Pub Date : 2024-10-09DOI: 10.1016/j.jpaa.2024.107816
Bivas Khan , Mainak Poddar
Let X be a complete variety over an algebraically closed field k of characteristic zero, equipped with an action of an algebraic group G. Let H be a reductive group. We study the notion of G-connection on a principal H-bundle. We give necessary and sufficient criteria for the existence of G-connections extending the Atiyah-Weil type criterion for holomorphic connections obtained by Azad and Biswas. We also establish a relationship between the existence of G-connection and equivariant structure on a principal H-bundle, under the assumption that G is semisimple and simply connected. These results have been obtained by Biswas et al. when the underlying variety is smooth.
设 X 是特征为零的代数闭域 k 上的一个完全杂化,具有一个代数群 G 的作用。设 H 是还原群。我们研究主 H 束上的 G 连接概念。我们给出了 G 连接存在的必要条件和充分条件,扩展了阿扎德和比斯沃斯获得的全形连接的 Atiyah-Weil 型判据。在 G 是半简单和简单连接的假设下,我们还建立了 G 连接的存在与主 H 束等变结构之间的关系。这些结果是 Biswas 等人在底层是光滑的情况下得到的。
{"title":"G-connections on principal bundles over complete G-varieties","authors":"Bivas Khan , Mainak Poddar","doi":"10.1016/j.jpaa.2024.107816","DOIUrl":"10.1016/j.jpaa.2024.107816","url":null,"abstract":"<div><div>Let <em>X</em> be a complete variety over an algebraically closed field <em>k</em> of characteristic zero, equipped with an action of an algebraic group <em>G</em>. Let <em>H</em> be a reductive group. We study the notion of <em>G</em>-connection on a principal <em>H</em>-bundle. We give necessary and sufficient criteria for the existence of <em>G</em>-connections extending the Atiyah-Weil type criterion for holomorphic connections obtained by Azad and Biswas. We also establish a relationship between the existence of <em>G</em>-connection and equivariant structure on a principal <em>H</em>-bundle, under the assumption that <em>G</em> is semisimple and simply connected. These results have been obtained by Biswas et al. when the underlying variety is smooth.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107816"},"PeriodicalIF":0.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426026","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-09DOI: 10.1016/j.jpaa.2024.107817
Hipolito Treffinger
In this article we study chains of torsion classes in an abelian category . We prove that chains of torsion classes satisfying mild technical conditions induce a Harder-Narasimhan filtration for every non-zero object M in , generalising a well-known property of stability conditions. We also characterise the slicings of in terms of chains of torsion classes. We finish the paper by showing that chains of torsion classes induce wall-crossing formulas in the completed Hall algebra of the category.
我们证明,满足温和技术条件的扭转类链会为 A 中的每个非零对象 M 诱导一个 Harder-Narasimhan 滤波,这是对稳定性条件的一个著名性质的推广。我们还用扭转类链描述了 A 的切分特征。最后,我们还证明了扭转类链会在该范畴的完备霍尔代数中诱发壁交公式。
{"title":"An algebraic approach to Harder-Narasimhan filtrations","authors":"Hipolito Treffinger","doi":"10.1016/j.jpaa.2024.107817","DOIUrl":"10.1016/j.jpaa.2024.107817","url":null,"abstract":"<div><div>In this article we study chains of torsion classes in an abelian category <span><math><mi>A</mi></math></span>. We prove that chains of torsion classes satisfying mild technical conditions induce a Harder-Narasimhan filtration for every non-zero object <em>M</em> in <span><math><mi>A</mi></math></span>, generalising a well-known property of stability conditions. We also characterise the slicings of <span><math><mi>A</mi></math></span> in terms of chains of torsion classes. We finish the paper by showing that chains of torsion classes induce wall-crossing formulas in the completed Hall algebra of the category.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107817"},"PeriodicalIF":0.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-09DOI: 10.1016/j.jpaa.2024.107822
Mousumi Mandal, Shruti Priya
Let be a Cohen-Macaulay local ring of dimension , and I an -primary ideal. Let be the reduction number of I, the postulation number and the stability index of the Ratliff-Rush filtration with respect to I. We prove that for , if , then , and if , then . For , assuming I is integrally closed, , and , we prove that . Our main result generalizes a result by Marley on the relation between the Hilbert-Samuel function and the Hilbert-Samuel polynomial by relaxing the condition on the depth of the associated graded ring to the good behavior of the Ratliff-Rush filtration with respect to I mod a superficial sequence. From this result, it follows that for Cohen-Macaulay local rings of dimension , if for some , then for all .
