Journal of Pure and Applied Algebra最新文献

英文中文

Almost ordered algebras and the Jacobson radical of tensor products 几乎有序代数与张量积的Jacobson根

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-09-02 DOI: 10.1016/j.jpaa.2025.108077

Piotr Grzeszczuk

This paper examines the Jacobson radical of tensor products of associative algebras over a field k focusing on algebras with a special type of ordering. The notion of almost ordered algebras is introduced. It is proved that for any k-algebra A and an almost ordered algebra B the Jacobson radical

J (A \otimes_{k} B) \subseteq N \otimes_{k} B

, where N is the nil radical of A. We apply this result to cases where the algebra A either satisfies a polynomial identity or is a Goldie algebra.

本文研究了结合代数张量积的Jacobson根在域k上的性质，重点研究了一类特殊排序代数。引入了几乎有序代数的概念。证明了对于任意k代数A和几乎有序代数B， Jacobson根J(A⊗kB)≠N⊗kB，其中N为A的零根。我们将此结果应用于代数A满足多项式恒等式或Goldie代数的情况。

引用次数: 0

Edge ideals and their asymptotic syzygies 边理想及其渐近合

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-09-02 DOI: 10.1016/j.jpaa.2025.108079

Antonino Ficarra , Ayesha Asloob Qureshi

Let G be a finite simple graph, and let

I (G)

denote its edge ideal. In this paper, we investigate the asymptotic behavior of the syzygies of powers of edge ideals through the lens of homological shift ideals

{HS}_{i} (I {(G)}^{k})

. We introduce the notion of the ith homological strong persistence property for monomial ideals I, providing an algebraic characterization that ensures the chain of inclusions

Ass {HS}_{i} (I) \subseteq Ass {HS}_{i} (I^{2}) \subseteq Ass {HS}_{i} (I^{3}) \subseteq \dots

. We prove that edge ideals possess both the 0th and 1st homological strong persistence properties. To this end, we explicitly describe the first homological shift algebra of

I (G)

and show that

{HS}_{1} (I {(G)}^{k + 1}) = I (G) \cdot {HS}_{1} (I {(G)}^{k})

for all

k \geq 1

. Finally, we conjecture that if

I (G)

has a linear resolution, then

{HS}_{i} (I {(G)}^{k})

also has a linear resolution for all

k ≫ 0

, and we present partial results supporting this conjecture.

设G是一个有限简单图，设I(G)表示它的边理想。本文通过同调移位理想HSi（I(G)k）透镜研究了边理想幂合子的渐近性质。引入单项式理想I的第I个同构强持久性的概念，给出了一种确保包含物链的代数表征，该包含物链的性质可满足下述条件：证明了边理想同时具有第0次和第1次同调强持久性。为此，我们明确地描述了I(G)的第一个同调移位代数，并证明了对于所有k≥1，HS1(I(G)k+1)=I(G)·HS1（I(G)k）。最后，我们推测如果I(G)具有线性分辨率，那么HSi（I(G)k）对于所有k > 0也具有线性分辨率，并给出了部分结果来支持这一猜想。

引用次数: 0

Multiplicative bases and commutative semiartinian von Neumann regular algebras 乘法基与交换半整数冯·诺伊曼正则代数

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-08-29 DOI: 10.1016/j.jpaa.2025.108075

Kateřina Fuková, Jan Trlifaj

Let R be a semiartinian (von Neumann) regular ring with primitive factors artinian. The dimension sequence

D_{R}

is an invariant that captures the various skew-fields and dimensions occurring in the layers of the socle sequence of R. Though

D_{R}

does not determine R up to an isomorphism even for rings of Loewy length 2, we prove that it does so when R is a commutative semiartinian regular K-algebra of countable type over a field K. The proof is constructive: given the sequence

D

, we construct the unique K-algebra of countable type

R = B_{α, n}

such that

D = D_{R}

by a transfinite iterative construction from the base case of the K-algebra

R (ℵ_{0}, K)

consisting of all eventually constant sequences in

K^{ℵ_{0}}

. Moreover, we prove that the K-algebras

B_{α, n}

possess conormed strong multiplicative bases despite the fact that the ambient K-algebras

K^{κ}

do not even have any bounded bases for any infinite cardinal κ.

