isotropy?

Key findings

• A $ \mathbf{G} $-torsor over $ R_f $ is trivial if it is rationally trivial, provided $ \mathbf{G} $ is isotropic over $ R $, $ R $ is a regular semilocal integral domain containing an infinite field, and $ f \notin \mathfrak{m}^2 $ for all maximal ideals $ \mathfrak{m} $.
• Without the isotropy assumption, the purity conjecture fails: a counterexample is constructed via a non-extendable morphism from $ \mathbb{A}^1_{B'} $ to the affine Grassmannian $ \mathrm{Gr}_G $, which corresponds to a non-extendable torsor.
• For commutative group schemes $ \mathbf{G} $ of multiplicative type over a regular semilocal ring containing a field of characteristic zero, an analogue of Nisnevich purity holds for higher étale cohomology groups.
• The existence of a closed $ \mathbb{P}^1 $-subscheme in $ \mathrm{Gr}_G $ for a split nontrivial semisimple group $ G $ is essential to the counterexample construction.
• The abstract framework of pointed presheaves satisfying certain extension and exactness conditions enables the generalization of the purity result beyond specific geometric settings.
• The obstruction to extending torsors from $ \mathbb{A}^1_{B'} $ to $ \mathbb{P}^1_{B'} $ arises from the failure of the corresponding morphism to $ \mathrm{Gr}_G $ to extend, due to the nontriviality of the cocharacter orbit.