论文标题
卓越，f-Simulunition和Solidity
Excellence, F-singularities, and solidity
论文作者
论文摘要
如果存在非零$ r $ linear map $ s $ s \ rightarrow r $，则$ r $ -Algebra $ s $是$ r $ - 固体。在特征$ p $中，$ f $ singularies（例如frobenius分裂）的研究隐含地依赖于$ r $ r $ s的$ r^{1/p} $。在前两位作者对Frobenius的最新结果之后，某些出色的$ f $ pure戒指的弗罗贝尼乌斯（Frobenius）不插分，在本文中，我们使用坚固性的概念系统地研究了卓越的概念，并着重于$ f $ singularies。我们表明，对于环$ r $，本质上是有限的类型，而完整的本地环$ p $，降低性意味着$ r $ - s的$ r^{1/p} $，$ f $ - purity意味着frobenius spraptitting和$ f $ - pure-pure-puredities noadities split $ f $ groumboltity。我们证明了henselization和poterions并不牢固，为任意优质戒指提供了$ r $ s的$ r $ salive the $ r^{1/p} $的障碍。这也对大科恩·麦克劳（Cohen-Macaulay）代数的坚固性产生负面影响，其中一个重要的例子是主要特征中出色局部环的绝对积分封闭。我们在绝对整体封闭的牢固性与日本环的概念之间建立了密切的关系。分析日本物业表明，$ r^{1/p} $是$ r $ -Solid的Dedekind域$ r $非常好，尽管我们最近示例了出色的欧几里得域，没有非零$ p^{ - 1} $ - 线性地图。此外，我们表明，尽管完美的封闭通常在代数几何情况下是坚固的，但存在具有固体完美封闭的本地出色域，其绝对积分闭合并不固体。在附录中，Karen E. Smith使用了绝对积分封闭的坚固性来表征一类主要特征的Gorenstein域的理想测试。
An $R$-algebra $S$ is $R$-solid if there exists a nonzero $R$-linear map $S \rightarrow R$. In characteristic $p$, the study of $F$-singularities such as Frobenius splittings implicitly rely on the $R$-solidity of $R^{1/p}$. Following recent results of the first two authors on the Frobenius non-splitting of certain excellent $F$-pure rings, in this paper we use the notion of solidity to systematically study the notion of excellence, with an emphasis on $F$-singularities. We show that for rings $R$ essentially of finite type over complete local rings of characteristic $p$, reducedness implies the $R$-solidity of $R^{1/p}$, $F$-purity implies Frobenius splitting, and $F$-pure regularity implies split $F$-regularity. We demonstrate that Henselizations and completions are not solid, providing obstructions for the $R$-solidity of $R^{1/p}$ for arbitrary excellent rings. This also has negative consequences for the solidity of big Cohen-Macaulay algebras, an important example of which are absolute integral closures of excellent local rings in prime characteristic. We establish a close relationship between the solidity of absolute integral closures and the notion of Japanese rings. Analyzing the Japanese property reveals that Dedekind domains $R$ for which $R^{1/p}$ is $R$-solid are excellent, despite our recent examples of excellent Euclidean domains with no nonzero $p^{-1}$-linear maps. Additionally, we show that while perfect closures are often solid in algebro-geometric situations, there exist locally excellent domains with solid perfect closures whose absolute integral closures are not solid. In an appendix, Karen E. Smith uses the solidity of absolute integral closures to characterize the test ideal for a large class of Gorenstein domains of prime characteristic.
