Serre's Problem on Projective Modules
Serre's Problem on Projective Modules
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DOI：
10.1007/978-3-540-34575-6
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发表时间：
2006-06
期刊：
影响因子：
--
通讯作者：
T. Lam
中科院分区：
文献类型：
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作者：
T. Lam
在世纪后半叶的大部分时间里，“塞尔猜想”被称为J. P.塞尔在1955年，大意是，一个不知道如果？有限生成的投射模在多项式环k [x，...，x]，其中k是a？eld.这一声明的动机是一个事实，即1在af？什么计划？由k [x，.，x]是代数几何模拟1 n的af？n-space over k.在拓扑学中，n-空间是可收缩的，所以在它上面只有平凡丛。代数几何中的n-空间是否也有类似的情形？由于Speck [x，...，[x] corre-1 n spond to？环k [x，.，x]，问题是1 n等于是否这样的投射模是自由的，任何基地？埃尔德河这是相当清楚，Serreintendedhisstatementasanopenproblem在她的理论框架的代数几何，这是刚刚开始出现在20世纪50年代中期。在他发表的著作中，没有任何地方以这样或那样的方式推测他的问题的可能结果。然而，几乎从一开始，一个猜测肯定的答案塞尔的问题成为众所周知的世界“塞尔的猜想”。后来，数学中两个新的（且密切相关的）学科的出现进一步提高了人们对这个“猜想”的兴趣：同调代数和代数K-理论。
“Serre’s Conjecture”, for the most part of the second half of the 20th century,-ferred to the famous statement made by J.-P. Serre in 1955, to the effect that one did not know if? nitely generated projective modules were free over a polynomial ring k [x,..., x], where k is a? eld. This statement was motivated by the fact that 1 n the af? ne scheme de? ned by k [x,..., x] is the algebro-geometric analogue of 1 n the af? ne n-space over k. In topology, the n-space is contractible, so there are only trivial bundles over it. Would the analogue of the latter also hold for the n-space in algebraic geometry? Since algebraic vector bundles over Speck [x,..., x] corre-1 n spond to? nitely generated projective modules over k [x,..., x], the question was 1 n tantamount to whether such projective modules were free, for any base? eld k. ItwasquiteclearthatSerreintendedhisstatementasanopenproblemintheshe-theoretic framework of algebraic geometry, which was just beginning to emerge in the mid-1950s. Nowhere in his published writings had Serre speculated, one way or another, upon the possible outcome of his problem. However, almost from the start, a surmised positive answer to Serre’s problem became known to the world as “Serre’s Conjecture”. Somewhat later, interest in this “Conjecture” was further heightened by the advent of two new (and closely related) subjects in mathematics: homological algebra, and algebraic K-theory.