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Karl Schwede

Publications and source records attributed to Karl Schwede.

At least 19 recordsLinked to original sources
arXiv2026

Thresholds of singularities in characteristic zero

We study properties of characteristic zero variants of Frobenius thresholds $c^J(\mathfrak a)$ inspired by the work of Epstein-McDonald-R.G. and the second author. Suppose $R$ is an excellent domain of equal characteristic zero with a dualizing complex and $\mathfrak a, J \subseteq R$ are nonzero ideals with $\mathfrak a \subseteq \sqrt{J}$. By using a log resolution of singularities $Y$ of $(R, \mathfrak a)$ as well as the derived global sections of a line bundle on $Y$, we construct two analogs of the Frobenius threshold, the Hironaka-threshold and the Koszul-Hironaka threshold. We show that these two thresholds agree for parameter ideals and are always rational numbers. We prove that these thresholds also share many of the properties of the classical Frobenius threshold, especially for parameter ideals. If $R$ is regular, we show that the set of thresholds coincides with the set of possible multiplier-ideal-jumping numbers. Finally, we show that for parameter ideals in a Kawamata log terminal ring, our thresholds coincide with the limit of the Frobenius thresholds of the mod-$p$-reductions as $p$ goes to infinity. There is even a version of this without that Kawamata log terminal hypothesis.

math.AC↗
arXiv2026

$F$-injectivity does not deform

We show that there exists an $F$-finite four-dimensional local domain $(R,\mathfrak{m})$ of characteristic two which is not $F$-injective but which admits a nonzerodivisor $f\in \mathfrak{m}$ such that $R/fR$ is $F$-injective.

math.AC↗
arXiv2026

Factoring maps to big Cohen-Macaulay algebras through blowups

We show that for a reduced universally catenary equidimensional Noetherian local ring $R$, any sufficiently functorial big Cohen-Macaulay algebra assignment $R \mapsto B_R$, and any proper birational map $Y \to \mathrm{Spec} R$, there exists a factorization $R \to {\bf R}\Gamma(Y, \mathcal{O}_Y) \to B_R$ in the derived category of $R$-modules.

math.AC↗
arXiv2026

$F$-finite schemes have a dualizing complex

In this paper we show that any Noetherian $F$-finite scheme has a dualizing complex $\omega^{\bullet}_{X}$ with the property that for all finite type maps $f \colon X \to Y$ between $F$-finite Noetherian schemes there is a canonical isomorphism $\omega^{\bullet}_{X} \xrightarrow{\cong} f^!\omega^{\bullet}_{Y}$ in $D^b_{coh}(X)$. This, in particular, applies to the Frobenius morphism $F \colon X \to X$ so that we obtain a canonical isomorphism $\omega^{\bullet}_{X} \xrightarrow{\cong} F^!\omega^{\bullet}_{X}$. To prove this, we rely on a result of Gabber that every Noetherian $F$-finite ring is a quotient of a regular ring, from which it follows that every $F$-finite Noetherian scheme has a (potentially non-canonical) dualizing complex. To make this canonical, we identify the dualizing complex of any $F$-finite Noetherian scheme as a unit of an alternate symmetric monoidal structure on $D^b_{coh}(X)$ we call the $!$-tensor product. We also sketch an alternate approach to finding this canonical dualizing complex following the more classical approach to Grothendieck duality.

math.AG↗
arXiv2026

Closure operations induced via resolutions of singularities in characteristic zero

Using the fact that the structure sheaf of a resolution of singularities, or regular alteration, pushes forward to a Cohen-Macaulay complex in equal characteristic zero with a differential graded algebra structure, we introduce a tight-closure-like operation on ideals in equal characteristic zero using the Koszul complex, which we call KH (Koszul-Hironaka). We prove it satisfies various strong colon capturing properties, a substantial case of the Briançon-Skoda theorem, and it behaves well under finite extensions. It detects rational singularities and is tighter than tight closure in equal characteristic zero. Furthermore, its formation commutes with localization and it can be computed effectively. On the other hand, the product of the KH closures of ideals is not always contained in the KH of the product, as one might expect. We also explore a related closure operation (canonical alteration closure), induced by canonical modules of regular alterations, which detects KLT-type singularities in equal characteristic zero and which is closely related to tight closure in characteristic $p > 0$. For parameter ideals we show both these closure operations coincide and reduce modulo $p \gg 0$ to tight closure. Finally, we explore an intermediate operation (Hironaka pre-closure) which which satisfies numerous desired properties, but for which we have not been able to prove idempotence.

