Related papers: Perfectoid pure singularities

We consider in-depth and characterize in certain aspects the class of so-called {\it strongly NUS-nil clean rings}, that are those rings whose non-units are {\it square nil-clean} in the sense that they are a sum of a nilpotent and a…

{Generalizing the notion of nil cleanness from \cite{D13}, in parallel to \cite{DM14}, we define the concept of {\it weak nil cleanness} for an arbitrary ring. Its comprehensive study in different ways is provided as well. A decomposition…

In this paper, new algebraic and topological results on purely-prime ideals of a commutative ring (pure spectrum) are obtained. Especially, Grothendieck type theorem is obtained which states that there is a canonical correspondence between…

We prove the following theorem characterizing Du Bois singularities. Suppose that $Y$ is smooth and that $X$ is a reduced closed subscheme. Let $\pi : \tld Y \to Y$ be a log resolution of $X$ in $Y$ that is an isomorphism outside of $X$. If…

In this paper we study singularities defined by the action of Frobenius in characteristic $p > 0$. We prove results analogous to inversion of adjunction along a center of log canonicity. For example, we show that if $X$ is a Gorenstein…

We compute a number of invariants of singularities defined via the Frobenius morphism for seminormal affine toric varieties over fields of characteristic p > 0. Our main technical tool is a combinatorial description of the potential…

We prove the following;Theorem:Let R be a prime noetherian ring with k.dimR = n, n a finite non-negative integer. We refer the reader to the definitions (1.1) of this paper.For a fixed non-negative integer m, m<n let Xm be the full set of…

Let $R$ be a commutative Noetherian ring of prime characteristic $p$. The main goal of this paper is to study in some detail when \[ \overline{W^R}:=\{\mathfrak{p}\in\operatorname{Spec} (R):\ \mathcal{F}^{E_{\mathfrak{p}}}\text{ is finitely…

If $(R,\mathfrak{m})$ is a complete local ring of mixed characteristic $(0,p)$ and $R/pR$ is an $F$-pure Gorenstein domain, we find a sufficient condition for $R$ to be perfectoid pure. This condition is related to the Cohen-Macaulayness of…

We compare a couple of notions of differential form on singular complex algebraic varieties, and relate them to the outermost associated graded spaces of the Hodge filtration of ordinary and intersection cohomology. In particular, we…

Consider a pair $(R, \ba^t)$ where $R$ is a ring of positive characteristic, $\ba$ is an ideal such that $a \cap $R^{\circ} \neq \emptyset$, and $t > 0$ is a real number. In this situation we have the ideal $\tau_R(\ba^t)$, the generalized…

An important classical result in ZFC asserts that every infinite cardinal number is idempotent. Using this fact, we obtain several algebraic results in this article. The first result asserts that an infinite Abelian group has a proper…

We discuss Matijevic-Roberts type theorem on strong $F$-regularity, $F$-purity, and Cohen-Macaulay $F$-injective (CMFI for short) property. Related to this problem, we also discuss the base change problem and the openness of loci of these…

We introduce new notions of $k$-Du Bois and $k$-rational singularities, extending the previous definitions in the case of local complete intersections (lci), to include natural examples outside of this setting. We study the stability of…

An element in a ring $R$ is called uniquely weakly nil-clean if every element in $R$ can be uniquely written as a sum or a difference of a nilpotent and an idempotent in the sense of very idempotents. The structure of the ring in which…

In this paper, we study the classes of rings in which every proper (regular) ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals. More precisely, we investigate the stability of these properties…

Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). A theorem of Lipman implies that I has a unique factorization as a *-product of special *-simple complete ideals with possibly negative exponents for…

Let $p$ be a prime, $P$ a finite p-group and $\cal F$ a Frobenius $P$-category. In "Existence, uniqueness and functoriality of the perfect locality over a Frobenius $P$-category", Algebra Colloquium, 23(2016) 541-622, we also claimed the…

Using the Frobenius map, we introduce a new invariant for a pair $(R,\a)$ of a ring $R$ and an ideal $\a \subset R$, which we call the F-pure threshold $\mathrm{c}(\a)$ of $\a$, and study its properties. We see that the F-pure threshold…

Students studying the Lasker-Noether theorem on primary decomposition of ideals may want to see an example of an ideal (necessarily in a non-Noetherian ring) which does not have a primary decomposition. The most well-known counterexample is…