平坦环满射与交换环的泛局部化
Flat ring epimorphisms and universal localisations of commutative rings
摘要：
研究交换诺特环的不同类型的局部化。更确切地说，我们提供的标准来决定：（a）如果一个给定的平坦环满射是一个普遍的定位在科恩和斯科菲尔德的意义上;和（b）当这样的普遍定位是经典的分数环。为了找到这样的标准，我们使用的理论支持，我们分析了专业化闭子集相关联的一个平坦的环满射。当基础环是局部阶乘环或克鲁尔维数为1时，我们证明了所有平坦环满射都是泛局部化的.此外，事实证明，普遍定域何时是经典的问题的答案取决于皮卡德群的结构。我们还讨论了正规环的情况下，其中的除子类群起着至关重要的作用，以决定如果一个给定的平坦环满射是一个普遍的本地化。最后，我们探讨了几个（反）例子，突出了我们的假设的必要性。
英文摘要：
We study different types of localisations of a commutative noetherian ring. More precisely, we provide criteria to decide: (a) if a given flat ring epimorphism is a universal localisation in the sense of Cohn and Schofield; and (b) when such universal localisations are classical rings of fractions. In order to find such criteria, we use the theory of support and we analyse the specialisation closed subset associated to a flat ring epimorphism. In case the underlying ring is locally factorial or of Krull dimension one, we show that all flat ring epimorphisms are universal localisations. Moreover, it turns out that an answer to the question of when universal localisations are classical depends on the structure of the Picard group. We furthermore discuss the case of normal rings, for which the divisor class group plays an essential role to decide if a given flat ring epimorphism is a universal localisation. Finally, we explore several (counter)examples which highlight the necessity of our assumptions.
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