math.AC↗
arXiv2025

Variants on Frobenius Intersection Flatness and Applications to Tate Algebras

The theory of singularities defined by Frobenius has been extensively developed for $F$-finite rings and for rings that are essentially of finite type over excellent local rings. However, important classes of non-local excellent rings, such as Tate algebras and their quotients (affinoid algebras) do not fit into either setting. We investigate here a framework for moving beyond the $F$-finite setting, developing the theory of three related classes of regular rings defined by properties of Frobenius. In increasing order of strength, these are Frobenius Ohm-Rush (FOR), Frobenius intersection flat, and Frobenius Ohm-Rush trace (FORT). We show that Tate algebras are Frobenius intersection flat, from which it follows that reduced affinoid algebras have test elements using a result of Sharp. We also deduce new cases of the openness of the $F$-pure locus.

math.AC↗
arXiv2025

The Brian\c{c}on-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings

Suppose $J = (f_1, \dots, f_n)$ is an $n$-generated ideal in any ring $R$. We prove a general Brian\c{c}on-Skoda-type containment relating the integral closure $\overline{J^{n+k-1}}$ with ordinary powers $J^k$. We prove that our result implies the full Brian\c{c}on-Skoda containment $\overline{J^{n+k-1}} \subseteq J^k$ for pseudo-rational singularities (for instance regular rings), and even for the weaker condition of birational derived splinters. Our methods also yield the containment $\overline{J^{n+k}} \subseteq J^k$ for Du Bois singularities and even for a characteristic-free generalization. Our Brian\c{c}on-Skoda-type theorem also implies well-known closure-based Brian\c{c}on-Skoda results $\overline{J^{n+k-1}} \subseteq (J^k)^{\mathrm{cl}}$ where, for instance, $\mathrm{cl}$ is tight or plus closure in characteristic $p > 0$, or $\mathrm{ep}$ closure or extension and contraction from $\widehat{R^+}$ in mixed characteristic. Our proof relies on a study of the tensor product of the derived image of the structure sheaf of a partially normalized blowup of $J$ with the Buchsbaum-Eisenbud complex (equivalently the Eagon-Northcott complex) associated to $(f_1,\dots,f_n)^k$. As an application of our results and methods above, we prove the uniform Artin-Rees theorem and the uniform Brian\c{c}on-Skoda theorem for quasi-excellent, respectively quasi-excellent reduced, rings of finite dimension, answering conjectures of Huneke.

math.AC↗
arXiv2025

Bounds on the plus-pure thresholds of some hypersurfaces in (ramified) regular rings

We study the plus-pure threshold (ppt) of hypersurfaces in mixed characteristic. We show that the ppt limits to the $F$-pure threshold (fpt) as we ramify the base DVR. Additionally, we show that analogs of some positive characteristic extremal singularities cannot attain the same `extremal' ppt values in the unramified setting. We also study equations which have controlled ramification when we adjoin their $p$-th roots as well as equations which admit $p$-th roots modulo $p^2$ (or modulo other values), bounding their ppts. In particular, given a complete unramified regular local ring of mixed characteristic $p>0$, $f^p + p^2 g$ does not define a perfectoid pure singularity for any $f$ and $g$. Finally, we compute bounds on the ppt of hypersurfaces related to elliptic curves. This gives examples where the ppt is neither the corresponding fpt in characteristic $p > 0$ nor the lct in characteristic zero. This also provides examples where $p$ times the ppt is not a jumping number, in stark contrast with the characteristic $p > 0$ picture.

math.AC↗
arXiv2025

Plus-pure thresholds of some cusp-like singularities in mixed characteristic

Log-canonical and $F$-pure thresholds of pairs in equal characteristic admit an analog in the recent theory of singularities in mixed characteristic, which is known as the plus-pure threshold. In this paper we study plus-pure thresholds for singularities of the form $p^a + x^b \in {\bf Z}_p [[ x ]]$, showing that in a number of cases this plus-pure threshold agrees with the $F$-pure threshold of the singularity $t^a + x^b \in {\bf F}_p [[ t, x ]]$. We also discuss a few other sporadic examples.

math.AG↗
arXiv2025

Test ideals in mixed characteristic: a unified theory up to perturbation

Let $X$ be an integral scheme of finite type over a complete DVR of mixed characteristic. We provide a definition of a test ideal which agrees with the multiplier ideal after inverting $p$, is computed from a sufficiently large alteration, agrees with previous mixed characteristic BCM test ideals after completing at any point of residue characteristic $p$ (up to small perturbation), and which satisfies the full suite of expected properties of a multiplier or test ideal. This object is obtained via the $p$-adic Riemann-Hilbert functor.

math.AG↗
arXiv2025

Perfectoid signature, perfectoid Hilbert-Kunz multiplicity, and an application to local fundamental groups

