Journal of Pure and Applied Algebra最新文献

英文中文

The F-pure threshold of a Schubert cycle 舒伯特循环的f -纯阈值

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-10-01 DOI: 10.1016/j.jpaa.2025.108104

Justin Fong , Mitsuhiro Miyazaki

The F-pure threshold is the characteristic p counter part of the log canonical threshold in characteristic zero. It is a numerical invariant associated to the singularities of a variety, hence computing its value is important. We give a closed formula for the F-pure threshold of the irrelevant maximal ideal of Schubert cycles, which are the homogeneous coordinate rings of Schubert subvarieties of a Grassmannian. The main point of the computation is to give an explicit formula for the a-invariant of a Schubert cycle. The derivation of both formulas is made possible through the combinatorics of the underlying poset of these rings.

f -纯阈值是特征0中对数正则阈值的特征p计数器部分。它是一个与各种奇点有关的数值不变量，因此计算它的值是很重要的。我们给出了无关极大理想Schubert环的f -纯阈值的一个封闭公式，Schubert环是一类Grassmannian的Schubert子变量的齐次坐标环。计算的重点是给出舒伯特循环的a不变量的显式公式。这两个公式的推导是通过对这些环的基本偏序集的组合而实现的。

引用次数: 0

Noether's normalization in skew polynomial rings 歪斜多项式环中的Noether归一化

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-10-01 DOI: 10.1016/j.jpaa.2025.108101

Elad Paran , Thieu N. Vo

We study Noether's normalization lemma for finitely generated algebras over a division algebra. In its classical form, the lemma states that if I is a proper ideal of the ring

R = F [t_{1}, \dots, t_{n}]

of polynomials over a field F, then the quotient ring

R / I

is a finite extension of a polynomial ring over F. We prove that the lemma holds when

R = D [t_{1}, \dots, t_{n}]

is the ring of polynomials in n central variables over a division algebra D. We provide examples demonstrating that Noether's normalization may fail for the skew polynomial ring

D [t_{1}, \dots, t_{n}; σ_{1}, \dots, σ_{n}]

with respect to commuting automorphisms

σ_{1}, \dots, σ_{n}

of D. We give a sufficient condition for

σ_{1}, \dots, σ_{n}

under which the normalization lemma holds for such ring. In the case where

D = F

is a field, this sufficient condition is proved to be necessary.

研究了除法代数上有限生成代数的Noether归一化引理。经典形式的引理表明，如果I是域F上多项式环R=F[t1，…，tn]的固有理想，则商环R/I是F上多项式环的有限扩展。我们证明了当R=D[t1，…，tn]是除法代数D上n个中心变量的多项式环时，引理成立。我们提供了例子证明了对于偏多项式环D[t1，…，tn] Noether的归一化可能失败；d的交换自同构σ1，…，σn]，给出了σ1，…，σn的一个充分条件，在这个条件下，这种环的归一化引理成立。在D=F为域的情况下，证明了这个充分条件是必要的。

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引用次数: 0

On the standard models of del Pezzo fibrations of degree four 关于四阶del Pezzo振动的标准模型

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-10-01 DOI: 10.1016/j.jpaa.2025.108103

Natsume Kitagawa

Corti defined the notion of standard models of del Pezzo fibrations, and studied their existence over

C

with a fixed generic fibre in [6]. In this paper, we prove the existence of standard models of del Pezzo fibrations of degree 4 in characteristic >2. To show this, we use the notion of Kollár stability, which was introduced in [12] and [1].

