scholarly journals Computing the invariants of intersection algebras of principal monomial ideals

2019 ◽  
Vol 29 (02) ◽  
pp. 309-332 ◽  
Author(s):  
Florian Enescu ◽  
Sandra Spiroff
Keyword(s):  
Polynomial Ring ◽  
Noetherian Ring ◽  
Class Group ◽  
Monomial Ideals ◽  
Semigroup Ring ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Commutative Noetherian Ring ◽  
New Class ◽  
Explicit Formulae

We continue the study of intersection algebras [Formula: see text] of two ideals [Formula: see text] in a commutative Noetherian ring [Formula: see text]. In particular, we exploit the semigroup ring and toric structures in order to calculate various invariants of the intersection algebra when [Formula: see text] is a polynomial ring over a field and [Formula: see text] are principal monomial ideals. Specifically, we calculate the [Formula: see text]-signature, divisor class group, and Hilbert–Samuel and Hilbert–Kunz multiplicities, sometimes restricting to certain cases in order to obtain explicit formulæ. This provides a new class of rings where formulæ for the [Formula: see text]-signature and Hilbert–Kunz multiplicity, dependent on families of parameters, are provided.

scholarly journals Flat ring epimorphisms and universal localizations of commutative rings

2020 ◽  
Vol 71 (4) ◽  
pp. 1489-1520
Author(s):  
Lidia Angeleri Hügel ◽  
Frederik Marks ◽  
Jan Št’ovíček ◽  
Ryo Takahashi ◽  
Jorge Vitória
Keyword(s):  
Essential Role ◽  
Class Group ◽  
Commutative Rings ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Commutative Noetherian Ring ◽  
Different Types ◽  
Flat Ring ◽  
Dimension One ◽  
Universal Localization

Abstract We study different types of localizations of a commutative noetherian ring. More precisely, we provide criteria to decide: (a) if a given flat ring epimorphism is a universal localization in the sense of Cohn and Schofield; and (b) when such universal localizations are classical rings of fractions. In order to find such criteria, we use the theory of support and we analyse the specialization closed subset associated to a flat ring epimorphism. In case the underlying ring is locally factorial or of Krull dimension one, we show that all flat ring epimorphisms are universal localizations. Moreover, it turns out that an answer to the question of when universal localizations are classical depends on the structure of the Picard group. We furthermore discuss the case of normal rings, for which the divisor class group plays an essential role to decide if a given flat ring epimorphism is a universal localization. Finally, we explore several (counter)examples which highlight the necessity of our assumptions.

scholarly journals Some factorization properties of Krull domains with infinite cyclic divisor class group

1994 ◽  
Vol 96 (2) ◽  
pp. 97-112 ◽  
Author(s):  
David F. Anderson ◽  
Scott T. Chapman ◽  
William W. Smith
Keyword(s):  
Class Group ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Krull Domains

scholarly journals The divisor class group of a Quot scheme

2010 ◽  
Vol 3 (0) ◽  
pp. 1-14
Author(s):  
Rafael Hernández ◽  
Daniel Ortega
Keyword(s):  
Class Group ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Quot Scheme

scholarly journals ON FACTORIZATION IN BLOCK MONOIDS FORMED BY $\{\bar{1},\bar{a}\}$ IN $\mathbb{Z}_{n}$

2003 ◽  
Vol 46 (2) ◽  
pp. 257-267 ◽  
Author(s):  
Scott T. Chapman ◽  
William W. Smith
Keyword(s):  
Class Group ◽  
Subject Classification ◽  
Division Algorithm ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Irreducible Elements ◽  
Complete Set ◽  
Cross Number ◽  
Block Monoids ◽  
Block Monoid

AbstractWe consider the factorization properties of block monoids on $\mathbb{Z}_n$ determined by subsets of the form $S_a=\{\bar{1},\bar{a}\}$. We denote such a block monoid by $\mathcal{B}_a(n)$. In §2, we provide a method based on the division algorithm for determining the irreducible elements of $\mathcal{B}_a(n)$. Section 3 offers a method to determine the elasticity of $\mathcal{B}_a(n)$ based solely on the cross number. Section 4 applies the results of §3 to investigate the complete set of elasticities of Krull monoids with divisor class group $\mathbb{Z}_n$.AMS 2000 Mathematics subject classification: Primary 20M14; 20D60; 13F05

scholarly journals A Theorem About Maximal Cohen–Macaulay Modules

2020 ◽  
Author(s):  
Thomas Polstra
Keyword(s):  
Natural Number ◽  
Regular Ring ◽  
Class Group ◽  
Torsion Subgroup ◽  
Finitely Generated ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Frobenius Endomorphism

Abstract It is shown that for any local strongly $F$-regular ring there exists natural number $e_0$ so that if $M$ is any finitely generated maximal Cohen–Macaulay module, then the pushforward of $M$ under the $e_0$th iterate of the Frobenius endomorphism contains a free summand. Consequently, the torsion subgroup of the divisor class group of a local strongly $F$-regular ring is finite.

scholarly journals The Noether–Lefschetz theorem for the divisor class group

2009 ◽  
Vol 322 (9) ◽  
pp. 3373-3391 ◽  
Author(s):  
G.V. Ravindra ◽  
V. Srinivas
Keyword(s):  
Class Group ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Lefschetz Theorem
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