We define a (perfectoid) mixed characteristic version of $F$-signature and Hilbert-Kunz multiplicity by utilizing the perfectoidization functor of Bhatt-Scholze and Faltings' normalized length (also developed in the work of Gabber-Ramero). We show that these definitions coincide with the classical theory in equal characteristic $p > 0$. We prove that a ring is regular if and only if either its perfectoid signature or perfectoid Hilbert-Kunz multiplicity is 1 and we show that perfectoid Hilbert-Kunz multiplicity characterizes BCM closure and extended plus closure of $m$-primary ideals. We demonstrate that perfectoid signature detects BCM-regularity and transforms similarly to $F$-signature or normalized volume under quasi-étale maps. As a consequence, we prove that BCM-regular rings have finite local étale fundamental group and also finite torsion part of their divisor class groups. Finally, we also define a mixed characteristic version of relative rational signature, and show it characterizes BCM-rational singularities.

math.AC↗
arXiv2025

Finite generation of split F-regular monoid algebras

Let $S$ be a submonoid of a free Abelian group of finite rank. We show that if $k$ is a field of prime characteristic such that the monoid $k$-algebra $k[S]$ is split $F$-regular, then $k[S]$ is a finitely generated $k$-algebra, or equivalently, that $S$ is a finitely generated monoid. Split $F$-regular rings are possibly non-Noetherian or non-$F$-finite rings that satisfy the defining property of strongly $F$-regular rings from the theories of tight closure and $F$-singularities. Our finite generation result provides evidence in favor of the conjecture that split $F$-regular rings in function fields over $k$ have to be Noetherian. The key tool is Diophantine approximation from convex geometry.

math.AC↗
arXiv2025

BCM-thresholds of non-principal ideals

Generalizing previous work of the first author, we introduce and study a characteristic free analog of the $F$-threshold for non-principal ideals, BCM-thresholds. We show that this coincides with the classical $F$-threshold for weakly $F$-regular rings and that the set of BCM-thresholds coincides with the set of BCM-jumping numbers in a complete local regular ring. We obtain results on $F$-thresholds of parameter ideals analogous to results of Huneke-\mustata-Takagi-Watanabe as well as a mixed characteristic version of one of their results on multiplicity. Instead of taking ordinary powers of an ideal, our definition uses fractional integral closure in an absolute integral closure of our ambient ring.

math.AC↗
arXiv2024

Perfectoid pure singularities

Fix a prime number $p$. Inspired by the notion of $F$-pure or $F$-split singularities, we study the condition that a Noetherian ring with $p$ in its Jacobson radical is pure inside some perfectoid (classical) ring, a condition we call perfectoid pure. We also study a related a priori weaker condition which asks that $R$ is pure in its absolute perfectoidization, a condition we call lim-perfectoid pure. We show that both these notions coincide when $R$ is LCI. Mixed characteristic analogs of $F$-injective and Du Bois singularities are also explored. We study these notions of singularity, proving that they are weakly normal and that they are Du Bois after inverting $p$. We also explore the behavior of \claperfdpure singularities under finite covers and their relation to log canonical singularities. Finally, we prove an inversion of adjunction result in the LCI setting, and use it to prove that many common examples are perfectoid pure.

math.AG↗
arXiv2024

The Briançon-Skoda Theorem via weak functoriality of big Cohen-Macaulay algebras

We prove that, given a sufficiently functorial assignment from rings to big Cohen-Macaulay algebras $R \mapsto B$, that the associated big Cohen-Macaulay closure operation on ideals $I \mapsto I B \cap R$ necessarily satisfies the Briançon-Skoda type property. The proof combines arguments of Lipman-Teissier, Hochster, Ma, and Hochster-Huneke. Specializing to mixed characteristic, and utilizing a result of Bhatt on absolute integral closures, this recovers a slight strengthening of a result of Heitmann.

math.AC↗
arXiv2022

Global generation of test ideals in mixed characteristic and applications

Suppose that $X$ is an integral scheme (quasi-)projective over a complete local ring of mixed characteristic. Using ideas of Takamatsu-Yoshikawa and Bhatt-Ma-et. al, we define a notion of a $+$-test ideal on $X$, including for divisors and linear series. We obtain global generation results in this setting that generalize the well known global generation results obtained via multiplier ideal sheaf techniques in characteristic $0$ and via test ideals in characteristic $p>0$. We also obtain applications to the order of vanishing of linear series and to the diminished base locus in mixed characteristic similar to results of Ein-Lazarsfeld-Mustata-Nakamaye-Popa, Nakayama, and Mustata in the equal characteristic case.

math.AG↗
arXiv2022

Compatible ideals in Gorenstein rings

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 $S\to R$.

math.AC↗