Corti定义了del Pezzo纤维的标准模型的概念，并在b[6]中研究了它们在C上的存在性。本文证明了特征度为4的del Pezzo振动标准模型的存在性。为了说明这一点，我们使用了在[12]和[1]中引入的Kollár稳定性的概念。

引用次数: 0

Orbits on a product of two flags and a line and the Bruhat order, I 轨道是由两面旗帜和一条线以及布鲁哈特命令组成的

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-09-29 DOI: 10.1016/j.jpaa.2025.108100

Mark Colarusso , Sam Evens

<div><div>Let <math><mi>G</mi><mo>=</mo><mi>G</mi><mi>L</mi><mo>(</mo><mi>n</mi><mo>)</mo></math> be the <math><mi>n</mi><mo>×</mo><mi>n</mi></math> complex general linear group and let <math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math> be its flag variety. The standard Borel subgroup B of upper triangular matrices acts on the product <math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math> diagonally with finitely many orbits. In this paper, we study the B-orbits on the subvarieties <math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>i</mi></mrow></msub></math>, where <math><msub><mrow><mi>O</mi></mrow><mrow><mi>i</mi></mrow></msub></math> is the B-orbit on <math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math> containing the line through the origin in the direction of the i-th standard basis vector of <math><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msup></math>. For each <math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></math>, we construct a bijection between B-orbits on <math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>i</mi></mrow></msub></math> and certain pairs of Schubert cells in <math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math>. We also show that this bijection can be used to understand the Richardson-Springer monoid action on such B-orbits in terms of the classical monoid action of the symmetric group on itself. We also develop combinatorial models of these orbits and use these models to compute exponential generating functions for the sequences <math><msub><mrow><mo>{</mo><mo>|</mo><mi>B</mi><mo>﹨</mo><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>|</mo><mo>}</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></math> and <math><msub><mrow><mo>{</mo><mo>|</mo><mi>B</mi><mo>﹨</mo><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo><mo>|</mo><mo>}</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></math>. In the sequel to this paper, we use the results of this paper to construct a correspondence between B-orbits on <math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn>

设G=GL(n)为n×n复一般线性群，设Bn为其标志簇。上三角矩阵的标准Borel子群B作用于乘积Bn×Pn−1与有限多轨道对角线。本文研究了子集Bn×Oi上的b轨道，其中Oi是Pn−1上的b轨道，其中包含了沿Cn的第i个标准基向量方向穿过原点的直线。对于每个i=1，…，n，我们在Bn×Oi上的b轨道和Bn×Bn上的某些Schubert细胞对之间构造一个双射。我们还证明了这种双射可以用对称群对自身的经典单群作用来理解b轨道上的Richardson-Springer单群作用。我们还建立了这些轨道的组合模型，并使用这些模型计算了序列{|B(Bn×Oi)|}n≥1和{|B(Bn×Pn−1)|}n≥1的指数生成函数。在本文的续文中，我们利用本文的结果构造了Bn×Pn−1上的b轨道与GL（n+1）的标志簇Bn+1上的b轨道集合之间的对应关系，并证明了这种对应关系尊重闭包关系并保留了单群作用。因此，利用我们在[1]中的结果，可以通过Bruhat阶来理解Bn×Pn−1上所有b轨道集合上的闭包关系和幺正作用。

引用次数: 0

On the spectrum of residual finiteness growth functions 残差有限生长函数的谱

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-09-25 DOI: 10.1016/j.jpaa.2025.108099

Henry Bradford

In [4] Bou-Rabee and Seward constructed examples of finitely generated residually finite groups G whose residual finiteness growth function

F_{G}

can be at least as fast as any prescribed function. In this note we describe a modified version of their construction, which allows us to give a complementary upper bound on

F_{G}

. As such, every nondecreasing function at least

\exp (n \log {(n)}^{2} \log \log {(n)}^{1 + ϵ})

is close to the residual finiteness growth function of some finitely generated group. We also have similar result for the full residual finiteness growth function and for the divisibility function.

在1996年，boub - rabee和Seward构造了有限生成的剩余有限群G的例子，其剩余有限生长函数FG至少可以与任何规定函数一样快。在本文中，我们描述了它们构造的一个改进版本，它允许我们给出FG上的一个互补上界。因此，每一个至少为exp (nlog （n)2log (log)1+ λ）的非递减函数都接近于某个有限生成群的残差有限生长函数。对于完全剩余有限生长函数和可整除函数，我们也有类似的结果。

引用次数: 0

Linear subspaces of the intersection of two quadrics via Kuznetsov component 通过库兹涅佐夫分量的两个二次曲面交点的线性子空间

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-09-19 DOI: 10.1016/j.jpaa.2025.108096

