Failure of a quadratic analogue of Serre’s conjecture

Parimala, S.

doi:10.1090/s0002-9904-1976-14234-6

Cited by 15 publications

(9 citation statements)

References 10 publications

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“…By a quadratic module I will mean a finitely generated projective module with a non-singular quadratic form. In this case the results are not so easy to describe, since Parimala [13] has shown that in general the answer is "No", but there are cases in which the answer is affirmative.…”

Section: Bass-quillen Conjecture If R Is a Regular Ring Is Every Fimentioning

confidence: 94%

Book Review: Serre's problem on projective modules

Swan¹

2008

Bull. Amer. Math. Soc.

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Section: Bass-quillen Conjecture If R Is a Regular Ring Is Every Fimentioning

confidence: 94%

Book Review: Serre's problem on projective modules

Swan¹

2008

Bull. Amer. Math. Soc.

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“…(ii) An example of a principal SO(4, R)-bundle over A 1 R not isomorphic to a pullback from Spec R has been constructed by Parimala, see [Par,Thm. 2.1].…”

Section: 3mentioning

confidence: 99%

Affine Grassmannians of group schemes and exotic principal bundles over ${\Bbb A}^1$

Fedorov¹

2016

American Journal of Mathematics

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Abstract. Let G be a simple simply-connected group scheme over a regular local scheme U . Let E be a principal G-bundle over A 1 U trivial away from a subscheme finite over U . We show that E is not necessarily trivial and give some criteria of triviality. To this end, we define affine Grassmannians for group schemes and study their Bruhat decompositions for semi-simple group schemes. We also give examples of principal G-bundles over A 1 U with split G such that the bundles are not isomorphic to pullbacks from U . IntroductionIn 1976 Daniel Quillen and Andrei Suslin independently proved a conjecture of Serre that an algebraic vector bundle over an affine space is algebraically trivial (see [Qui, Sus]). A few years earlier, Hyman Bass (see [Bas, Problem IX]) asked a more general question (see also [Qui]): Let R be a regular ring, is every vector bundle over A 1 R := Spec R[t] isomorphic to the pullback of a vector bundle over Spec R? This is now known as Bass-Quillen problem. Note that it is enough to consider the case when R is a regular local ring (see [Qui, Thm. 1]). The problem was solved by Hartmut Lindel in [Lin] in the geometric case, that is, when R is a localization of a k-algebra of finite type, where k is a field.We can ask a more general question: consider a regular local k-algebra R and let G be a simple simply-connected group scheme over R. Question 1. Let E be a principal G-bundle over the affine line A 1 R . Is E isomorphic to the pullback of a principal G-bundle over Spec R?Using the standard relation between principal SL(n, R)-bundles and rank n vector bundles, we see that the above question reduces to the Bass-Quillen problem (=Lindel's Theorem if R is of essentially finite type), when G is the special linear group.Note that the answer to Question 1 is positive if R is a perfect field by a theorem of Raghunathan and Ramanathan (see [RR] and [Gil1]). We will see that the answer is in general negative even if we assume that G is a split group, and k is the field of complex numbers (see Theorem 3 and Example 2.4, where G = Spin(7, C)). To the best of my knowledge such examples were not known before (while examples with algebraically non-closed k, e.g. k being the field of real numbers, were known before, see Remark 2.6(ii)).The principal bundles over A 1 R we construct have the following property: they are isomorphic to the pullback of principal bundles over Spec R on the complement

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“…To put our result in context, we note the following Example 1.3 (Parimala). In the remarkable paper [Par78] constructed infinitely many, explicit rank 4 inner product spaces on R[x, y] which are not extended from R. Translating this problem to torsors, she showed that the map…”

Section: Introductionmentioning

confidence: 99%

$\mathbb{A}^1$-connected components of classifying spaces and purity for torsors

Elmanto,

Kulkarni,

Wendt

2021

Preprint

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In this paper, we study the Nisnevich sheafification H 1 ét (G) of the presheaf associating to a smooth scheme the set of isomorphism classes of G-torsors, for a reductive group G. We show that if G-torsors on affine lines are extended, then H 1 ét (G) is homotopy invariant and show that the sheaf is unramified if and only if Nisnevich-local purity holds for G-torsors. We also identify the sheaf H 1 ét (G) with the sheaf of A 1 -connected components of the classifying space B ét G. This establishes the homotopy invariance of the sheaves of components as conjectured by Morel. It moreover provides a computation of the sheaf of A 1 -connected components in terms of unramified G-torsors over function fields whenever Nisnevich-local purity holds for G-torsors.

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Failure of a quadratic analogue of Serre’s conjecture

Cited by 15 publications

References 10 publications

Book Review: Serre's problem on projective modules

Book Review: Serre's problem on projective modules

Affine Grassmannians of group schemes and exotic principal bundles over ${\Bbb A}^1$

$\mathbb{A}^1$-connected components of classifying spaces and purity for torsors

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