Shimomoto, Kazuma On the semicontinuity problem of fibers and global \(F\)-regularity. (English) Zbl 1453.13005 Commun. Algebra 45, No. 3, 1057-1075 (2017). Summary: 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. Cited in 2 Documents MSC: 13A02 Graded rings 13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure 14A15 Schemes and morphisms Keywords:lifting problem; localization problem; \(\mathbf{P}\)-homomorphism; semicontinuity PDFBibTeX XMLCite \textit{K. Shimomoto}, Commun. Algebra 45, No. 3, 1057--1075 (2017; Zbl 1453.13005) Full Text: DOI arXiv References: [1] DOI: 10.1090/conm/159/01498 · doi:10.1090/conm/159/01498 [2] DOI: 10.1006/aima.1997.1684 · Zbl 0935.13008 · doi:10.1006/aima.1997.1684 [3] DOI: 10.4007/annals.2010.171.571 · Zbl 1239.13008 · doi:10.4007/annals.2010.171.571 [4] DOI: 10.1017/CBO9780511608681 · doi:10.1017/CBO9780511608681 [5] DOI: 10.1016/j.jalgebra.2003.11.013 · Zbl 1057.14062 · doi:10.1016/j.jalgebra.2003.11.013 [6] DOI: 10.1007/BF02698644 · Zbl 0916.14005 · doi:10.1007/BF02698644 [7] Fedder R., Trans. Am. Math. Soc. 278 pp 461– (1983) [8] DOI: 10.1090/S1056-3911-2014-00641-X · Zbl 1444.14044 · doi:10.1090/S1056-3911-2014-00641-X [9] DOI: 10.1007/978-3-8348-9722-0 · doi:10.1007/978-3-8348-9722-0 [10] DOI: 10.2969/jmsj/03020179 · Zbl 0371.13017 · doi:10.2969/jmsj/03020179 [11] Greco S., Compos. Math. 40 pp 325– (1980) [12] Grothendieck A., Publ. Math. I.H.E.S. 11 pp 17– (1961) [13] Grothendieck A., Publ. Math. I.H.E.S. 20 (1964) [14] DOI: 10.1007/978-1-4757-3849-0 · doi:10.1007/978-1-4757-3849-0 [15] DOI: 10.1201/9780203908013.ch21 · doi:10.1201/9780203908013.ch21 [16] DOI: 10.2748/tmj/1163775133 · Zbl 1112.14057 · doi:10.2748/tmj/1163775133 [17] Hashimoto M., Commun. Algebra 39 pp 1– (2011) [18] DOI: 10.1307/mmj/1220879417 · Zbl 1181.13017 · doi:10.1307/mmj/1220879417 [19] Hochster M., Mem. Soc. Math. France (N.C.) 38 pp 31– (1989) [20] DOI: 10.2307/2154942 · Zbl 0844.13002 · doi:10.2307/2154942 [21] DOI: 10.1090/cbms/088 · doi:10.1090/cbms/088 [22] Kirti J., Int. Math. Res. Notices pp 109– (2003) [23] Kollár, J., Mori, S. (1998). Birational geometry of algebraic varieties. Cambridge Tracts in Mathematics. New York: Cambridge University Press. · doi:10.1017/CBO9780511662560 [24] Kollár, J. Moduli of Surfaces. Book in Preparation. Available at: http://www.math.princeton.edu/ kollar/. [25] DOI: 10.1016/0021-8693(84)90164-9 · Zbl 0544.13011 · doi:10.1016/0021-8693(84)90164-9 [26] Matsumura H., Commutative Ring Theory 8 (1986) · Zbl 0603.13001 [27] DOI: 10.2307/1971368 · Zbl 0601.14043 · doi:10.2307/1971368 [28] Mumford D., Abelian Varieties (1974) [29] DOI: 10.1016/j.aim.2009.12.020 · Zbl 1193.13004 · doi:10.1016/j.aim.2009.12.020 [30] DOI: 10.1016/j.jpaa.2008.11.047 · Zbl 1177.13014 · doi:10.1016/j.jpaa.2008.11.047 [31] DOI: 10.1016/j.jpaa.2013.06.018 · Zbl 1280.13001 · doi:10.1016/j.jpaa.2013.06.018 [32] DOI: 10.1353/ajm.1999.0029 · Zbl 0946.13002 · doi:10.1353/ajm.1999.0029 [33] DOI: 10.1307/mmj/1030132733 · Zbl 0994.14012 · doi:10.1307/mmj/1030132733 [34] DOI: 10.1016/j.aim.2008.05.006 · Zbl 1146.14009 · doi:10.1016/j.aim.2008.05.006 [35] DOI: 10.1007/978-1-4419-6990-3_17 · Zbl 1256.13009 · doi:10.1007/978-1-4419-6990-3_17 [36] DOI: 10.1017/S0027763000019498 · Zbl 0518.13003 · doi:10.1017/S0027763000019498 [37] Watanabe, K.I. Private communication. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.