arXiv subjects
Publications and source records attributed to Karl Schwede.
Suppose $R$ is a $\mathbb{Q}$-Gorenstein $F$-finite and $F$-pure ring of prime characteristic $p>0$. We show that if $I\subseteq R$ is a compatible ideal (with all $p^{-e}$-linear maps) then there exists a module finite extension $R\to S$ such that the ideal $I$ is the sum of images of all $R$-linear maps $S\to R$.
We further the classification of rational surface singularities. Suppose $(S, \mathfrak{n}, \mathcal{k})$ is a strictly Henselian regular local ring of mixed characteristic $(0, p > 5)$. We classify functions $f$ for which $S/(f)$ has an isolated rational singularity at the maximal ideal $\mathfrak{n}$. The classification of such functions are used to show that if $(R, \mathfrak{m}, \mathcal{k})$ is an excellent, strictly Henselian, Gorenstein rational singularity of dimension $2$ and mixed characteristic $(0, p > 5)$, then there exists a split finite cover of $\mbox{Spec}(R)$ by a regular scheme. We give an application of our result to the study of $2$-dimensional BCM-regular singularities in mixed characteristic.
The containment problem for symbolic and ordinary powers of ideals asks for what values of $a$ and $b$ we have $I^{(a)} \subseteq I^b$. Over a regular ring, a result by Ein-Lazarsfeld-Smith, Hochster-Huneke, and Ma-Schwede partially answers this question, but the containments it provides are not always best possible. In particular, a tighter containment conjectured by Harbourne has been shown to hold for interesting classes of ideals - although it does not hold in general. In this paper, we develop a Fedder (respectively, Glassbrenner) type criterion for $F$-purity (respectively, strong $F$-regularity) for ideals of finite projective dimension over $F$-finite Gorenstein rings and use our criteria to extend the prime characteristic results of Grifo-Huneke to singular ambient rings. For ideals of infinite projective dimension, we prove that a variation of the containment still holds, in the spirit of work by Hochster-Huneke and Takagi.
We use the framework of perfectoid big Cohen-Macaulay algebras to define a class of singularities for pairs in mixed characteristic, which we call purely BCM-regular singularities, and a corresponding adjoint ideal. We prove that these satisfy adjunction and inversion of adjunction with respect to the notion of BCM-regularity and the BCM test ideal defined by the first two authors. We compare them with the existing equal characteristic PLT and purely $F$-regular singularities and adjoint ideals. As an application, we obtain a uniform version of the Briançon-Skoda theorem in mixed characteristic. We also use our theory to prove that two-dimensional KLT singularities are BCM-regular if the residue characteristic $p>5$, which implies an inversion of adjunction for three-dimensional PLT pairs of residue characteristic $p>5$. In particular, divisorial centers of PLT pairs in dimension three are normal when $p > 5$. Furthermore, in the appendix we provide a streamlined construction of perfectoid big Cohen-Macaulay algebras and show new functoriality properties for them using the perfectoidization functor of Bhatt and Scholze.
This paper describes the RationalMaps package for Macaulay2. This package provides functionality for computing several aspects of rational maps such as whether a map is birational, or a closed embedding.
In this article, we present FastMinors.m2, a package in Macaulay2 designed to introduce new methods focused on computations in function field linear algebra. Some key functionality that our package offers includes: finding a submatrix of a given rank in a provided matrix (when present), verifying that a ring is regular in codimension n, recursively computing the ideals of minors in a matrix, and finding an upper bound of the projective dimension of a module.
We present {\tt RandomPoints}, a package in \emph{Macaulay2} designed mainly to identify rational and geometric points in a variety over a finite field. We provide tools to estimate the dimension of a variety. We also present methods to obtain non-vanishing minors of a given size in a given matrix, by evaluating the matrix at a point.
This work introduces a notion of complexes of maximal depth, and maximal Cohen-Macaulay complexes, over a commutative noetherian local ring. The existence of such complexes is closely tied to the Hochster's ``homological conjectures", most of which were recently settled by André. Various constructions of maximal Cohen-Macaulay complexes are described, and their existence is applied to give new proofs of some of the homological conjectures, and also of certain results in birational geometry.
We show that Bertini theorems hold for $F$-signature and Hilbert--Kunz multiplicity. In particular, if $X \subseteq \mathbb{P}^n$ is normal and quasi-projective with $F$-signature greater than $λ$ (respectively the Hilbert--Kunz multiplicity is less than $λ$) at all points $x \in X$, then for a general hyperplane $H \subseteq \mathbb{P}^n$ the $F$-signature (respectively Hilbert--Kunz multiplicity) of $X \cap H$ is greater than $λ$ (respectively less than $λ$) at all points $x \in X \cap H$.
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 BCM-variant 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.
We prove that a local domain $R$, essentially of finite type over a field, is regular if and only if for every regular alteration $π: X \to Spec R$, we have that $R π_* \mathcal{O}_X$ has finite (equivalently zero in characteristic zero) projective dimension.
This note describes a \emph{Macaulay2} package for computations in prime characteristic commutative algebra. This includes Frobenius powers and roots, $p^{-e}$-linear and $p^{e}$-linear maps, singularities defined in terms of these maps, different types of test ideals and modules, and ideals compatible with a given $p^{-e}$-linear map.
This article describes the \emph{Macaulay2} package \emph{FrobeniusThresholds}, designed to estimate and calculate $F$-pure thresholds, more general $F$-thresholds, and related numerical invariants arising in the study of singularities in prime characteristic commutative algebra.
We study $F$-signature under proper birational morphisms $π: Y \to X$, showing that $F$-signature strictly increases for small morphisms or if $ K_Y \geq π^*K_X$. In certain cases, we can even show that the $F$-signature of $Y$ is at least twice as that of $X$. We also provide examples of $F$-signature dropping and Hilbert-Kunz multiplicity increasing under birational maps without these hypotheses.
This note describes a package for computing seminormalization of rings within Macaulay2.
This note describes a Macaulay2 package for handling divisors. Group operations for divisors are included. There are methods for converting divisors to reflexive or invertible sheaves. Additionally, there are methods for checking whether divisors are Cartier, $Q$-Cartier, simple normal crossings, generate base point free linear systems, or satisfy numerous other conditions.
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.
We study the behavior of test ideals and F-singularities in families. In particular, we obtain generic (and non-generic) restriction theorems for test ideals and non-F-pure ideals. Additionally, we study the global behavior of certain canonical linear systems (induced by Frobenius) associated to adjoint line bundles, in families. As a consequence, we obtain some positivity results for pushforwards of some adjoint line bundles and for certain subsheaves of these.