摘要翻译：
从^1-同伦理论的角度讨论了光滑k-格式X可收缩的代数向量丛；当k=C时，光滑流形X(C)是可收缩的拓扑空间。这类格式的积分代数K-理论和积分模态上同调都是K格式的积分代数K-理论和积分模态上同调。人们可能希望，此外，与流形上的拓扑向量丛的分类类似，这类方案上的代数向量丛都同构于平凡丛；当方案是仿射时，这几乎可以肯定是正确的。然而，在非仿射情形下，这是错误的：我们证明了（本质上）每一个光滑的a^1-可收缩严格拟仿射格式，它允许一个总空间是仿射的U-torsor,对于U是一个幂次群，它拥有一个非平凡的向量丛。事实上，我们产生了这种非同构格式的显式任意维族，族中的每一个格式都配备了“尽可能多的”（即任意维模）具有足够大秩n的非同构向量束，这是我们所希望的；这些格式和它们上的向量丛都不是代数K-理论所能区分的。我们还讨论了某些光滑复仿射簇的向量丛的平凡性，其下复流形是可收缩的，但不一定是^1-可收缩的。
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英文标题：
《Vector bundles on contractible smooth schemes》
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作者：
Aravind Asok, Brent Doran
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics        数学
二级分类：K-Theory and Homology        K-理论与同调
分类描述：Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras
代数和拓扑K-理论，与拓扑的关系，交换代数和算子代数
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英文摘要：
  We discuss algebraic vector bundles on smooth k-schemes X contractible from the standpoint of A^1-homotopy theory; when k = C, the smooth manifolds X(C) are contractible as topological spaces. The integral algebraic K-theory and integral motivic cohomology of such schemes are that of Spec k. One might hope that furthermore, and in analogy with the classification of topological vector bundles on manifolds, algebraic vector bundles on such schemes are all isomorphic to trivial bundles; this is almost certainly true when the scheme is affine. However, in the non-affine case this is false: we show that (essentially) every smooth A^1-contractible strictly quasi-affine scheme that admits a U-torsor whose total space is affine, for U a unipotent group, possesses a non-trivial vector bundle. Indeed we produce explicit arbitrary dimensional families of non-isomorphic such schemes, with each scheme in the family equipped with "as many" (i.e., arbitrary dimensional moduli of) non-isomorphic vector bundles, of every sufficiently large rank n, as one desires; neither the schemes nor the vector bundles on them are distinguishable by algebraic K-theory. We also discuss the triviality of vector bundles for certain smooth complex affine varieties whose underlying complex manifolds are contractible, but that are not necessarily A^1-contractible.
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PDF链接：
https://arxiv.org/pdf/0710.3607