基于多项式展开的一致符号拓扑
Uniform Symbolic Topologies via Multinomial Expansions
摘要：
什么时候诺特交换环$R$在素数上有一致的符号拓扑--读，什么时候存在整数$D>0$使得所有素理想$P \subseteq R$和所有$r >0$的符号幂$P^{(Dr)} \subseteq P^r$？开创性的工作艾因Lazarsfeld史密斯，延长Hochster和Huneke，并由马和Schwede反过来，提供了一个美丽的答案，在设置有限维优秀的正规环。然后自然地寻找类似物，其中环$R$是不规则的，或者其中上述理想包含可以使用增长速率较慢的线性函数来改进。这份手稿属于这些研究方向的重叠部分。工作与规定类型的素理想$Q$内的张量积的域的有限型在代数闭域$\mathbb{F}$，我们提出了二项式和多项式的扩展标准包含的类型$Q^{(E r)} \subseteq Q^r$，甚至更好，类型$Q^{(E (r-1)+1)} \subseteq Q^r$的所有$r>0$。最后一节综合了关于我们使用这些标准的频率的评论，并提供了一个例子。
英文摘要：
When does a Noetherian commutative ring $R$ have uniform symbolic topologies on primes--read, when does there exist an integer $D>0$ such that the symbolic power $P^{(Dr)} \subseteq P^r$ for all prime ideals $P \subseteq R$ and all $r >0$? Groundbreaking work of Ein-Lazarsfeld-Smith, as extended by Hochster and Huneke, and by Ma and Schwede in turn, provides a beautiful answer in the setting of finite-dimensional excellent regular rings. It is natural to then sleuth for analogues where the ring $R$ is non-regular, or where the above ideal containments can be improved using a linear function whose growth rate is slower. This manuscript falls under the overlap of these research directions. Working with a prescribed type of prime ideal $Q$ inside of tensor products of domains of finite type over an algebraically closed field $\mathbb{F}$, we present binomial- and multinomial expansion criteria for containments of type $Q^{(E r)} \subseteq Q^r$, or even better, of type $Q^{(E (r-1)+1)} \subseteq Q^r$ for all $r>0$. The final section consolidates remarks on how often we can utilize these criteria, presenting an example.
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