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We prove the relative Grauert–Riemenschneider vanishing, Kawamata–Viehweg vanishing, and Kollár injectivity theorems for proper morphisms of schemes of equal characteristic zero, solving conjectures of Boutot and Kawakita. Our proof uses the Grothendieck limit theorem for sheaf cohomology and Zariski–Riemann spaces. We also show that these vanishing and injectivity theorems hold for locally Moishezon (respectively, projective) morphisms of quasi-excellent algebraic spaces and semianalytic germs of complex-analytic spaces (respectively, quasi-excellent formal schemes and non-Archimedean analytic spaces), all in equal characteristic zero. We give many applications of our vanishing results. For example, we extend Boutot’s theorem to all Noetherian Q-algebras by showing that pseudo-rationality descends under pure maps of Q-algebras. This solves a conjecture of Boutot and answers a question of Schoutens. The proofs of this Boutot-type result and of our vanishing and injectivity theorems all use a new characterization of rational singularities using Zariski–Riemann spaces. 

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. 

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. 

An excellent ring of prime characteristic for which the Frobenius map is pure is also Frobenius split in many commonly occurring situations in positive characteristic commutative algebra and algebraic geometry. However, using a fundamental construction from rigid geometry, we show that excellent $$F$$-pure rings of prime characteristic are not Frobenius split in general, even for Euclidean domains. Our construction uses the existence of a complete non-Archimedean field $$k$$ of characteristic $$p$$ with no nonzero continuous $$k$$-linear maps $$k^{1/p} \to k$$. An explicit example of such a field is given based on ideas of Gabber, and may be of independent interest. Our examples settle a long-standing open question in the theory of $$F$$-singularities whose origin can be traced back to when Hochster and Roberts introduced the notion of $$F$$-purity. The excellent Euclidean domains we construct also admit no nonzero $$R$$-linear maps $$R^{1/p} \rightarrow R$$. These are the first examples that illustrate that $$F$$-purity and Frobenius splitting define different classes of singularities for excellent domains, and are also the first examples of excellent domains with no nonzero $$p^{-1}$$-linear maps. The latter is particularly interesting from the perspective of the theory of test ideals. 

In this expository paper, we show that the Deligne–Mumford moduli space of stable curves is projective over Spec(Z). The proof we present is due to Kollár. Ampleness of a line bundle is deduced from the nefness of a related vector bundle via the ampleness lemma, a classifying map construction. The main positivity result concerns the pushforward of relative dualizing sheaves on families of stable curves over a smooth projective curve. 

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.