Related papers: Characteristic-free test ideals
We introduce a certain class of so-called perfectoid rings and spaces, which give a natural framework for Faltings' almost purity theorem, and for which there is a natural tilting operation which exchanges characteristic 0 and…
Let $(A,\mathfrak{m})$ be an excellent normal domain of dimension two containing a field $k \cong A/\mathfrak{m}$. An $\mathfrak{m}$-primary ideal $I$ to be a $p_g$-ideal if the Rees algebra $A[It]$ is a Cohen-Macaulay normal domain. If $k$…
The present paper introduces and studies some new types of rings and ideals such as completely nilary rings ( resp. completely nilary ideals ), weakly nilary ideals. Some properties of each are obtained and some characterizations of each…
A new notion of typicality for arbitrary probability measures on standard Borel spaces is proposed, which encompasses the classical notions of weak and strong typicality as special cases. Useful lemmas about strong typical sets, including…
Dedicated to the memory of Professor Tze Leung Lai, this paper introduces three multi-hypothesis sequential tests. These tests are derived from one-sided versions of the sequential probability ratio test and its modifications. They are…
This paper represents the main portion of the Ph.D. Thesis of the author, and is the first of the series of four papers, which is a joint work with K. Matsuki as a whole. We present a program toward constructing an algorithm for resolution…
Let R be a local Noetherian domain of positive characteristic. A theorem of Hochster and Huneke (1992) states that if R is excellent, then the absolute integral closure of R is a big Cohen-Macaulay algebra. We prove that if R is the…
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…
Over a complete Noetherian local domain of mixed characteristic with perfect residue field, we construct a perfectoid ring which is similar to an explicit representation of a perfect closure in positive characteristic. Then we demonstrate…
In this paper, we introduce and study families of squarefree monomial ideals called clique ideals and independence ideals that can be associated to a finite graph. A family of clique ideals with linear resolutions has been characterized.…
We prove positive characteristic analogues of certain measure rigidity theorems in characteristic zero. More specifically we give a classification result for positive entropy measures on quotients of $\operatorname{SL}_d$ and a…
For a Noetherian local domain $R$ let $R^+$ be the absolute integral closure of $R$ and let $R_{\infty}$ be the perfect closure of $R$, when $R$ has prime characteristic. In this paper we investigate the projective dimension of residue…
The existence of a maximal ideal in a general nontrivial commutative ring is tied together with the axiom of choice. Following Berardi, Valentini and thus Krivine but using the relative interpretation of negation (that is, as "implies 0 =…
Let $R$ be a commutative, local, Noetherian ring. In a past article, the first author developed a theory of $R$-algebras, termed seeds, that can be mapped to balanced big Cohen-Macaulay $R$-algebras. In prime characteristic $p$, seeds can…
We study the relationship between the sigma-ideal generated by closed measure zero sets and the ideals of null and meager sets. We show that the additivity of the ideal of closed measure zero sets is not bigger than covering for category.…
A central question in mathematics and computer science is the question of determining whether a given ideal $I$ is prime, which geometrically corresponds to the zero set of $I$, denoted $Z(I)$, being irreducible. The case of principal…
To a simplicial complex, we associate a square-free monomial ideal in the polynomial ring generated by its vertex set over a field. We study algebraic properties of this ideal via combinatorial properties of the simplicial complex. By…
Motivated by the concept of clean ideals, we introduce the notion of weakly clean ideals. We define an ideal $I$ of a ring $R$ to be weakly clean ideal if for any $x\in I$, $x=u+e$ or $x=u-e$, where $u$ is a unit in $R$ and $e$ is an…
It was pointed out in my last paper that there are rings whose real closure * are not unique. In [4] we also discussed some example of rings by which there is a unique real closure * (mainly the real closed rings). Now we want to determine…
Given a reduced, local ring $R$ and an ideal $\mathfrak{a}$ of positive height, we give a decomposition of the test module, $\tau(\omega_T, t^{-\lambda})$, of the extended Rees algebra, $T =R[\mathfrak{a} t, t^{-1}]$. In particular, the…