𝔸¹-connectedness in reductive algebraic groups

Balwe, Chetan; Sawant, Anand

doi:10.1090/tran/7090

Cited by 9 publications

(15 citation statements)

References 25 publications

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“…, t n ), but this follows immediately from the fact that a field has no nontrivial Nisnevich covering sieves. Remark 3.3.8 At least if k is an infinite perfect field, Theorem 3.3.7 admits a converse: if G is a reductive k-group such that H 1 Nis (−, G) is A 1 -invariant on Sm aff k , then G is isotropic, see Balwe and Sawant [14,Theorem 1]. In fact, for G reductive, the following three conditions are equivalent:…”

Section: Isotropic Reductive Groupsmentioning

confidence: 99%

Affine representability results in 𝔸1–homotopy theory, II : Principal bundles and homogeneous spaces

2018

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We establish a relative version of the abstract "affine representability" theorem in A 1 -homotopy theory from Part I of this paper. We then prove some A 1 -invariance statements for generically trivial torsors under isotropic reductive groups over infinite fields analogous to the Bass-Quillen conjecture for vector bundles. Putting these ingredients together, we deduce representability theorems for generically trivial torsors under isotropic reductive groups and for associated homogeneous spaces in A 1 -homotopy theory. 14F42; 14L10, 55R15, 20G15Published: XX Xxxember 20XX All rings considered in this paper will be assumed unital. We use the symbol S for a quasi-compact, quasi-separated base scheme, Sm S for the category of finitely presented smooth S-schemes, and Sm aff S ⊂ Sm S for the full subcategory of affine schemes (in the absolute sense). We also reuse some terminology and notation introduced in [9],

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Section: Isotropic Reductive Groupsmentioning

confidence: 99%

Affine representability results in 𝔸1–homotopy theory, II : Principal bundles and homogeneous spaces

2018

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“…In this case, it is known that Sing A 1 ˚G fails to be A 1 -local [5,Theorem 4.7]. We now assume that G is a semisimple anisotropic group.…”

Section: Definition 23mentioning

confidence: 99%

“…A result of Borel-Tits implies in this case that SpGqpkq » Gpkq (see [4,Lemma 3.7] for details). However, one has the following result (see [4,Theorem 4.2], [5,Theorem 3.6]): if G is a semisimple, simply connected group over an infinite perfect field k which does not satisfy the above isotropy hypothesis, then one has canonical isomorphisms π A 1 0 pGqpkq » S 2 pGqpkq » Gpkq{R. It is worthwhile to mention here that we do not yet know whether π A 1 0 pGq agrees with S 2 pGq as a sheaf.…”

Section: Definition 23mentioning

confidence: 99%

Naive vs. genuine A^1-connectedness

Sawant¹

2017

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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 triviality of the sheaf of its A 1 -chain connected components, contrary to the situation with genuine A 1 -connected components. As a consequence, we show that there exists an A 1 -connected scheme for which the Morel-Voevodsky singular construction is not A 1 -local.

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“…We now briefly discuss the case of non-split, semisimple algebraic groups. As mentioned in Remark 5, the sheaf π A 1 1 (G) depends upon the A 1 -connected component of the neutral element of G. Note that at least over a field of characteristic 0, a reductive algebraic group G is A 1 -connected if and only if G is semisimple, simply connected and every almost k-simple factor of G is R-trivial [4,Theorem 5.2]. Using this criterion, one can see that over special classes of fields (such as local fields and global fields) isotropic, semisimple, simply connected groups are A 1 -connected (see [14] for a survey of known positive results and a counterexample in general due to Platonov).…”

Section: Introductionmentioning

confidence: 98%

Cellular $\mathbb A^1$-homology and the motivic version of Matsumoto's theorem

Morel¹,

Sawant²

2020

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We define a new version of A 1 -homology, called cellular A 1 -homology, for smooth schemes over a field that admit an increasing filtration by open subschemes with cohomologically trivial closed strata. We provide several explicit computations of cellular A 1 -homology and use them to determine the A 1 -fundamental group of a split reductive group over an arbitrary field, thereby obtaining the motivic version of Matsumoto's theorem on universal central extensions of split, semisimple, simply connected algebraic groups. As applications, we uniformly explain and generalize results due to Brylinski-Deligne and Esnault-Kahn-Levine-Viehweg, determine the isomorphism classes of central extensions of such an algebraic group by an arbitrary strictly A 1 -invariant sheaf and also reprove classical results of E. Cartan on homotopy groups of complex Lie groups.

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𝔸¹-connectedness in reductive algebraic groups

Cited by 9 publications

References 25 publications

Affine representability results in 𝔸1–homotopy theory, II : Principal bundles and homogeneous spaces

Affine representability results in 𝔸1–homotopy theory, II : Principal bundles and homogeneous spaces

Naive vs. genuine A^1-connectedness

Cellular $\mathbb A^1$-homology and the motivic version of Matsumoto's theorem

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