# A uniform treatment of Grothendieck's localization problem

@article{Murayama2022AUT, title={A uniform treatment of Grothendieck's localization problem}, author={Takumi Murayama}, journal={Compositio Mathematica}, year={2022}, volume={158}, pages={57 - 88} }

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…

## One Citation

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- 2020

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## References

SHOWING 1-10 OF 142 REFERENCES

Permanence properties of $F$-injectivity

- Mathematics
- 2019

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…

Openness of the F-rational locus and smooth base change

- Mathematics
- 1995

Throughout this paper rings are assumed to be commutative, associative, with identity, and Noetherian of prime characteristic p. For the most part we will also assume that rings are reduced. In [HH0]…

F-Pure Homomorphisms, Strong F-regularity, and F-injectivity

- Mathematics
- 2009

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…

Singularities in mixed characteristic via perfectoid big Cohen–Macaulay algebras

- Mathematics
- 2018

We utilize recent results of Andre and Gabber on the existence of weakly functorial integral perfectoid big Cohen-Macaulay (BCM) algebras to study singularities of local rings in mixed…

On arithmetic Macaulayfication of Noetherian rings

- Mathematics
- 2001

The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A…

F-rational rings have rational singularities

- Mathematics
- 1997

<abstract abstract-type="TeX"><p>It is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be…

On ideal-adic completion of noetherian rings

- Mathematics
- 1981

In commutative (noetherian) ring theory, complete local rings p lay many important roles. T h a n k s t o the efforts m ade by K rull, Z ariski, N agata and Grothendieck, a lot of marvelous…

Desingularization of two-dimensional schemes

- Mathematics
- 1978

We present a new proof of the existence of a desingularization for any excellent surface (where "surface" means "two-dimensional reduced noetherian scheme"). The problem of resolution of…

Resolution of singularities of arithmetical threefolds

- MathematicsJournal of Algebra
- 2019

We prove Grothendieck's Conjecture on Resolution of Singulari-ties for quasi-excellent schemes X of dimension three and of arbitrary characteristic. This applies in particular to X = SpecA, A a…

Cohen-Macaulay rings

- Mathematics
- 1993

In this chapter we introduce the class of Cohen–Macaulay rings and two subclasses, the regular rings and the complete intersections. The definition of Cohen–Macaulay ring is sufficiently general to…