Birings and plethories of integer-valued polynomials

Jesse Elliott

Displaying similar documents to “Birings and plethories of integer-valued polynomials”

Generalized E-algebras via λ-calculus I

Rüdiger Göbel, Saharon Shelah (2006)

Fundamenta Mathematicae

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An R-algebra A is called an E(R)-algebra if the canonical homomorphism from A to the endomorphism algebra $E n d_{R} A$ of the R-module $_{R} A$ , taking any a ∈ A to the right multiplication $a_{r} \in E n d_{R} A$ by a, is an isomorphism of algebras. In this case $_{R} A$ is called an E(R)-module. There is a proper class of examples constructed in [4]. E(R)-algebras arise naturally in various topics of algebra. So it is not surprising that they were investigated thoroughly in the last decades; see [3, 5, 7, 8, 10, 13, 14, 15, 18,...

The categories of presheaves containing any category of algebras

V. Trnková, J. Reiterman

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ContentsIntroduction.................................................................................................................................................. 5I. Preliminaries........................................................................................................................................... 6II. Main theorem.......................................................................................................................................... 8III. The...

Derived endo-discrete artin algebras

Raymundo Bautista (2006)

Colloquium Mathematicae

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Let Λ be an artin algebra. We prove that for each sequence ${(h_{i})}_{i \in ℤ}$ of non-negative integers there are only a finite number of isomorphism classes of indecomposables $X \in^{b} (Λ)$ , the bounded derived category of Λ, with $l e n g t h_{E (X)} H^{i} (X) = h_{i}$ for all i ∈ ℤ and E(X) the endomorphism ring of X in $^{b} (Λ)$ if and only if $^{b} (M o d Λ)$ , the bounded derived category of the category $M o d Λ$ of all left Λ-modules, has no generic objects in the sense of [4].

$n$ -angulated quotient categories induced by mutation pairs

Zengqiang Lin (2015)

Czechoslovak Mathematical Journal

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Geiss, Keller and Oppermann (2013) introduced the notion of $n$ -angulated category, which is a “higher dimensional” analogue of triangulated category, and showed that certain $(n - 2)$ -cluster tilting subcategories of triangulated categories give rise to $n$ -angulated categories. We define mutation pairs in $n$ -angulated categories and prove that given such a mutation pair, the corresponding quotient category carries a natural $n$ -angulated structure. This result generalizes a theorem of Iyama-Yoshino...

Schur-Finite Motives and Trace Identities

Alessio Del Padrone, Carlo Mazza (2009)

Bollettino dell'Unione Matematica Italiana

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We provide a sufficient condition that ensures the nilpotency of endomorphisms universally of trace zero of Schur-finite objects in a category of homological type, i.e., a $\mathbb{Q}$ -linear $\otimes$ -category with a tensor functor to super vector spaces. We present some applications in the category of motives, where our result generalizes previous results about finite-dimensional objects, in particular by Kimura. We also present some facts which suggest that this might be the best generalization possible...

$A F$ -algebras and topology of mapping tori

Igor Nikolaev (2015)

Czechoslovak Mathematical Journal

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The paper studies applications of $C^{*}$ -algebras in geometric topology. Namely, a covariant functor from the category of mapping tori to a category of $A F$ -algebras is constructed; the functor takes continuous maps between such manifolds to stable homomorphisms between the corresponding $A F$ -algebras. We use this functor to develop an obstruction theory for the torus bundles of dimension $2$ , $3$ and $4$ . In conclusion, we consider two numerical examples illustrating our main results.

A Generalization of Baer's Lemma

Molly Dunkum (2009)

Czechoslovak Mathematical Journal

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There is a classical result known as Baer’s Lemma that states that an $R$ -module $E$ is injective if it is injective for $R$ . This means that if a map from a submodule of $R$ , that is, from a left ideal $L$ of $R$ to $E$ can always be extended to $R$ , then a map to $E$ from a submodule $A$ of any $R$ -module $B$ can be extended to $B$ ; in other words, $E$ is injective. In this paper, we generalize this result to the category $q_{ω}$ consisting of the representations of an infinite line quiver. This generalization of Baer’s...

The importance of rational extensions

Frans Loonstra (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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The rational completion $\bar{M}$ of an $R$ -module $M$ can be characterized as a $\tau_{M}$ -injective hull of $M$ with respect to a (hereditary) torsion functor $\tau_{M}$ depending on $M$ . Properties of a torsion functor depending on an $R$ -module $M$ are studied.

About G-rings

Najib Mahdou (2017)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we are concerned with G-rings. We generalize the Kaplansky’s theorem to rings with zero-divisors. Also, we assert that if $R \subseteq T$ is a ring extension such that $m T \subseteq R$ for some regular element $m$ of $T$ , then $T$ is a G-ring if and only if so is $R$ . Also, we examine the transfer of the G-ring property to trivial ring extensions. Finally, we conclude the paper with illustrative examples discussing the utility and limits of our results.

