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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 March 1, 2026
Variants of Normality and Steadfastness Deform
The cancellation problem asks whether A[X1,X2,…,Xn] ≅ B[Y1, Y2, . . . , Yn] implies A ≅ B. Hamann introduced the class of steadfast rings as the rings for which a version of the cancellation problem considered by Abhyankar, Eakin, and Heinzer holds. By work of Asanuma, Hamann, and Swan, steadfastness can be characterized in terms of p-seminormality, which is a variant of normality introduced by Swan. We prove that p-seminormality and steadfastness deform for reduced Noetherian local rings. We also prove that p-seminormality and steadfastness are stable under adjoining formal power series variables for reduced (not necessarily Noetherian) rings. Our methods also give new proofs of the facts that normality and weak normality deform, which are of independent interest.
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- Award ID(s):
- 1902616
- PAR ID:
- 10585464
- Publisher / Repository:
- The University of Michigan
- Date Published:
- Journal Name:
- Michigan Mathematical Journal
- Volume:
- 75
- Issue:
- 1
- ISSN:
- 0026-2285
- Page Range / eLocation ID:
- 145-172
- Subject(s) / Keyword(s):
- 13F45 13B25 14B07 13B22 13F25 p-seminormality steadfastness deformation
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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