Locally polynomial algebras are symmetric algebras
Locally polynomial algebras are symmetric algebras
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局部多项式代数是对称代数
DOI：
10.1007/bf01403135
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发表时间：
1976
影响因子：
3.1
通讯作者：
D. Wright
中科院分区：
文献类型：
--
作者：
H. Bass;E. Connell;D. Wright
我们在一个可交换环k上固定，设a是一个有限表示的k代数。假设，对于K的每一个极大理想m， K…-代数A…同构于多项式K， -代数。则A与有限生成的射影k模P的对称代数S (P)是同构的，这个结果，即标题所指的，包含在下面的定理（4.4）中。在几何上，它断言在spec (K)上的每一个具有仿射空间纤维的局部平凡纤维空间都是由一个矢量束产生的。如果A局部是单变量多项式代数这个定理是平凡的？我们已知的唯一的另一种情况是在[W]中处理的情况，当K是一个主理想定义域。该定理解决了[EH] p. 67和[W]， w6.2中提出的一个问题。本文[ES]包含了许多关于局部多项式代数的结果，但没有我们的有限表示性假设。[ES]的例子（3.15）给出了一个n代数a，它是一个noetherian UFD，局部是在Z上的一个变量多项式环，但不是在~上有限生成的，特别是，它不是任何7z模的对称代数。
We fix throughout a commutative ring K. Let A be a finitely presented K-algebra. Suppose that, for each maximal ideal m of K, the K,,,-algebra A,,, is isomorphic to a polynomial K,,-algebra. Then A is isomorphic to the symmetric algebra S (P) of a finitely generated projective K-module P. This result, to which the title refers, is contained in Theorem (4.4) below. Geometrically it asserts that every locally trivial fibre space over spec (K) with affine space fibres arises from a vector bundle. The theorem is trivial if A is locally a polynomial algebra in one variable? The only other case previously known to us is the case, treated in [W], when K is a principal ideal domain. The theorem solves a problem posed in [EH] p. 67, and in [W], w 6.2The paper [ES] contains many results on locally polynomial algebras, but without our finite presentability assumption. The example (3.15) of [ES] furnishes a N-algebra A which is a noetherian UFD, locally a polynomial ring in one variable over Z, yet not finitely generated over~, and, in particular, not the symmetric algebra of any 7Z-module.