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Let $$f\colon Y \to X$$ be a proper flat morphism of locally noetherian schemes. Then the locus in $$X$$ over which $$f$$ is smooth is stable under generization. We prove that, under suitable assumptions on the formal fibers of $$X$$ , the same property holds for other local properties of morphisms, even if $$f$$ is only closed and flat. Our proof of this statement reduces to a purely local question known as Grothendieck's localization problem. To answer Grothendieck's problem, we provide a general framework that gives a uniform treatment of previously known cases of this problem, and also solves this problem in new cases, namely for weak normality, seminormality, $$F$$ -rationality, and the ‘Cohen–Macaulay and $$F$$ -injective’ property. For the weak normality statement, we prove that weak normality always lifts from Cartier divisors. We also solve Grothendieck's localization problem for terminal, canonical, and rational singularities in equal characteristic zero.
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This content will become publicly available on November 18, 2025
Permanence properties of F-injectivity
We prove that F-injectivity localizes, descends under faithfully flat homomorphisms, and ascends under flat homomorphisms with Cohen–Macaulay and geometrically F-injective fibers, all for arbitrary Noetherian rings of prime characteristic. As a consequence, we show that the F-injective locus is open on most rings arising in arithmetic and geometry. As a geometric application, we prove that over an algebraically closed field of characteristic p > 3, generic projection hypersurfaces associated to suitably embedded smooth projective varieties of dimension ≤5 are F-pure, and hence F-injective. This geometric result is the positive characteristic analogue of a theorem of Doherty.
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- Award ID(s):
- 1902616
- PAR ID:
- 10585463
- Publisher / Repository:
- International Press of Boston, Inc.
- Date Published:
- Journal Name:
- Mathematical Research Letters
- Volume:
- 31
- Issue:
- 4
- ISSN:
- 1073-2780
- Page Range / eLocation ID:
- 985 to 1027
- Subject(s) / Keyword(s):
- 13A35 13D45 13H10 14J17 F-injective ring local cohomology base change generic projection
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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