Journal of Algebra最新文献

英文中文

An extension of Schur's irreducibility result 舒尔不可还原性结果的扩展

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-11-19 DOI: 10.1016/j.jalgebra.2024.10.047

Ankita Jindal , Sudesh Kaur Khanduja

<div><div>Let <math><mi>n</mi><mo>≥</mo><mn>2</mn></math> be an integer. Let <math><mi>ϕ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> belonging to <math><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math> be a monic polynomial which is irreducible modulo all primes less than or equal to n. Let <math><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math> be polynomials in <math><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math> each having degree less than <math><mi>deg</mi><mo>⁡</mo><mi>ϕ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> and <math><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math> be an integer. Assume that <math><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math> and the content of <math><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math> are coprime with n!. In the present paper, we prove that the polynomial <math><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mfrac><mrow><mi>ϕ</mi><msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>i</mi></mrow></msup></mrow><mrow><mi>i</mi><mo>!</mo></mrow></mfrac><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mfrac><mrow><mi>ϕ</mi><msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup></mrow><mrow><mi>n</mi><mo>!</mo></mrow></mfrac></math> is irreducible over the field <math><mi>Q</mi></math> of rational numbers. This generalizes a well known result of Schur which states that the polynomial <math><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></munderover><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mfrac><mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msup></mrow><mrow><mi>i</mi><mo>!</mo></mrow></mfrac></math> is irreducible over <math><mi>Q</mi></math> for all <math><mi>n</mi><mo>≥</mo><mn>1</mn></math> when each <math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>Z</mi></math> and <math><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>|</mo><mo>=</mo><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>|</mo><mo>=</mo><mn>1</mn></math>. The present paper also extends a result of Filaseta thereby leading to a generalization of the classical Schönemann Irreducibility Criterion.</div></d

设 n≥2 为整数。设属于 Z[x] 的 j(x)是一个一元多项式，它在所有小于或等于 n 的素数模中是不可约的。设 a0(x),a1(x),...,an-1(x)是 Z[x] 中的多项式，每个多项式的度数都小于 degj(x)，且 an 是整数。假设 an 和 a0(x) 的内容与 n!在本文中，我们将证明在有理数域 Q 上的多项式 ∑i=0n-1ai(x)ϕ(x)ii!+anϕ(x)nn! 是不可约的。这概括了舒尔的一个著名结果，即当每个 ai∈Z 和 |a0|=|an|=1 时，多项式∑i=0naixii!本文还扩展了菲拉塞塔的一个结果，从而引出了经典的舍内曼不可还原性准则的一般化。

引用次数: 0

The non-decreasing condition on g-vectors g向量的非递减条件

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-11-19 DOI: 10.1016/j.jalgebra.2024.11.005

Mohamad Haerizadeh, Siamak Yassemi

The non-decreasing condition on g-vectors is introduced. Our study shows that this condition is both necessary and sufficient to ensure that the generically indecomposable direct summands of a given g-vector are linearly independent. Additionally, we prove that for any finite dimensional algebra Λ, under the non-decreasing condition, the number of generically indecomposable irreducible components that appear in the decomposition of a given generically τ-reduced component is lower than or equal to

| Λ |

. This solves the conjecture concerning the cardinality of component clusters by Cerulli-Labardini-Schröer, in a reasonable generality. Lastly, we study numerical criteria to check the wildness of g-vectors.

介绍了g向量的非递减条件。我们的研究表明，这个条件对于保证给定g向量的一般不可分解的直接和是线性无关的是充分必要的。此外，我们证明了对于任意有限维代数Λ，在非递减条件下，给定的广义τ约化分量的分解中出现的广义不可分解不可约分量的个数小于或等于|Λ|。这通过Cerulli-Labardini-Schröer以合理的通用性解决了有关组件簇基数的猜想。最后，研究了判别g向量野性的数值准则。

引用次数: 0

The commuting variety of pgln pgln 的换向变种

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-11-19 DOI: 10.1016/j.jalgebra.2024.10.034

Vlad Roman

We are considering the commuting variety of the Lie algebra

{pgl}_{n}

over an algebraically closed field of characteristic

p > 0

, namely the set of pairs

{(A, B) \in {pgl}_{n} \times {pgl}_{n} | [A, B] = 0}

. We prove that if

n = p r

, then there are precisely two irreducible components, of dimensions

n^{2} + r - 1

and

n^{2} + n - 2

. We also prove that the variety

{(x, y) \in G L_{n} (k) \times G L_{n} (k) | [x, y] = ζ I}

is irreducible of dimension

n^{2} + n / d

, where ζ is a root of unity of order d with d dividing n.