Recently, a study of the number of limit models in AECs of modules [9] has raised interest in the question of existence of strictly λ-injective modules for arbitrary infinite cardinals λ. In the final section, we construct examples of such modules over the K-algebra

R (κ, K)

for each cardinal

κ \geq λ

设R是一个具有原始因子artinian的半artinian （von Neumann）正则环。维数序列DR是一个不变量，它捕获了在R的社会序列的各层中出现的各种偏域和维数。尽管DR不能决定R是否同构，即使对于Loewy长度为2的环，我们证明了当R是域k上可数型的可交换半整数正则k代数时，它可以这样做。给定序列D，我们从由K的所有最终常数列组成的K代数R（ρ 0, ρ K）的基本情况出发，通过超限迭代构造了唯一的可计数型R=Bα，n使得D=DR的K代数。此外，我们证明了k -代数Bα，n具有符合的强乘法基，尽管周围k -代数Kκ甚至对任何无限基数κ都没有任何有界基。最近，对模[9]的AECs中极限模型数目的研究引起了人们对任意无限基λ的严格λ内射模的存在性问题的兴趣。在最后一节中，我们在K代数R（κ，K）上为每个基数κ≥λ构造这样的模块的例子。

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引用次数: 0

Bass-Serre theory for groupoids 类群的Bass-Serre理论

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-08-29 DOI: 10.1016/j.jpaa.2025.108074

Giulia dal Verme , Thomas Weigel

In this paper, Bass-Serre theory is developed in the groupoid framework, and a structure theorem is established. We show that, when a groupoid action on a forest is without inversion of edges, it induces a graph of groupoids. Conversely, a graph of groupoids that satisfies certain conditions admits a canonical associated groupoid, which we call the fundamental groupoid, and a forest, which we call the Bass-Serre forest. The fundamental groupoid acts on the Bass-Serre forest. The structure theorem asserts that these processes are mutually inverse.

本文在群样框架中发展了Bass-Serre理论，建立了一个结构定理。我们证明了，当群拟作用于森林时，其边不存在反转时，可以得到一个群拟图。相反地，满足一定条件的群类群图包含一个正则关联群类群，我们称之为基本群类群，和一个林，我们称之为Bass-Serre林。基本类群作用于Bass-Serre森林。结构定理断言这些过程是相互逆的。

引用次数: 0

Topoi with enough points and topological groupoids 具有足够点和拓扑群的拓扑

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-08-26 DOI: 10.1016/j.jpaa.2025.108073

J.L. Wrigley

We establish a bi-equivalence between the bi-category of topoi with enough points and a localisation of a bi-subcategory of topological groupoids.

建立了具有足够点的拓扑双范畴与拓扑群类双子范畴的局部化之间的双等价性。

引用次数: 0

Continuous and algebraic domains in univalent foundations 一元基上的连续和代数域

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-08-26 DOI: 10.1016/j.jpaa.2025.108072

Tom de Jong , Martín Hötzel Escardó

We develop the theory of continuous and algebraic domains in constructive and predicative univalent foundations, building upon our earlier work on basic domain theory in this setting. That we work predicatively means that we do not assume Voevodsky's propositional resizing axioms. Our work is constructive in the sense that we do not rely on excluded middle or the axiom of (countable) choice. To deal with size issues and give a predicatively suitable definition of continuity of a dcpo, we follow Johnstone and Joyal's work on continuous categories. Adhering to the univalent perspective, we explicitly distinguish between data and property. To ensure that being continuous is a property of a dcpo, we turn to the propositional truncation, although we explain that some care is needed to avoid needing the axiom of choice. We also adapt the notion of a domain-theoretic basis to the predicative setting by imposing suitable smallness conditions, analogous to the categorical concept of an accessible category. All our running examples of continuous dcpos are then actually examples of dcpos with small bases which we show to be well behaved predicatively. In particular, such dcpos are exactly those presented by small ideals. As an application of the theory, we show that Scott's

D_{\infty}

model of the untyped λ-calculus is an example of an algebraic dcpo with a small basis. Our work is formalised in the Agda proof assistant and its ability to infer universe levels has been invaluable for our purposes.

我们在构造和谓词一元基础上发展了连续域和代数域的理论，建立在我们早期的基本域理论的基础上。我们的工作是预言性的，这意味着我们不假设Voevodsky的命题大小调整公理。我们的工作是建设性的，因为我们不依赖于排除中间或（可数）选择公理。为了处理大小问题并给出dcpo连续性的一个可预测的合适定义，我们遵循Johnstone和Joyal关于连续类别的工作。坚持一元视角，我们明确区分数据和属性。为了确保连续是dcpo的一个属性，我们转向命题截断，尽管我们解释了一些需要避免需要选择公理的注意。我们还通过施加适当的小条件来适应领域理论基础的概念，类似于可访问范畴的范畴概念。我们所有连续dcpos的运行示例实际上都是具有小基数的dcpos的示例，我们证明了它们具有良好的预测性。特别是，这样的dpos正是由小理想呈现的。作为该理论的一个应用，我们证明了无类型λ-微积分的Scott's D∞模型是一个小基代数dcpo的例子。我们的工作在Agda证明助手中被形式化，它推断宇宙水平的能力对我们的目的来说是无价的。

