The number of generators of a module
The number of generators of a module
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一个模块的生成器数量
DOI：
10.1007/bf01110912
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发表时间：
1967
期刊：
影响因子：
--
通讯作者：
R. G. Swan
中科院分区：
文献类型：
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作者：
R. G. Swan
设R是交换Noether环，M是R-生成的R-模.如果已知M的所有局部化Mr都可以由r个元素生成，那么就有理由寻找M的生成元的最小数目的界限。FORSTER [2]解决了这个问题，他证明了M可以由r+ n个元素生成，其中n是R的Krull维数。然而，在许多情况下，这种结果可以得到改善。例如，如果R是半局部的，则M可以由r个元素生成。SERR [3]证明了M实际上可以由r+ d个元素生成，其中d是R的极大理想谱的维数[1]。在这里，我将用福斯特论证的一个修正来证明情况确实如此。同时，对非对易情形也作了推广。设R是交换环，A是R-代数，它是R-模的逆生成的。设X是R的极大理想谱。假设X是一个维数为d的诺特空间。设M是A-模，M是A-生成的A-模.如果对R的每个极大理想92 l，M~由A~上的r个元生成，则M可以由A上的r+ d个元生成.
Suppose R is a commutative noetherian ring and M is a finitely generated R-module. If it is known that all localizations M r of M can be generated by r elements, it is reasonable to look for a bound on the minimal number of generators of M in terms of r. This problem was solved by FORSTER [2] who showed that M can be generated by r+ n elements where n is the Krull dimension of R. In many cases, however, this result can be improved. For example, if R is semilocal, M can be generated by r elements. SERR~ has conjectured [3] that in fact M can be generated by r+ d elements where d is the dimension of the maximal ideal spectrum of R [1]. I will show here that is indeed the case, by using a modification of FORSTER'S argument. At the same time, I will give a generalization to the non-commutative case.Theorem 1. Let R be a commutative ring and A an R-algebra which is finitely generated as an R-module. Let X be the maximal ideal spectrum of R. Assume X is a noetherian space of dimension d. Let M be a finitely generated A-module. If for each maximal ideal 92l of R, M~ is generated by r elements over A~, then M can be generated over A by r+ d elements.