Schedule for: 22w5174 - Advances in Mixed Characteristic Commutative Algebra & Geometric Connections

What allowed for many developments in algebraic geometry and commutative algebra was a discovery of the notion of a Frobenius splitting, which, briefly speaking, detects how pathological singularities or projective varieties can be. Recently, Yobuko introduced a more general concept, a quasi-F-splitting, which captures much more refined arithmetic invariants. In my talk, I will discuss on-going projects in which we develop the theory of quasi-F-splittings in the context of birational geometry and derive applications, for example, to liftability of singularities. This is joint work with Tatsuro Kawakami, Hiromu Tanaka, Teppei Takamatsu, Fuetaro Yobuko, and Shou Yoshikawa.

A number of the tools used in studying singularities in commutative algebra have come from the study of tight closure and its test ideal in rings of equal characteristic. In replicating these results in mixed characteristic, it has been useful to find test ideal-like structures for a variety of new closure operations. We apply the closure-interior duality laid out by the speaker and Neil Epstein to describe properties shared by many test ideals, and use these to give results on particular test ideals.

Let $X$ be a perfectoid space with tilt $X^\flat$. We build a natural map $\theta:\Pic X^\flat\to\lim\Pic X$ where the (inverse) limit is taken over the $p$-power map, and show that $\theta$ is an isomorphism if $R = \Gamma(X,\sO_X)$ is a perfectoid ring. As a consequence we obtain a characterization of when the Picard groups of $X$ and $X^\flat$ agree in terms of the $p$-divisibility of $\Pic X$. The main technical ingredient is the vanishing of certain higher derived limits of the unit group $R^*$, whence the main result follows from the Grothendieck spectral sequence.

We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel coefficients of the Frobenius powers of an $m$-primary ideal exists in a Noetherian local ring $(R,m)$ with prime characteristic $p>0.$ This, in turn, gives an expression of the Hilbert-Kunz multiplicity of powers of the ideal. We also prove that for a face ring $R$ of a simplicial complex and an ideal $J$ generated by pure powers of the variables, the generalized Hilbert-Kunz function $\ell(R/(J^{[q]})^k)$ is a polynomial for all $q,k$ and also give an expression of the generalized Hilbert-Kunz multiplicity of powers of $J$ in terms of Hilbert-Samuel multiplicity of $J.$ We conclude by giving a counter-example to a conjecture proposed by I. Smirnov which connects the stability of an ideal with the asymptotic limit of the first Hilbert coefficient of the Frobenius power of the ideal.

In a polynomial ring over a perfect field, a classical theorem of Zariski and Nagata says that the symbolic powers of a radical ideal I -- which are defined by taking the minimal primary components of I^n -- coincide with its differential powers. This description fails in mixed characteristic: we need to also consider differential-like operators that decrease p-adic order for a prime integer p. The solution turns out to be Buium and Joyal's p-derivations. This is joint work with Alessandro De Stefani and Jack Jeffries.

Let R be a graded direct summand of a positively graded polynomial ring over the p-adic integers. We exhibit an explicit constant D such that, for any positive integer n and any homogeneous prime ideal Q of R, the Dn-th symbolic power of Q is contained in the n-th power of the homogeneous maximal ideal (p)R + R_+. The strategy relies on a Zariski-Nagata type of theorem that works in mixed characteristic, together with the introduction of a new type of differential powers which do not require the existence of a p-derivation on R. The talk will be based on joint work with E. Grifo and J. Jeffries.

A Noetherian ring is a splinter if it is a direct summand of every finite cover. In this talk, I will discuss some recent work on splinter rings in both positive and mixed characteristics. In particular, inspired by the result of Bhatt on the Cohen-Macaulayness of the absolute integral closure, I will describe a global notion of splinter in the mixed characteristic setting called global +-regularity with applications to birational geometry in mixed characteristic. This can be seen as a generalization of the theory of globally F-regular pairs from positive to mixed characteristic. This is based on the joint work arXiv:2012.15801 with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Joe Waldron, and Jakub Witaszek.

