Related papers: Multiplier Ideals, Vanishing Theorem and Applicati…
This article describes recent applications of algebraic geometry to noncommutative algebra. These techniques have been particularly successful in describing graded algebras of small dimension.
Two correspondences have been provided that associate any linear code over a finite field with a binomial ideal. In this paper, algorithms for computing their Graver bases and universal Gr\"obner bases are given. To this end, a connection…
We show that all integrally closed ideals on log terminal surfaces are multiplier ideals by extending an existing proof for smooth surfaces.
We introduce the notion of strong test module and show that a large number of such modules appear in the tight closure theory of complete domains: the test ideal (this has already been known), the parameter test module, and the module of…
In this expositional paper, we discuss commutative algebra -- a study inspired by the properties of integers, rational numbers, and real numbers. In particular, we investigate rings and ideals, and their various properties. After, we…
We consider the notion of multiple gap as a finite set of ideals that cannot be separated. We study the different types of such objects that can be found in the Boolean algebra of subsets of the natural numbers modulo finite sets.
We use multiplication maps to give a characteristic-free approach to vanishing theorems on toric varieties. Our approach is very elementary but is enough powerful to prove vanishing theorems.
Symmetries and reductions of some algebraic equations are considered. Transformations that preserve the form of several algebraic equations, as well as transformations that reduce the degree of these equations, are described. Illustrative…
This paper is a review of concepts from graded commutative algebra with specific attention given to length and multiplicity. The author's motivation for this paper comes from the study of equivariant cohomology in algebraic topology where…
The main purpose of this paper is to investigate prime, primary, and maximal ideals of semirings. The localization and primary decomposition of ideals in semirings are also studied.
This is an attempt to present axioms for Euclidean geometry, aiming at the following goals: to work with geometric notions (thus not merely identify points with pairs of numbers, giving a special status to a particular coordinate system);…
We introduce the concept of a Gr\"obner nice pair of ideals in a polynomial ring and we present some applications.
Some notions of algebraic geometry can be defined for arbitrary varieties of algebras. This leads to universal algebraic geometry. The main idea of the presented theory is to consider interactions between algebra, logic and geometry in…
An exposition of the basic geometry of twistor integrals, intended for mathematicians.
Reduced ideals have been defined in the context of integer rings in quadratic number fields, and they are closely tied to the continued fraction algorithm. The notion of this type of ideal extends naturally to number fields of higher…
After sketching the basic theory of injective ideals of homogeneous polynomials, we characterize injective polynomial ideals by means of a domination property and applications of this characterization to some classical operator ideals and…
By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are…
By using Mather-Jacobian multiplier ideals, we first prove a formula on comparing Grauert-Riemenschneider canonical sheaf with canonical sheaf of a variety over an algebraically closed field of characteristic zero. Then we turn to study…
The purpose of this note is twofold. First, we survey results on the construction of large class groups of number fields by specialization of finite covers of curves. Then we give examples of applications of these techniques.
We provide formulas and algorithms for computing the excess numbers of certain ideals. The solution for monomial ideals is given by the mixed volumes of certain polytopes. These results enable us to design specific homotopies for numerical…