关于纤维的半连续性问题与整体$F$-正则性
On the semicontinuity problem of fibers and global $F$-regularity
AI总结：
本文研究概形态射下纤维性质的半连续性问题，从局部层面通过解决局部环的提升与局部化问题得到半正规环、$F$-有理环的局部化定理，又从整体层面以整体$F$-正则性的形变为测试示例开展研究，推进了相关代数几何问题的探索。
摘要：
本文讨论了概形态射下纤维的若干性质的半连续性问题。该问题的一个方面是局部的：我们在局部环层面考虑概形的性质，通过解决局部环的提升问题和局部化问题建立了主要结果，特别地分别得到了半正规环和$F$-有理环情形下的局部化定理。该问题的另一个方面是整体的，通常与某些高阶直像层的零化问题相关，作为测试示例，我们研究了整体$F$-正则性的形变。
英文摘要：
In this article, we discuss the semicontinuity problem of certain properties on fibers for a morphism of schemes. One aspect of this problem is local. Namely, we consider properties of schemes at the level of local rings, in which the main results are established by solving the lifting and localization problems for local rings. In particular, we obtain the localization theorems in the case of seminormal and $F$-rational rings, respectively. Another aspect of this problem is global, which is often related to the vanishing problem of certain higher direct image sheaves. As a test example, we consider the deformation of the global $F$-regularity.
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