Yanjie Li , Shizhuo Zhang

Let

Q_{i} (i = 1, 2)

be 2g dimensional quadrics in

P^{2 g + 1}

and let Y be the smooth intersection

Q_{1} \cap Q_{2}

. We associate the linear subspaces in Y with vector bundles on the hyperelliptic curve C of genus g via categorical methods. As an application, we give a different proof of the classification of line bundles and stable bundles of rank 2 on hyperelliptic curves given by Desale and Ramanan. When

g = 3

, we show that the projection functor induces a closed embedding

α : Y \to S U_{C}^{s} (4, h)

into the moduli space of stable bundles on C of rank 4 of fixed determinant.

设Qi（i=1,2）为P2g+1中的2g次元，设Y为光滑交集Q1∩Q2。通过分类方法将Y中的线性子空间与g属的超椭圆曲线C上的向量束联系起来。作为应用，我们给出了Desale和Ramanan给出的超椭圆曲线上2阶线束和稳定束分类的另一种证明。当g=3时，我们证明了投影函子在固定行列式的4阶C上的稳定束的模空间中诱导出一个闭合嵌入α:Y→SUCs（4,h）。

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引用次数: 0

On the average degree of characters with odd or even degrees 奇数度或偶数度字符的平均度

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-09-19 DOI: 10.1016/j.jpaa.2025.108098

Kamal Aziziheris , Necat Gorentas , Zinah Naser Sulaiman

Let

{acd}_{2^{'}} (G)

be the average degree of irreducible characters of G with odd degree. It has been proved that if

{acd}_{2^{'}} (G) < {acd}_{2^{'}} (A_{5}) = 3

, then G is a solvable group. On the other hand, let

{acd}_{2} (G)

be the average degree of linear characters and irreducible characters of G with even degree. It has been shown that if

{acd}_{2} (G) < {acd}_{2} (A_{5}) = 5 / 2

, then G is a solvable group. In this paper, we improve these bounds and we show that if G is a finite group with

{acd}_{2^{'}} (G) < {acd}_{2^{'}} (PSL (2, 7)) = 7 / 2

, then either G is a solvable group or G has a chief factor isomorphic to

A_{5}

. Also, we prove that if G is a finite group with

{acd}_{2} (G) < {acd}_{2} (PSL (2, 8)) = 9 / 2

, then either G is a solvable group or all minimal normal subgroups of G are abelian or isomorphic to

A_{5}

. Clearly, these bounds are the best.

设add ' (G)为G的奇次不可约字符的平均次。证明了如果acd2 ' (G)<acd2 ' (A5)=3，则G是一个可解群。另一方面，设acd2(G)为G的偶数次线性字符和不可约字符的平均次数。证明了如果acd2(G)<acd2(A5)=5/2，则G是一个可解群。本文改进了这些边界，证明了如果G是一个有限群，且acd2 ' (G)<acd2 ' (PSL(2,7))=7/2，则G是一个可解群或G有一个主因子同态于A5。同时证明了如果G是一个有限群，且acd2(G)<acd2(PSL(2,8))=9/2，则G是一个可解群，或者G的所有最小正规子群都与A5是阿贝的或同构的。显然，这些边界是最好的。

引用次数: 0

On the Picard number and the extension degree of period matrices of complex tori 复环面周期矩阵的皮卡德数及其可拓度

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-09-19 DOI: 10.1016/j.jpaa.2025.108097

Robert Auffarth , Jorge Duque Franco

The rank ρ of the Néron-Severi group of a complex torus X of dimension g satisfies

0 \leq ρ \leq g^{2} = h^{1, 1}

. The degree

d

of the extension field generated over

Q

by the entries of a period matrix of X imposes constraints on its Picard number ρ and, consequently, on the structure of X. In this paper, we show that when

d

is 2, 3, or 4, the Picard number ρ is necessarily large. Moreover, for an abelian variety X of dimension g with

d = 3

, we establish a structure-type result: X must be isogenous to

E^{g}

, where E is an elliptic curve without complex multiplication. In this case, the Picard number satisfies

