R-Equivalence and A1-Connectedness in Anisotropic Groups

Balwe, Chetan; Sawant, Anand

doi:10.1093/imrn/rnv044

Cited by 12 publications

(38 citation statements)

References 10 publications

(11 reference statements)

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“…Therefore, there exists a pair of distinct R-equivalent elements in G(k). Since G is anisotropic, we have G(k) ≃ S(G)(k), by [4,Lemma 3.7]. Thus, the map S(G)(k) → π A 1 0 (G)(k) is not a bijection.…”

Section: Algebraic Groups and Their A 1 -Connected Componentsmentioning

confidence: 97%

“…First assume that G is a semisimple, simply connected and absolutely almost simple group over k. If G is isotropic, then by [2, Theorem 4.3.1], Sing A 1 * G is A 1 -local and it follows that π A 1 0 (G)(F ) ≃ G(F )/R, for every field extension F/k (see [4,Theorem 3.4]). If G is anisotropic, this is [4, Theorem 4.2] (although it is stated there with the assumption that the base field is of characteristic 0, it is easy to see that the proof works over any infinite perfect field).…”

Section: Algebraic Groups and Their A 1 -Connected Componentsmentioning

confidence: 99%

“…Examples of anisotropic groups G for which Sing A 1 * G fails to be A 1 -local were obtained in [4]. For these examples, the failure of affine homotopy invariance of G-torsors was noted in [2].…”

Section: Introductionmentioning

confidence: 99%

“…In [4], it was shown that sections over fields of the sheaf of A 1 -connected components of a semisimple simply connected group agree with the group of its R-equivalence classes. However, this is not the case if we drop the hypothesis of simple connectedness.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2, we recollect preliminaries about A 1 -connectedness. In Section 3, we recall basic notions about reductive algebraic groups and describe A 1 -connected components of semisimple, simply connected algebraic groups as an immediate consequence of results of [4]. Section 4 is devoted to the proof of Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

𝔸¹-connectedness in reductive algebraic groups

Balwe

Sawant

2017

Trans. Amer. Math. Soc.

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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 hypotheses. As a consequence, we show the failure of A 1 -invariance of torsors for such groups on smooth affine schemes over infinite perfect fields. We also characterize A 1connected reductive algebraic groups over a field of characteristic 0.

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Section: Algebraic Groups and Their A 1 -Connected Componentsmentioning

confidence: 97%

Section: Algebraic Groups and Their A 1 -Connected Componentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

𝔸¹-connectedness in reductive algebraic groups

Balwe

Sawant

2017

Trans. Amer. Math. Soc.

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$${\mathbb A}^1$$ -homotopy Theory and Contractible Varieties: A Survey

Asok

Østvær

2021

Lecture Notes in Mathematics

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A1-connected components of schemes

Balwe

Hogadi

Sawant

2015

Advances in Mathematics

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A conjecture of Morel asserts that the sheaf π A 1 0 (X ) of A 1 -connected components of a simplicial sheaf X is A 1 -invariant. A conjecture of Asok and Morel asserts that the sheaves of A 1 -connected components of smooth schemes over a field coincide with the sheaves of their A 1 -chain-connected components. Another conjecture of Asok and Morel states that the sheaf of A 1 -connected components is a birational invariant of smooth proper schemes. In this article, we exhibit examples of schemes for which conjectures of Asok and Morel fail to hold and whose Sing * is not A 1 -local. We also give equivalent conditions for Morel's conjecture to hold and obtain an explicit conjectural description of π A 1 0 (X ). A method suggested by these results is then used to prove Morel's conjecture for non-uniruled surfaces over a field.

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R-Equivalence and A1-Connectedness in Anisotropic Groups

Cited by 12 publications

References 10 publications

𝔸¹-connectedness in reductive algebraic groups

𝔸¹-connectedness in reductive algebraic groups

$${\mathbb A}^1$$ -homotopy Theory and Contractible Varieties: A Survey

A1-connected components of schemes

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