论文标题
主要理想戒指的多元代数
Multi-Rees algebras on principal ideal rings
论文作者
论文摘要
当$ r $是一个Noetherian环时，我们拥有一个理想的家族，每个理想都包含一个非零的除数，那么已经知道，这些理想的多元素代数的定义理想等于饱和的理想。在这种情况下，要获得多元代数的定义理想，我们只需要饱和该理想家族的直接总和的第一个syzygies即可。但是，当这些理想中至少一个不包含任何非零的除数时，这个事实并非如此。在本文中，我们表明，在主理想环上，多项式环的多元代数的定义理想等于另一种饱和理想，在这种理想中，我们除了首先syzygies以外，我们更饱和更明确的多项式。请注意，其中一些理想中的一些可能不包含非零的除数。鉴于此明确的公式，我们可以使用消除顺序计算定义理想的Gröbner基础，我们在本文中谈到了这一定义。
When $R$ is a Noetherian ring and we have a family of ideals in which every ideal contains at least one nonzero divisor, then it is already known that the defining ideal of the multi-Rees algebra of these ideals is equal to a saturated ideal. In such a case to get the defining ideal of the multi-Rees algebra we only need to saturate the first syzygies of direct sum of this family of ideals. However, this fact is not true, when at least one of these ideals does not contain any nonzero divisor. In this paper we show that the defining ideal of the multi-Rees algebra of a family of ideals of a polynomial ring over a principal ideal ring, is equal to another kind of saturated ideal, where we saturate more explicit polynomials other than just first syzygies. Please notice that in general some of these ideals may not contain nonzero divisors. Given this explicit formula, we can compute Gröbner basis of the defining ideal using an elimination order, which we talk about in the present paper.
