Cancellation over affine varieties

A. A. Suslin

doi:10.1007/BF01410752

Cancellation over affine varieties

A. A. Suslin

Mathematics

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Abstract

It is proved that if X is a smooth affine curve over a field F of characteristic ≠ℓ, then the group SK₁(X)/ℓ SK₁(X) is isomorphic to a subgroup of the étale cohomology group H_et³(X,Μ_e^F{cyrillic}2) and if F is algebraically closed, then SK₁(X) is a uniquely divisible group. The following cancellation theorem is obtained from results about SK₁ for curves: If X is a normal affine variety of dimension n over a field F, and if char F > n and C.d._e(F)≤1 for any prime ℓ>/n then any stably trivial vector bundle of rank n over X is trivial.

Original language	English (US)
Pages (from-to)	2974-2980
Number of pages	7
Journal	Journal of Soviet Mathematics
Volume	27
Issue number	4
DOIs	https://doi.org/10.1007/BF01410752
State	Published - Nov 1984

ASJC Scopus subject areas

Statistics and Probability
General Mathematics
Applied Mathematics

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10.1007/BF01410752

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title = "Cancellation over affine varieties",

abstract = "It is proved that if X is a smooth affine curve over a field F of characteristic ≠ℓ, then the group SK1(X)/ℓ SK1(X) is isomorphic to a subgroup of the {\'e}tale cohomology group Het3(X,ΜeF{cyrillic}2) and if F is algebraically closed, then SK1(X) is a uniquely divisible group. The following cancellation theorem is obtained from results about SK1 for curves: If X is a normal affine variety of dimension n over a field F, and if char F > n and C.d.e(F)≤1 for any prime ℓ>/n then any stably trivial vector bundle of rank n over X is trivial.",

author = "Suslin, {A. A.}",

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T1 - Cancellation over affine varieties

AU - Suslin, A. A.

PY - 1984/11

Y1 - 1984/11

N2 - It is proved that if X is a smooth affine curve over a field F of characteristic ≠ℓ, then the group SK1(X)/ℓ SK1(X) is isomorphic to a subgroup of the étale cohomology group Het3(X,ΜeF{cyrillic}2) and if F is algebraically closed, then SK1(X) is a uniquely divisible group. The following cancellation theorem is obtained from results about SK1 for curves: If X is a normal affine variety of dimension n over a field F, and if char F > n and C.d.e(F)≤1 for any prime ℓ>/n then any stably trivial vector bundle of rank n over X is trivial.

AB - It is proved that if X is a smooth affine curve over a field F of characteristic ≠ℓ, then the group SK1(X)/ℓ SK1(X) is isomorphic to a subgroup of the étale cohomology group Het3(X,ΜeF{cyrillic}2) and if F is algebraically closed, then SK1(X) is a uniquely divisible group. The following cancellation theorem is obtained from results about SK1 for curves: If X is a normal affine variety of dimension n over a field F, and if char F > n and C.d.e(F)≤1 for any prime ℓ>/n then any stably trivial vector bundle of rank n over X is trivial.

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JO - Journal of Soviet Mathematics

JF - Journal of Soviet Mathematics

IS - 4

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Cancellation over affine varieties

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