Projective modules over polynomial rings
Projective modules over polynomial rings
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DOI：
10.1007/bf01390008
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发表时间：
1976-12
影响因子：
3.1
通讯作者：
D. Quillen
中科院分区：
文献类型：
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作者：
D. Quillen
本文证明了多项式环k [T~.....]上的所有n-生成投射模都是n-生成投射模.当k是一个域，或者更一般地说是一个主理想整环时，T，]是自由的。这肯定地解决了塞尔在他的论文FAC（[5]，第243页）中提出的一个问题，这个问题引起了很多关注（例如，见[1，2]）。通过Murthy的一个论证，这个结果来自于这样的断言，即当限制到仿射线Spec（A [T]）时，P~上的任何向量丛都是Spec（A）上向量丛的基扩张。本文将这一论断归结为Horrocks [3]所讨论的局部Noether环的情形，除非另有说明，否则所有环都是有单位元的交换环。设R是非交换环，我们用（1+ TRIT]）”表示多项式环R1，T]中与1模T全等的可逆元群.
In this paper we prove that all finitely generated projective modules over a polynomial ring k [T~..... T,] are free when k is a field, or more generally a principal ideal domain. This settles affirmatively a question posed by Serre in his paper FAC ([5], p. 243) which has attracted much attention (see for example [1, 2]). By an argument due to Murthy, this result follows from the assertion that any vector bundle over P~ when restricted to the affine line Spec (A [T]) is the base extension of a vector bundle over Spec (A). We reduce this assertion to the case where A is local noetherian, which has been treated by Horrocks [3].All rings are supposed to be commutative with identity unless specified otherwise. If R is a not necessarily commutative ring, we let (1+ TR IT])" denote the group of invertible elements in the polynomial ring R l, T] which are congruent to 1 modulo T.