我们考虑的是特征 p>0 的代数闭域上的李代数 pgln 的换元杂集，即{(A,B)∈pgln×pgln|[A,B]=0}对的集合。我们证明，如果 n=pr，那么恰好有两个不可还原成分，维数分别为 n2+r-1 和 n2+n-2。我们还证明了{(x,y)∈GLn(k)×GLn(k)|[x,y]=ζI}是维数为n2+n/d的不可约分项，其中ζ是阶数为d且d除以n的统一根。

引用次数: 0

On stable equivalences of Morita type and nilpotent blocks 论莫里塔类型的稳定等价物和零能块

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-11-19 DOI: 10.1016/j.jalgebra.2024.10.039

Conghui Li

In this note, we give a new proof by module-theoretic methods for a result of Puig asserting that blocks which are stable equivalent of Morita type to nilpotent blocks are also nilpotent.

在本注释中，我们用模块理论方法对普依格的一个结果给出了新的证明，这个结果断言，莫里塔类型稳定等价于零能块的块也是零能块。

引用次数: 0

Rationality of weighted hypersurfaces of special degree 特级加权超曲面的合理性

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-11-19 DOI: 10.1016/j.jalgebra.2024.10.032

Michael Chitayat

Let

X \subset P (w_{0}, w_{1}, w_{2}, w_{3})

be a quasismooth well-formed weighted projective hypersurface and let

L = lcm (w_{0}, w_{1}, w_{2}, w_{3})

. We characterize when X is rational under the assumption that L divides

\deg (X)

. Furthermore, we give a new family of normal rational weighted projective hypersurfaces with ample canonical divisor, valid in all dimensions, adding to the list of examples discovered by Kollár. Finally, we determine precisely which affine Pham-Brieskorn threefolds are rational, answering a question of Rajendra V. Gurjar.

设 X⊂P(w0,w1,w2,w3)是一个准光滑的完形加权投影超曲面，设 L=lcm(w0,w1,w2,w3).假设 L 平分 deg(X)，我们将描述当 X 为有理时的特征。此外，我们还给出了在所有维度上都有效的、具有充裕典范除数的正常有理加权投影超曲面的一个新族，为 Kollár 发现的例子列表增添了新的内容。最后，我们精确地确定了哪些仿射 Pham-Brieskorn 三折是有理的，回答了 Rajendra V. Gurjar 的一个问题。

引用次数: 0

Higher torsion-free Auslander-Reiten sequences and the dominant dimension of algebras 更高的无扭 Auslander-Reiten 序列和代数的主维

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-11-19 DOI: 10.1016/j.jalgebra.2024.11.004

Tiago Cruz , René Marczinzik

We generalise a theorem of Tachikawa about reflexive Auslander-Reiten sequences. We apply this to give a new characterisation of the dominant dimension of gendo-symmetric algebras. We also generalise a formula due to Reiten about the dominant dimension of an algebra A and grades of torsion A-modules.

我们概括了立川关于反身奥斯兰德-莱腾序列的定理。我们以此给出了元对称代数的主维的新特征。我们还推广了赖特恩提出的一个关于代数 A 的主维和扭转 A 模量级数的公式。

引用次数: 0

On the number of quaternion and dihedral braces and Hopf–Galois structures 论四元数、二面括号和霍普夫-伽罗瓦结构

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-11-19 DOI: 10.1016/j.jalgebra.2024.11.011

Nigel P. Byott , Fabio Ferri

We prove a conjecture of Guarnieri and Vendramin on the number of braces of a given order whose multiplicative group is a generalised quaternion group. At the same time, we give a similar result where the multiplicative group is dihedral. We also enumerate Hopf-Galois structures of abelian type on Galois extensions with generalised quaternion or dihedral Galois group.