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引用次数: 0

A convenient category to study asymptotic primes and related questions 研究渐近素数及相关问题的方便范畴

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-08-19 DOI: 10.1016/j.jpaa.2025.108071

Tony J. Puthenpurakal

Let A be a Noetherian ring and let

R, S

be standard graded A-algebras with

R_{0} = S_{0} = A

and assume we have a homogeneous inclusion

R \subseteq S

of graded rings. Assume M is a finitely generated graded

R

-module and N is a finitely generated

S

module. Also assume

M \subseteq N

as graded

R

-modules. Let F be a covariant, coherent functor on the category of finitely generated A-modules. Then we show

(1)
${Ass}_{A} F (N_{n} / M_{n})$ is stable for $n ≫ 0$ .
(2)
if J is an ideal in A then $grade (J, F (N_{n} / M_{n}))$ is constant for $n ≫ 0$ .
(3)
if A is local and $F (N_{n} / M_{n})$ has finite length for all n then the function $n \mapsto ℓ (F (N_{n} / M_{n}))$ is of polynomial type.

Our technique to prove the above assertions is categorical. We define a category

A (R)

where we show all the above properties hold for

X \in A (R)

. Finally we show

M / N \in A (R)

设A为noether环，设R、S为标准的分级A代数，且R0=S0=A，并设有一个分级环的齐次包体R≤S。设M为有限生成的分级r模，N为有限生成的S模。并设M≤N为梯度r模。设F是有限生成a模范畴上的协变相干函子。然后我们证明(1)AssAF（Nn/Mn）在n < 0时是稳定的。(2)如果J是A中的理想，则等级（J，F(Nn/Mn)）在n < 0时是常数。(3)如果A是局部的，并且F（Nn/Mn）对于所有n都有有限的长度，则函数n∈r （F(Nn/Mn)）是多项式型的。我们证明上述断言的技术是绝对的。我们定义了一个范畴a (R)，在这里我们展示了X∈a (R)的所有上述性质。最后我们得到M/N∈A(R)

引用次数: 0

Corrigendum to “Image of ideals under linear K-derivations and the LNED conjecture” [Journal of Pure and Applied Algebra 229 (2025) 108041] “线性k导下的理想象和LNED猜想”的勘误表[Journal of Pure and Applied Algebra] 229 (2025) 108041]

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-08-14 DOI: 10.1016/j.jpaa.2025.108067

Sakshi Gupta

引用次数: 0

Strong generation for module categories 模块类别的强生成

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-08-13 DOI: 10.1016/j.jpaa.2025.108070

Souvik Dey , Pat Lank , Ryo Takahashi

This article investigates strong generation within the module category of a commutative Noetherian ring. We establish a criterion for such rings to possess strong generators within their module category, addressing a question raised by Iyengar and Takahashi. As a consequence, this not only demonstrates that any Noetherian quasi-excellent ring of finite Krull dimension satisfies this criterion, but applies to rings outside this class. Additionally, we identify explicit strong generators within the module category for rings of prime characteristic, and establish upper bounds on Rouquier dimension in terms of classical numerical invariants for modules.

本文研究了交换noether环模范畴内的强生成。我们建立了这样的环在其模类内具有强发生器的准则，解决了Iyengar和Takahashi提出的问题。因此，这不仅证明了任何有限Krull维的noether拟优环都满足这一准则，而且也适用于该类以外的环。此外，我们在模范畴内确定了素特征环的显式强生成器，并根据模的经典数值不变量建立了Rouquier维的上界。

引用次数: 0

A geometric construction of U(n) for affine Kac-Moody algebras of type C˜n C ~ n型仿射Kac-Moody代数U(n)的几何构造

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-08-12 DOI: 10.1016/j.jpaa.2025.108069

Alberto Castillo-Gómez, Christof Geiss

Inspired by the work of Geiss, Leclerc and Schröer [6] we realize the enveloping algebra of the positive part of an affine Kac-Moody Lie algebra of Dynkin type

{\tilde{C}}_{n}

as a generalized composition algebra of constructible functions on the varieties of locally free representations of the corresponding 1-Iwanaga-Gorenstein algebra

H = H_{C} (C, D, Ω)

with minimal symmetrizer D. To this end, we exploit in several ways the fact that in this situation H is a string algebra.

受Geiss， Leclerc和Schröer[6]的工作启发，我们将Dynkin型的仿射Kac-Moody Lie代数的正部分的包络代数实现为相应的1-Iwanaga-Gorenstein代数H=HC（C，D，Ω）具有最小对称子D的局部自由表示的变体上的可构造函数的广义复合代数。为此，我们利用了在这种情况下H是弦代数的事实。

引用次数: 0

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