Vector bundles on contractible smooth schemes
Vector bundles on contractible smooth schemes
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可收缩光滑方案上的向量丛
DOI：
10.1215/00127094-2008-027
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
B. Doran
中科院分区：
文献类型：
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作者：
A. Asok;B. Doran
从A^1-同伦理论的角度讨论光滑k-方案X上的代数向量丛的可压缩性;当k = C时，光滑流形X（C）作为拓扑空间是可压缩的。这类格式的积分代数K-理论和积分动机上同调是Spec k的积分代数K-理论。此外，人们可能希望，类似于流形上拓扑向量丛的分类，这种概型上的代数向量丛都同构于平凡丛;当概型是仿射的时，这几乎肯定是真的。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-可收缩严格拟仿射概型，它允许一个U-扭子，其全空间是仿射的，对于U是一个幂幺群，拥有一个非平凡向量丛。实际上，我们产生了非同构的这种方案的显式任意维族，其中族中的每个方案配备有“尽可能多”（即，任意维模的）非同构向量丛，每一个足够大的秩n，作为一个愿望;无论是计划，也不是向量丛对他们是可区分的代数K-理论。我们还讨论了某些光滑复仿射簇的向量丛的平凡性，这些簇的基础复流形是可收缩的，但不一定是A^1-可收缩的。
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.