On the structure of triangulated categories with finitely many indecomposables

Claire Amiot (2007)

Bulletin de la Société Mathématique de France

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We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field $k$ . We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category $𝒯$ is of the form $ℤ Δ / G$ where $Δ$ is a disjoint union of simply-laced Dynkin diagrams and $G$ a weakly admissible group of automorphisms of $ℤ Δ$ . Then we prove that for ‘most’ groups $G$ , the category $𝒯$ is standard, ...

Two results of $n$ -exangulated categories

Jian He, Jing He, Panyue Zhou (2024)

Czechoslovak Mathematical Journal

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M. Herschend, Y. Liu, H. Nakaoka introduced $n$ -exangulated categories, which are a simultaneous generalization of $n$ -exact categories and $(n + 2)$ -angulated categories. This paper consists of two results on $n$ -exangulated categories: (1) we give an equivalent characterization of axiom (EA2); (2) we provide a new way to construct a closed subfunctor of an $n$ -exangulated category.

Yetter-Drinfeld-Long bimodules are modules

Daowei Lu, Shuan Hong Wang (2017)

Czechoslovak Mathematical Journal

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Let $H$ be a finite-dimensional bialgebra. In this paper, we prove that the category $ℒℛ (H)$ of Yetter-Drinfeld-Long bimodules, introduced by F. Panaite, F. Van Oystaeyen (2008), is isomorphic to the Yetter-Drinfeld category $_{H \otimes H^{*}}^{H \otimes H^{*}} 𝒴𝒟$ over the tensor product bialgebra $H \otimes H^{*}$ as monoidal categories. Moreover if $H$ is a finite-dimensional Hopf algebra with bijective antipode, the isomorphism is braided. Finally, as an application of this category isomorphism, we give two results.

Torsion points in families of Drinfeld modules

Dragos Ghioca, Liang-Chung Hsia (2013)

Acta Arithmetica

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Let $Φ^{λ}$ be an algebraic family of Drinfeld modules defined over a field K of characteristic p, and let a,b ∈ K[λ]. Assume that neither a(λ) nor b(λ) is a torsion point for $Φ^{λ}$ for all λ. If there exist infinitely many λ ∈ K̅ such that both a(λ) and b(λ) are torsion points for $Φ^{λ}$ , then we show that for each λ ∈ K̅, a(λ) is torsion for $Φ^{λ}$ if and only if b(λ) is torsion for $Φ^{λ}$ . In the case a,b ∈ K, we prove in addition that a and b must be $̅_{p}$ -linearly dependent.

The importance of rational extensions

Frans Loonstra (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Singularity categories of skewed-gentle algebras

Xinhong Chen, Ming Lu (2015)

Colloquium Mathematicae

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Let K be an algebraically closed field. Let (Q,Sp,I) be a skewed-gentle triple, and let $(Q^{s g}, I^{s g})$ and $(Q^{g}, I^{g})$ be the corresponding skewed-gentle pair and the associated gentle pair, respectively. We prove that the skewed-gentle algebra $K Q^{s g} / ⟨ I^{s g} ⟩$ is singularity equivalent to KQ/⟨I⟩. Moreover, we use (Q,Sp,I) to describe the singularity category of $K Q^{g} / ⟨ I^{g} ⟩$ . As a corollary, we find that $g l d i m K Q^{s g} / ⟨ I^{s g} ⟩ < \infty$ if and only if $g l d i m K Q / ⟨ I ⟩ < \infty$ if and only if $g l d i m K Q^{g} / ⟨ I^{g} ⟩ < \infty$ .

On matrix Lie rings over a commutative ring that contain the special linear Lie ring

Evgenii L. Bashkirov, Esra Pekönür (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $K$ be an associative and commutative ring with $1$ , $k$ a subring of $K$ such that $1 \in k$ , $n \geq 2$ an integer. The paper describes subrings of the general linear Lie ring $g l_{n} (K)$ that contain the Lie ring of all traceless matrices over $k$ .

Hall algebras of two equivalent extriangulated categories

Shiquan Ruan, Li Wang, Haicheng Zhang (2024)

Czechoslovak Mathematical Journal

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For any positive integer $n$ , let $A_{n}$ be a linearly oriented quiver of type $A$ with $n$ vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories $ℳ_{n + 1}$ and $ℱ_{n}$ , where $ℳ_{n + 1}$ and $ℱ_{n}$ are the two extriangulated categories corresponding to the representation category of $A_{n + 1}$ and the morphism category of projective representations of $A_{n}$ , respectively. As a...