$F$ -纯同态、强$F$ -正则性和$F$ -内射性
$F$-pure homomorphisms, strong $F$-regularity, and $F$-injectivity
摘要：
讨论了强$F$ -正则性、$F$ -纯性和Cohen-Macaulay $F$ -内射（简称CMFI）性质的Matijevic-Roberts型定理.与此相关，我们还讨论了基变问题和这些性质的轨迹的开放性。特别地，我们利用Radu-Andre同态定义了同态的$F$ -纯性的概念，证明了它的基本性质，讨论了强$F$ -正则性（非常强$F$ -正则性）的一个强版本，并比较了强$F$ -正则性的两个版本.结果证明了局部环、$F$ -有限环和优良局部环上的本质有限型代数的强$F$-正则性和非常强$F$ -正则性是一致的.证明了强$F$ -正则性的$F$ -纯基变换.
英文摘要：
We discuss Matijevic-Roberts type theorem on strong $F$-regularity, $F$-purity, and Cohen-Macaulay $F$-injective (CMFI for short) property. Related to this problem, we also discuss the base change problem and the openness of loci of these properties. In particular, we define the notion of $F$-purity of homomorphisms using Radu-Andre homomorphisms, and prove basic properties of it. We also discuss a strong version of strong $F$-regularity (very strong $F$-regularity), and compare these two versions of strong $F$-regularity. As a result, strong $F$-regularity and very strong $F$-regularity agree for local rings, $F$-finite rings, and essentially finite-type algebras over an excellent local rings. We prove the $F$-pure base change of strong $F$-regularity.
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