Related papers: On the Equality of Test Ideals

We prove an assortment of results on (commutative and unital) NIP rings, especially $\mathbb{F}_p$-algebras. Let $R$ be a NIP ring. Then every prime ideal or radical ideal of $R$ is externally definable, and every localization $S^{-1}R$ is…

We prove that if M is a finitely-generated module of dimension d with finite local cohomologies over a Noetherian local ring, and if the ith local cohomology module of M is zero unless i = d, i = 0, and i = r for some r strictly between 0…

This article focuses on the existence of asymptotic colengths for families of $\fm_{R}$-primary ideals in a Noetherian local ring $(R,\fm)$. In any characteristic, we generalize graded families to weakly graded families of ideals, and in…

Let $(R,\frak m)$ be a Noetherian local ring of prime characteristic $p>0$, and $t$ an integer such that $H_{\frak m}^j(R)/0^F_{H^j_{\frak m}(R)}$ has finite length for all $j<t$. The aim of this paper is to show that there exists an…

The aim of this work is to study duality of fractional ideals with respect to a fixed ideal and to investigate the relationship between value sets of pairs of dual ideals in admissible rings, a class of rings that contains the local rings…

We explore the singularity classes $F$-nilpotent, weakly $F$-nilpotent, and generalized weakly $F$-nilpotent under faithfully flat local ring maps. As an application, we show that the loci of primes in a Noetherian ring of prime…

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d\geq 2$. We prove that if $e(\widehat{R}_{red})>1$, then the classical Lech's inequality can be improved uniformly for all $\mathfrak{m}$-primary ideals, that is, there exists…

Let $(R,\m,k)$ be a local (Noetherian) ring of positive prime characteristic $p$ and dimension $d$. Let $G_\dt$ be a minimal resolution of the residue field $k$, and for each $i\ge 0$, let $\gothic t_i(R) = \lim_{e\to \8}…

We show that there exists a complete local Noetherian normal domain of prime characteristic whose perfection is a non-coherent GCD domain, answering a question of Patankar in the negative concerning characterizations of $F$-coherent rings.…

Let $\bar{I}$ denote the integral closure of an ideal in a Noetherian ring $R$. The main result of this paper asserts that $R$ is locally quasi-unmixed if and only if, the topologies defined by $\overline{I^n}$ and $I^{\langle n\rangle}$,…

We show that the F-signature of an F-finite local ring R of characteristic p >0 exists when R is either the localization of an $\mathbf{N}$-graded ring at its irrelevant ideal or $\mathbf{Q}$-Gorenstein on its punctured spectrum. This…

Let $T$ be a complete local (Noetherian) ring. For each $i \in \mathbb{N}$, let $C_i$ be a nonempty countable set of nonmaximal pairwise incomparable prime ideals of $T$, and suppose that if $i \neq j$, then either $C_i = C_j$ or no element…

In this paper, we say a ring $R$ is Nil$_{\ast}$-Noetherian provided that any nil ideal is finitely generated. First, we show that the Hilbert basis theorem holds for Nil$_{\ast}$-Noetherian rings, that is, $R$ is Nil$_{\ast}$-Noetherian if…

A ring $R$ is called left strictly $(<\aleph_{\alpha})$-noetherian if $\aleph_{\alpha}$ is the minimum cardinal such that every ideal of $R$ is $(<\aleph_{\alpha})$-generated. In this note, we show that for every singular (resp., regular)…

Let $(R,\frak m)$ be an excellent generalized Cohen-Macaulay local ring of dimension $d$ that is $F$-injective on the punctured spectrum. Let $\frak q$ be a standard parameter ideal of $R$. The aim of the paper is to prove that…

A commutative noetherian local ring $(R,\mathfrak{m})$ is Gorenstein if and only if every parameter ideal of $R$ is irreducible. Although irreducible parameter ideals may exist in non-Gorenstein rings, Marley, Rogers, and Sakurai show there…

We establish a series of results showing that the Jacobian ideal is contained in the test ideal. We first prove a new result in characteristic $p$ for complete rings over a field $K$. Then we prove some results showing that Jacobian ideals…

In this paper, we consider whether parameter test ideals, conductors, $F$-ideals, and trace ideals are contained in an ideal whose quotient ring has finite phantom projective dimension (for example, ideals generated by a system of…

In this note, we consider a corollary of the ACC conjecture for F-pure thresholds. Specifically, we show that the F-pure threshold (and more generally, the test ideals) associated to a polynomial with an isolated singularity are locally…

For an ideal $I$ of a Noetherian local ring $(R,\fm,k)$ we show that $\bt_1^R(I)-\bt_0^R(I)\geq -1$. It is demonstrated that some residual intersections of an ideal $I$ for which $\bt_1^R(I)-\bt_0^R(I)= -1\;\text{or}\;0$ are perfect. Some…