我们证明了瓜尔尼里和文德拉明关于乘法群为广义四元数的给定阶的括号数的猜想。同时，我们还给出了乘法群为二面体群时的类似结果。我们还列举了具有广义四元组或二面体伽罗瓦群的伽罗瓦扩展上的无边型霍普夫-伽罗瓦结构。

引用次数: 0

Non-associative Frobenius algebras of type E61 with trivial Tits algebras E61 型非共轭弗罗贝尼斯代数与微不足道的 Tits 代数

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-11-19 DOI: 10.1016/j.jalgebra.2024.10.035

Jari Desmet

Very recently, Maurice Chayet and Skip Garibaldi have introduced a class of commutative non-associative algebras, for each simple linear algebraic group over an arbitrary field (with some minor restriction on the characteristic). In a previous paper, we gave an explicit description of these algebras for groups of type

G_{2}

and

F_{4}

in terms of the octonion algebras and the Albert algebras, respectively. In this paper, we attempt a similar approach for type

E_{6}

最近，莫里斯-查耶（Maurice Chayet）和斯基普-加里巴尔迪（Skip Garibaldi）为任意域上的每个简单线性代数群（对特征有一些小限制）引入了一类交换非共轭布拉。在前一篇论文中，我们分别用八分子代数和阿尔伯特代数对这些代数的 G2 和 F4 型群进行了明确描述。在本文中，我们尝试用类似的方法来描述 E6 型。

引用次数: 0

The diagonal set of the self-product of an algebraic curve 代数曲线自积的对角集

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-11-19 DOI: 10.1016/j.jalgebra.2024.10.050

Masayoshi Miyanishi

Let C be a smooth projective curve of genus g defined over an algebraically closed field of characteristic

p \neq 2

and let Δ be the diagonal of

C \times C

. We observe the complement

X : = (C \times C) ∖ Δ

. If

g = 0

, X is an affine hypersurface

x y = z^{2} - 1

A^{3}

which is the simplest example of Danielewski surfaces. If

g = 1

then Δ is a fiber of an elliptic fibration over C so that

(Δ^{2}) = 0

and

\overline{κ} (X) = 1

, and if

g > 1

(Δ^{2}) = 2 - 2 g

and Δ is contractible. In the case C has genus

g > 1

, X is embedded bijectively into the Jacobian variety if C is non-hyperelliptic, though X is generically a double covering of a surface in the Jacobian variety if C is hyperelliptic. Some observations will be made in the case k has characteristic 2.

设 C 是定义在特征 p≠2 的代数闭域上的属 g 的光滑投影曲线，设 Δ 是 C×C 的对角线。我们观察补集 X:=(C×C)∖Δ。如果 g=0，X 是 A3 中的仿射超曲面 xy=z2-1，这是达尼埃夫斯基曲面最简单的例子。如果 g=1，那么 Δ 是 C 上椭圆纤维的纤维，因此 (Δ2)=0 和 κ‾(X)=1 ；如果 g>1，(Δ2)=2-2g 和 Δ 是可收缩的。在 C 具有 g>1 属性的情况下，如果 C 是非超椭圆形，X 将被双射嵌入雅各布综中，但如果 C 是超椭圆形，X 通常是雅各布综中曲面的双覆盖。在 k 的特征为 2 的情况下，我们将提出一些看法。

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引用次数: 0

On the regularity of squarefree part of symbolic powers of edge ideals 论边理想符号幂的无平方部分的规则性

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-11-19 DOI: 10.1016/j.jalgebra.2024.10.046

S.A. Seyed Fakhari

Assume that G is a graph with edge ideal

I (G)

. For every integer

s \geq 1

, we denote the squarefree part of the s-th symbolic power of

I (G)

I {(G)}^{{s}}

. We determine an upper bound for the regularity of

I {(G)}^{{s}}

when G is a chordal graph. If G is a Cameron-Walker graph, we compute

reg (I {(G)}^{{s}})

in terms of the induced matching number of G. Moreover, for any graph G, we provide sharp upper bounds for

reg (I {(G)}^{{2}})

and

reg (I {(G)}^{{3}})

假设 G 是一个具有边理想 I(G) 的图。对于每一个整数 s≥1，我们用 I(G){s} 表示 I(G) 的 s 次符号幂的无平方部分。当 G 是弦图时，我们将确定 I(G){s} 的正则性上限。此外，对于任何图 G，我们都提供了 reg(I(G){2}) 和 reg(I(G){3}) 的尖锐上限。

引用次数: 0

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