Using perfectoid algebras we introduce a mixed characteristic analog of the multiplier ideal, respectively test ideal, from characteristic zero, respectively p > 0, in the case of a regular ambient ring. We prove several properties about this ideal such as subadditivity. We then use these techniques to derive a uniform bound on the growth of symbolic powers of radical ideals in all excellent regular rings. The analogous result was shown in equal characteristic by Ein-Lazarsfeld-Smith and Hochster-Huneke. . 1 Q (mh) = Q mh R Q ∩ R, i.e., the elements of R which vanish generically to order mh at Q. 1 2 LINQUAN MA AND KARL SCHWEDEThe main ideas of our proof come from the recent solution of the direct summand conjecture and its derived variant [And16a, Bha18]: we introduce a mixed characteristic analog of the multiplier ideal or test ideal using perfectoid algebras, prove many properties of it, and finally (and analogously to the strategy of [ELS01], see also [Har05]) use those properties to deduce the Main Theorem above.1.1. Multiplier and test ideals. Suppose that R is an equal characteristic regular domain satisfying mild geometric assumptions 2 . Further suppose that a ⊆ R is an ideal and t ∈ R ≥0 a formal exponent for a. In this setting we can form the test ideal τ (R, a t ) in characteristic p > 0 or the multiplier ideal J (R, a t ) in characteristic 0. This is an ideal of R which measures the singularities of V (a) ⊆ Spec R, scaled by t. Roughly speaking, for relevant values of t, the multiplier or test ideal of (R, a t ) is smaller/deeper than that of (R, b t ) if V (a) has the same dimension as V (b) and is more singular than V (b). Crucially for the applications to symbolic powers, the multiplier or test ideal satisfies the following list of properties, see for example [HH90, HY03, Laz04]. We state them for the multiplier ideal J (R, a t ) but they also hold for the test ideal τ (R, a t ).(A) Basic containments: If a ⊆ b is a containment of ideals, then(B) Unambiguity of exponent: For any positive integer n, J (R, a tn ) = J (R, (a n ) t ). (C) Not too small: a ⊆ J (R, a). (D) Not too big: If a is prime of height h, J (R, (a (lh) ) 1 l ) ⊆ a. (E) Subadditivity: If b is another ideal and if s ≥ 0 is another real number, then J (a s b t ) ⊆ J (a s ) · J (b t ).In particular we have J (a tn ) ⊆ J (a t ) n .Combining these results, the application to the growth of symbolic powers follows from a clever asymptotic construction of multiplier ideals [ELS01], see also [Har05]. We aim to do the same thing in mixed characteristic.Very roughly, the multiplier ideal and test ideal of a regular local ring (R, m) of dimension d can be defined in the following way:(J /τ )(R, a t ) = Ann R {η ∈ H d m (R) | η's image in H d m (B) is "annihilated" by a t }. Here B and "annihilated" are made precise as follows:2 For example, of essentially finite type over a field, or complete, or F -finite in characteristic p > 0.

We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global F -regularity to mixed characteristic and identify certain stable sections of adjoint line bundles. Finally, by passing to graded rings, we generalize a special case of Fujita's conjecture to mixed characteristic.

We utilize recent results of André and Gabber on the existence of weakly functorial integral perfectoid big Cohen-Macaulay (BCM) algebras to study singularities of local rings in mixed characteristic. In particular, we introduce a mixed characteristic BCMvariant of rational/F -rational singularities, of log terminal/F -regular singularities and of multiplier/test ideals of divisor pairs. We prove a number of results about these objects including a restriction theorem for perfectoid BCM multiplier/test ideals and deformation statements for perfectoid BCM-regular and BCM-rational singularities. As an application, we obtain results on the behavior of F -regular and F -rational singularities in arithmetic families. Contents 1. Introduction 1 2. Preliminaries 5 3. Big Cohen-Macaulay rational singularities 10 4. Dominating integral perfectoid big Cohen-Macaulay R + -algebras 13 5. Perfectoid big Cohen-Macaulay parameter test submodules 21 6. Perfectoid big Cohen-Macaulay test ideals of pairs 30 7. Application: F -rational and strongly F -regular singularities in families 43 8. Discussion of algorithmic consequences 47 References 48

No abstract

In this paper we study the local cohomology modules of Du Bois singularities. Let $(R,m)$ be a local ring, we prove that if $R_{red}$ is Du Bois, then $H_m^i(R)\to H_m^i(R_{red})$ is surjective for every $i$. We find many applications of this result. For example we answer a question of Kov\'acs and the second author on the Cohen-Macaulay property of Du Bois singularities. We obtain results on the injectivity of $Ext$ that provide substantial partial answers of questions of Eisenbud-Mustata-Stillman in characteristic $0$, and these results can be viewed as generalizations of the Kodaira vanishing theorem for Cohen-Macaulay Du Bois varieties. We prove results on the set-theoretic Cohen-Macaulayness of the defining ideal of Du Bois singularities, which are characteristic $0$ analog of results of Singh-Walther and answer some of their questions. We extend results of Hochster-Roberts on the relation between Koszul cohomology and local cohomology for $F$-injective and Du Bois singularities, see Hochster-Roberts. We also prove that singularities of dense $F$-injective type deform.Comment: 24 pages, minor changes. To appear in Compositio Mathematic