We study general properties of the perfectoidization of finite algebras over a perfectoid ring, which helps to understand some precise and explicit descriptions. For example, we prove that if $A=R[t]/(m(t))$ where $m(t)$ is monic, $R$ is perfectoid and the discriminant $d$ of $m(t)$ is a non-zero divisor of $R$ satisfying a bounded torsion condition, then $dA_{\mathrm{pfd}}\subset A$. We also prove a density criterion reducing the construction of the perfectoidization to adjoining suitable $p$-power roots modulo $p$. In the second part of the paper, we compute perfectoidizations in several families of examples, including Kummer-type extensions and split finite algebras.

In this paper, we study $S$-prime elements and $S$-primary elements within the framework of multiplicative lattices. Furthermore, we define and explore weakly $S$-prime elements and weakly $S$-primary elements, which generalize weakly prime elements and weakly primary elements in multiplicative lattices respectively. We show that the weakly $S$-prime ideals (weakly $S$-primary ideals) of a commutative ring $R$ with $1$ correspond precisely to the weakly $S_L$-prime elements (weakly $S$-primary elements) of the ideal lattice $Id(R)$ of $R$, where $S_L = \{(s) \mid s \in S\}$.

We prove a conjecture on the minimal number of generators of licci zero-dimensional ideals that are either monomial or have small Loewy colength.

A point of a scheme has an associated tangent cone, the spectrum of a standard graded algebra encoding the local singularity. Its homological complexity can be measured by its graded Betti table: a matrix that records a part of the structure of its graded, minimal free resolution over a polynomial ring. A natural question is whether the homological complexity of the tangent cones varies arbitrarily across a scheme. In this paper, we show that this is not the case for a quasi-excellent Noetherian scheme; over such schemes, only finitely many graded Betti tables can occur. More generally, we show that a coherent sheaf over a quasi-excellent Noetherian scheme admits finitely many graded Betti tables, and that the constancy loci for the graded Betti table are constructible. As an immediate consequence, regularity is bounded on a quasi-excellent Noetherian scheme.

Configuration polynomials generalize the Kirchhoff polynomial of a graph, as well as the Symanzik polynomials that appear in the denominators of Feynman integrands. The configuration hypersurfaces cut out by such polynomials are typically highly singular, which poses a challenge for the evaluation of Feynman integrals even in simplified settings. In this paper, we provide a two-step recipe for a resolution of singularities of any irreducible configuration hypersurface. We first consider the normalization of the Nash blow-up, which we identify with an incidence variety introduced by Bloch. This variety is typically still not smooth, but it is the closure of a smooth subvariety of a torus. The latter then a smooth, tropical compactification, using work of Tevelev. We construct explicitly such a compactification and a morphism to the normalized Nash blow-up for every configuration, described in terms of bipermutohedral matroid combinatorics introduced by Ardila, Denham and Huh. Along the way, we find that the normalized Nash blow-up of the configuration hypersurface has strongly $F$-regular singularities in positive characteristic. We deduce this by certifying $F$-rationality of its biprojective cone, and infer from it that the normalized Nash blow-up has rational singularities over the complex numbers.

Let $R$ be a noetherian algebra over a Cohen--Macaulay ring $S$ admitting a canonical module $\omega$, and assume that $R$ is maximal Cohen--Macaulay over $S$. We prove that the category of finitely generated Gorenstein projective $R$-modules coincides with the left $\mathrm Ext$-orthogonal class of the thick subcategory generated by $R$ and ${\mathrm Hom}_S(R,\omega)$. As an application, finitely generated Gorenstein projective $R$-modules form the left half of a hereditary cotorsion pair. In the case of Cohen--Macaulay local rings, this yields an affirmative answer to a question of R. Takahashi. We further characterize when $R$ is left weakly Gorenstein. Finally, we prove that a Cohen--Macaulay local ring is Gorenstein if and only if the right $\mathrm Ext$-orthogonal class of finitely generated Gorenstein projective modules coincides with the category of finitely generated modules of finite projective dimension.

Suppose $J = (f_1, \dots, f_n)$ is an $n$-generated ideal in any ring $R$. We prove a general Briançon-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 Briançon-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 Briançon-Skoda-type theorem also implies well-known closure-based Briançon-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 Briançon-Skoda theorem for quasi-excellent, respectively quasi-excellent reduced, rings of finite dimension, answering conjectures of Huneke.

Inspired by Terao's freeness conjecture, we examine Ziegler pairs, which are pairs of hyperplane arrangements that share the same underlying matroid but have different modules of logarithmic derivations. In this paper, we present a general construction that yields the first known families of Ziegler pairs in arbitrary dimension and size, starting from examples in the complex projective plane.

Under mild hypotheses, given a scheme $U$ and an open subset $V$ whose complement has codimension at least two, the pushforward of a torsion-free coherent sheaf on $V$ is coherent on $U$, and in particular is finite. We prove an analog of this finiteness assertion in the context of formal schemes over a complete discrete valuation ring, but show that coherence does not always hold. We then relate this to the problem of gluing formal functions, where the patches do not cover the entire scheme.

In this paper, we introduce the notions of tight closure of ideals on Witt rings and quasi-tightly closedness of system of parameters. By using the notions, we obtain a characterization of quasi-$F$-rationality. Furthermore, we study the relationship between the closure operator and integrally closure.