ρ (X) = \frac{g (g + 1)}{2}

. As a byproduct, we obtain that if

d

is odd, then

ρ (X) \leq \frac{g (g + 1)}{2}

g维的复环体X的nsamron - severi群的秩ρ满足0≤ρ≤g2=h1,1。由周期矩阵X的元素在Q上生成的扩展域的阶d对它的皮卡德数ρ施加了约束，从而对X的结构施加了约束。在本文中，我们证明了当d为2、3或4时，皮卡德数ρ必然很大。此外，对于d=3的维数为g的阿贝尔变量X，我们建立了一个结构型结果：X必须同Eg同构，其中E是一条没有复乘法的椭圆曲线。在这种情况下，皮卡德数满足ρ(X)=g(g+1)2。作为副产品，我们得到如果d是奇数，那么ρ(X)≤g(g+1)2。

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引用次数: 0

On isotropy groups of quantum plane 关于量子平面的各向同性群

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-09-17 DOI: 10.1016/j.jpaa.2025.108095

Adriano De Santana , Rene Baltazar , Robson Vinciguerra , Wilian De Araujo

This paper investigates the isotropy groups of derivations on the Quantum Plane

k_{q} [x, y]

, defined by the relation

y x = q x y

, where

q \in k^{⁎}

, with

q^{2} \neq 1

. The main goal is to determine the automorphisms of the Quantum Plane that commutes with a fixed derivation δ. We describe conditions under which the isotropy group

{Aut}_{δ} (A)

is trivial, finite, or infinite, depending on the structure of δ and whether q is a root of unity: additionally, we present the structure of the group in the finite case. A key tool is the analysis of polynomial equations of the form

μ_{1}^{a} μ_{2}^{b} = 1

, arising from monomials in the inner part of δ. We also make explicit which finite subgroups of

Aut (k_{q} [x, y])

are isotropy groups of some derivation: either q root of unity or not. Techniques from algebraic geometry, such as intersection multiplicity, are also employed in the classification of the finite case.

研究了量子平面kq[x，y]上的导数的各向同性群，由关系yx=qxy定义，其中q∈k _，且q2≠1。主要目标是确定量子平面的自同构，该量子平面具有固定的导数δ。我们描述了各向同性群Autδ(A)是平凡的、有限的或无限的条件，这取决于δ的结构和q是否为单位的根；另外，我们给出了有限情况下群的结构。一个关键的工具是分析μ1aμ2b=1形式的多项式方程，这些方程是由δ内部的单项式引起的。我们还明确了Aut（kq[x,y]）的有限子群是具有某种导数的各向同性群：q是否是单位根。从代数几何的技术，如交集多重性，也被用于有限情况的分类。

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引用次数: 0

Normalizer of twisted Chevalley groups over commutative rings 交换环上扭曲Chevalley群的归一化器

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Pure and Applied Algebra

Pub Date : 2025-09-15 DOI: 10.1016/j.jpaa.2025.108094

Shripad M. Garge , Deep H. Makadiya

Let R be a commutative ring with unity. Consider the twisted Chevalley group

G_{π, σ} (Φ, R)

of type Φ over R and its elementary subgroup

E_{π, σ}^{'} (Φ, R)

. This paper investigates the normalizers of

E_{π, σ}^{'} (Φ, R)

and

G_{π, σ} (Φ, R)

in the larger group

G_{π, σ} (Φ, S)

, where S is an extension ring of R. We establish that under certain conditions on R these normalizers coincide. Moreover, in the case of adjoint type groups, we show that they are precisely equal to

G_{π, σ} (Φ, R)

设R是一个有单位的交换环。考虑R上Φ型的扭曲Chevalley群Gπ，σ（Φ，R）及其初等子群Eπ，σ ' （Φ，R）。本文研究了大群Gπ，σ（Φ，S）中ε，σ′（Φ，R）和Gπ，σ（Φ，R）的归一化器，其中S是R的一个扩展环。在一定条件下，我们证明了这些归一化器在R上是一致的。此外，对于伴随型群，我们证明了它们精确地等于Gπ，σ（Φ，R）。

引用次数: 0

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