Cohen–Macaulay Rings: Preface to the revised edition
Cohen–Macaulay Rings: Preface to the revised edition
复制标题
科恩-麦考利环：修订版序言
DOI：
10.1017/cbo9780511608681.001
复制
发表时间：
1998
期刊：
影响因子：
--
通讯作者：
A. Gottschalk
中科院分区：
文献类型：
--
作者：
Cornelia Schmitt;C. Schultheis;N. Pokala;S. Husson;J. Liewald;Cori Bargmann;A. Gottschalk
修订版的主要变化是新增了关于紧密封闭的第10章。该理论由梅尔·霍赫斯特（Mel Hochster）和克雷格·胡内克（Craig Huneke）大约十年前创建，并且仍在大力扩展。我们讨论基本思想，F-正则环和F-有理环，包括史密斯定理，其中F-有理性隐含着伪有理性。在紧闭包的众多应用中，我们选择了 Briancon-Skoda 定理以及 Hochster 和 Huneke 定理，认为正则环的等特征直被加数是 Cohen-Macaulay。为了涵盖这些应用，必须重写第 8.4 节，该节开发了简化为特征 p 的技术。第三部分的标题不再合适，已更改。另一个值得注意的补充是新 4.3 节中的 Gotzmann 定理。我们相信第四章现在讨论了希尔伯特函数的所有基本定理。此外，本章还进行了轻微的重组。新的第 5.5 节包含了关于 Stanley-Reisner 环的 Betti 数的 Hochster 公式的证明，因为此类环的自由解析最近受到了广泛关注。在第一版中，该公式的使用未经证明。我们感谢第一版的所有读者提出的更正和改进建议。我们特别感谢 L. Avramov、A. Conca、S. Iyengar、R. Y. Sharp、B. Ulrich 和 K.-i。渡边。
The main change in the revised edition is the new Chapter 10 on tight closure. This theory was created by Mel Hochster and Craig Huneke about ten years ago and is still strongly expanding. We treat the basic ideas, F -regular rings, and F -rational rings, including Smith's theorem by which F -rationality implies pseudo-rationality. Among the numerous applications of tight closure we have selected the Briancon–Skoda theorem and the theorem of Hochster and Huneke saying that equicharacteristic direct summands of regular rings are Cohen–Macaulay. To cover these applications, Section 8.4, which develops the technique of reduction to characteristic p , had to be rewritten. The title of Part III, no longer appropriate, has been changed. Another noteworthy addition are the theorems of Gotzmann in the new Section 4.3. We believe that Chapter 4 now treats all the basic theorems on Hilbert functions. Moreover, this chapter has been slightly reorganized. The new Section 5.5 contains a proof of Hochster's formula for the Betti numbers of a Stanley–Reisner ring since the free resolutions of such rings have recently received much attention. In the first edition the formula was used without proof. We are grateful to all the readers of the first edition who have suggested corrections and improvements. Our special thanks go to L. Avramov, A. Conca, S. Iyengar, R. Y. Sharp, B. Ulrich, and K.-i. Watanabe.