R-Equivalence and 1-Connectedness in Anisotropic Groups

@article{Balwe2014REquivalenceA1,
  title={R-Equivalence and 1-Connectedness in Anisotropic Groups},
  author={Chetan T. Balwe and Anand Sawant},
  journal={International Mathematics Research Notices},
  year={2014},
  volume={2015},
  pages={11816-11827}
}
We show that if G is an anisotropic, semisimple, absolutely almost simple, simply connected group over a field k, then two elements of G over any field extension of k are R-equivalent if and only if they are A^1-equivalent. As a consequence, we see that Sing_*(G) cannot be A^1-local for such groups. This implies that the A^1-connected components of a semisimple, absolutely almost simple, simply connected group over a field k form a sheaf of abelian groups. 
A^1-connectedness in reductive algebraic groups
Using sheaves of A^1-connected components, we prove that the Morel-Voevodsky singular construction on a reductive algebraic group fails to be A^1-local if the group does not satisfy suitable isotropy
Corrigendum to “¹-connectedness in reductive algebraic groups”
Using sheaves of A^1-connected components, we prove that the Morel-Voevodsky singular construction on a reductive algebraic group fails to be A^1-local if the group does not satisfy suitable isotropy
Strong $\mathbb A^1$-invariance of $\mathbb A^1$-connected components of reductive algebraic groups
. We show that the sheaf of A 1 -connected components of a reductive algebraic group over a perfect field is strongly A 1 -invariant. As a consequence, torsors under such groups give rise to A 1 -fiber
Naive vs. genuine A^1-connectedness
We show that the triviality of sections of the sheaf of A^1-chain connected components of a space over finitely generated separable field extensions of the base field is not sufficient to ensure the
𝔸1–connected components of ruled surfaces
A conjecture of Morel asserts that the sheaf of $\mathbb A^1$-connected components of a space is $\mathbb A^1$-invariant. Using purely algebro-geometric methods, we determine the sheaf of $\mathbb
Affine representability results in 1–homotopy theory, II : Principal bundles and homogeneous spaces
We establish a relative version of the abstract "affine representability" theorem in ${\mathbb A}^1$--homotopy theory from Part I of this paper. We then prove some ${\mathbb A}^1$--invariance
Naive $\mathbb A^1$-homotopies on ruled surfaces
We explicitly describe the $\mathbb A^1$-chain homotopy classes of morphisms from a smooth henselian local scheme into a smooth projective surface, which is birationally ruled over a curve of genus
$\mathbb{A}^1$-connected components of blow-up of threefolds fibered over a surface
Over a perfect field, we determine the sheaf of $\mathbb{A}^1$-connected components of a class of threefolds given by the Blow-up of a variety admitting a $\mathbb{P}^1$-fibration over either an
$R$-triviality of some exceptional groups
The main aim of this paper is to prove $R$-triviality for simple, simply connected algebraic groups with Tits index $E_{8,2}^{78}$ or $E_{7,1}^{78}$, defined over a field $k$ of arbitrary
$${\mathbb A}^1$$ -homotopy Theory and Contractible Varieties: A Survey
We survey some topics in ${\mathbb A}^1$-homotopy theory. Our main goal is to highlight the interplay between ${\mathbb A}^1$-homotopy theory and affine algebraic geometry, focusing on the varieties

References

SHOWING 1-10 OF 15 REFERENCES
On A1-fundamental groups of isotropic reductive groups
R-Equivalence and Special Unitary Groups
Abstract A norm homomorphism for the group of R -equivalence classes of all simply connected semisimple classical algebraic groups is constructed. The group of R -equivalence classes for special
A^1-connected components of schemes
R-equivalence in spinor groups
The notion of R-equivalence in the set X(F) of F-points of an algebraic variety X defined over a field F was introduced by Manin in [11] and studied for linear algebraic groups by Colliot-Thelene and
Algebraic Groups and Their Birational Invariants
Since the late 1960s, methods of birational geometry have been used successfully in the theory of linear algebraic groups, especially in arithmetic problems. This book--which can be viewed as a
Connectivity of motivic H–spaces
In this note we prove that the $\mathbb{A}^1$-connected component sheaf $a_{Nis}(\pi_0^{\mathbb{A}^1}(\mathcal{X}))$ of an $H$-group $\mathcal{X}$ is $\mathbb{A}^1$-invariant.
Introduction to Affine Group Schemes
I The Basic Subject Matter.- 1 Affine Group Schemes.- 1.1 What We Are Talking About.- 1.2 Representable Functors.- 1.3 Natural Maps and Yoneda's Lemma.- 1.4 Hopf Algebras.- 1.5 Translating from
A1-homotopy theory of schemes
© Publications mathématiques de l’I.H.É.S., 1999, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http://
Linear Algebraic Groups
Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality
...
...