We present a combinatorial description of Nash blowups of toric varieties in positive characteristic. We then show that normalized Nash blowups solve the singularities of normal toric surfaces over fields of positive characteristic.

The classical Jacobian criterion is an important tool for finding singular points on a variety over a (perfect) field. How can we find the singular locus over the p-adics or over the integers? In this talk, I'll discuss a new analogue of the Jacobian criterion that gives a simple description of the singular locus in this setting. This criterion uses a curious notion of differentiation by a prime number called p-derivations. If time permits, we will also discuss an extension of the theory of Kahler differentials to this mixed characteristic setting. This is based on joint work with Melvin Hochster.

Recently, using perfectoid techniques, Ma and Schwede developed a new theory of singularities in mixed characteristic. As an application of this theory, they proved that a Q-Gorenstein affine domain over a field of characteristic zero has log terminal singularities if its mod p reduction is F-regular for one single prime p. In this talk, as a new application of their theory, I will discuss the analog of their result for log canonical singularities. This talk is based on joint work with Shunsuke Takagi.

I will discuss recent joint work with Christopher Hacon and Karl Schwede in which we define a notion  of test ideals for rings of finite type over a complete local Noetherian ring that commutes with localization.  I will also discuss applications of our definition.

Splinters were introduced by Frank Ma in the late '80s in order to understand Hochster's direct summand conjecture (now a theorem). Perhaps owing to their general definition, basic properties of splinters seem devilishly difficult to establish. In this talk we will discuss recent advances in our understanding of splinters from the perspective of permanence properties. These properties are at the heart of deep notions and results in commutative algebra and algebraic geometry such as excellent rings and the existence of test elements. Our talk will complement the one by Kevin Tucker by showcasing local aspects of this fascinating class of singularities and is based on joint work with him. We will also highlight recent work of Shiji Lyu.

I will revisit a characterization of rational singularities and discuss some new developments.

In algebraic geometry of positive characteristic, singularities defined by the Frobenius map, including the notion of Frobenius-splitting, have played a crucial role. Moreover, there are powerful criteria, so-called Fedder's criteria, to confirm such properties. Yobuko recently introduced the notion of quasi-F-splitting and F-split heights, which generalize and quantify the notion of Frobenius-splitting, and proved that F-split heights coincide with Artin-Mazur heights for Calabi-Yau varieties. This notion is defined for purely positive characteristic varieties, but the ring of Witt vectors, which is a mixed characteristic object, makes an essential role in the definition. In this talk, I will give a generalization of Fedder's criteria to quasi-Frobenius-splitting, and introduce examples and applications of such criteria. This talk is based on a joint paper with Tatsuro Kawakami and Shou Yoshikawa.

Lech's inequality can be viewed as a uniform, independent of I, upper bound on the ratio of the colength and the multiplicity of an m-primary ideal I. A recent work (joint with Huneke, Ma, and Quy in equicharacteristic and with Ma in mixed characteristic) showed that the uniform bound can be always improved under mild assumptions. I will explain the strategy of the proof and present the number of arising problems, open in mixed characteristic, regarding various ways of sharpening the Lech inequality.

Classical valuation theory has proved to be a valuable tool in number theory, algebraic geometry and singularity theory. Krull valuations defined on rings induce topologies on them, since these are maps taking values in totally ordered abelian groups, which carry an intrinsic topology themselves. One then uses these ring topologies to enrich algebraic varieties with new points coming from valuations. In this talk we consider valuations that might take values in certain lattice-ordered abelian monoids known as lattice-ordered semirings. For example, it has been used to construct geometric objects out of (commutative) rings endowed with a valuative topology. As an application of this set of ideas, we show how to associate an idempotent version of the structure sheaf of an algebraic variety over an arbitrary field, which behaves particularly well with respect to idempotization of closed subschemes. This is a joint work with S. Falkensteiner, M. Haiech, M. P. Noordman and L. Bossinger.