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首页> 外文期刊>The Journal of the London Mathematical Society >The analogue of Izumi's Theorem for Abhyankar valuations
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The analogue of Izumi's Theorem for Abhyankar valuations

机译：Izumi定理对Abhyankar估值的类似物

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摘要

A well-known theorem of Izumi, strengthened by Rees, asserts that all the divisorial valuations centered in an analytically irreducible local noetherian ring (R,m) are linearly comparable to each other. This is equivalent to saying that any divisorial valuation ν centered in R is linearly comparable to the m-adic order. In the present paper, we generalize this theorem to the case of Abhyankar valuations ν with archimedian value semigroup Φ. Indeed, we prove that in a certain sense linear equivalence of topologies characterizes Abhyankar valuations with archimedian semigroups, centered in analytically irreducible local noetherian rings. In other words, saying that R is analytically irreducible, ν is Abhyankar and Φ is archimedian is equivalent to linear equivalence of topologies plus another condition called weak noetherianity of the graded algebra gr_νR. We give some applications of Izumi's Theorem and of Lemma 2.8, which is a crucial step in our proof of the main theorem. We show that some of the classical results on equivalence of topologies in noetherian rings can be strengthened to include linear equivalence of topologies. We also prove a new comparison result between the m-adic topology and the topology defined by the symbolic powers of an arbitrary ideal.
机译：Rees强化了一个著名的Izumi定理，该定理断言，所有以分析不可约的局部noetherian环（R，m）为中心的除数估值在线性上可比。这等效于说，以R为中心的除数估值ν与m-adic阶线性可比。在本文中，我们将该定理推广到具有阿基米德值半群Φ的Abhyankar估值ν的情况。的确，我们证明，在某种意义上，拓扑的线性等价特征是阿奇延卡尔估值与阿基米德半群有关，其集中在分析不可约的局部noetherian环上。换句话说，说R是解析不可约的，ν是Abhyankar，而Φ是阿基米德，则等于拓扑的线性等价加上另一条件，称为渐变代数gr_νR的弱noetherianity。我们给出了Izumi定理和引理2.8的一些应用，这是我们证明主定理的关键步骤。我们表明，在noetherian环中有关拓扑等效性的一些经典结果可以得到增强，以包括拓扑的线性等效性。我们还证明了m-adic拓扑与由任意理想的符号幂定义的拓扑之间的新比较结果。

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