设 (R,m) 是维数 d≥2 的科恩-麦考莱局部环,I 是一个 m 初等理想。我们将证明,对于 d=2,如果 n(I)=ρ(I)-1,那么 r(I)≤n(I)+2;如果 n(I)≠ρ(I)-1,那么 r(I)≥n(I)+2。对于 d≥3,假设 I 是整闭的,depthgr(I)=d-2,n(I)=-(d-3),我们证明 r(I)≥n(I)+d。我们的主要结果概括了马利关于希尔伯特-萨缪尔函数和希尔伯特-萨缪尔多项式之间关系的结果,把相关分级环的深度条件放宽到了拉特利夫-拉什滤波关于 I mod a superficial sequence 的良好行为。从这个结果可以得出,对于维数 d≥2 的科恩-麦考莱局部环,如果对于某个 k≥ρ(I),PI(k)=HI(k),那么对于所有 n≥k,PI(n)=HI(n)。
{"title":"A note on Ratliff-Rush filtration, reduction number and postulation number of m-primary ideals","authors":"Mousumi Mandal, Shruti Priya","doi":"10.1016/j.jpaa.2024.107822","DOIUrl":"10.1016/j.jpaa.2024.107822","url":null,"abstract":"<div><div>Let <span><math><mo>(</mo><mi>R</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> be a Cohen-Macaulay local ring of dimension <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>, and <em>I</em> an <span><math><mi>m</mi></math></span>-primary ideal. Let <span><math><mi>r</mi><mo>(</mo><mi>I</mi><mo>)</mo></math></span> be the reduction number of <em>I</em>, <span><math><mi>n</mi><mo>(</mo><mi>I</mi><mo>)</mo></math></span> the postulation number and <span><math><mi>ρ</mi><mo>(</mo><mi>I</mi><mo>)</mo></math></span> the stability index of the Ratliff-Rush filtration with respect to <em>I</em>. We prove that for <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>, if <span><math><mi>n</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>=</mo><mi>ρ</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span>, then <span><math><mi>r</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>≤</mo><mi>n</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>+</mo><mn>2</mn></math></span>, and if <span><math><mi>n</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>≠</mo><mi>ρ</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span>, then <span><math><mi>r</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>≥</mo><mi>n</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>+</mo><mn>2</mn></math></span>. For <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>, assuming <em>I</em> is integrally closed, <span><math><mi>depth</mi><mspace></mspace><mi>gr</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>=</mo><mi>d</mi><mo>−</mo><mn>2</mn></math></span>, and <span><math><mi>n</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>=</mo><mo>−</mo><mo>(</mo><mi>d</mi><mo>−</mo><mn>3</mn><mo>)</mo></math></span>, we prove that <span><math><mi>r</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>≥</mo><mi>n</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>+</mo><mi>d</mi></math></span>. Our main result generalizes a result by Marley on the relation between the Hilbert-Samuel function and the Hilbert-Samuel polynomial by relaxing the condition on the depth of the associated graded ring to the good behavior of the Ratliff-Rush filtration with respect to <em>I</em> mod a superficial sequence. From this result, it follows that for Cohen-Macaulay local rings of dimension <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>, if <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo></math></span> for some <span><math><mi>k</mi><mo>≥</mo><mi>ρ</mi><mo>(</mo><mi>I</mi><mo>)</mo></math></span>, then <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for all <span><math><mi>n</mi><mo>≥</mo><mi>k</mi></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107822"},"PeriodicalIF":0.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142441733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-05DOI: 10.1016/j.jpaa.2024.107820
Alexander S. Duncan , Wenbo Niu , Jinhyung Park
The gonality conjecture, proved by Ein–Lazarsfeld, asserts that the gonality of a nonsingular projective curve of genus g can be detected from its syzygies in the embedding given by a line bundle of sufficiently large degree. An effective result obtained by Rathmann says that any line bundle of degree at least would work in the gonality theorem. In this note, we develop a new method to improve the degree bound to with two exceptional cases.
由艾因-拉扎斯菲尔德证明的冈性猜想认为,可以通过由足够大阶数的线束给出的嵌入中的共轭来检测属数为 g 的非共轭投影曲线的冈性。拉特曼得到的一个有效结果表明,任何阶数至少为 4g-3 的线束都可以用于贡性定理。在本注释中,我们开发了一种新方法,通过两种特殊情况将阶数约束提高到 4g-4。
{"title":"A note on an effective bound for the gonality conjecture","authors":"Alexander S. Duncan , Wenbo Niu , Jinhyung Park","doi":"10.1016/j.jpaa.2024.107820","DOIUrl":"10.1016/j.jpaa.2024.107820","url":null,"abstract":"<div><div>The gonality conjecture, proved by Ein–Lazarsfeld, asserts that the gonality of a nonsingular projective curve of genus <em>g</em> can be detected from its syzygies in the embedding given by a line bundle of sufficiently large degree. An effective result obtained by Rathmann says that any line bundle of degree at least <span><math><mn>4</mn><mi>g</mi><mo>−</mo><mn>3</mn></math></span> would work in the gonality theorem. In this note, we develop a new method to improve the degree bound to <span><math><mn>4</mn><mi>g</mi><mo>−</mo><mn>4</mn></math></span> with two exceptional cases.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107820"},"PeriodicalIF":0.7,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-04DOI: 10.1016/j.jpaa.2024.107814
A.S. Gordienko
Using the braided version of Lawvere's algebraic theories and Mac Lane's PROPs, we introduce polynomial identities for arbitrary algebraic structures in a braided monoidal category as well as their codimensions in the case when is linear over some field. The new cases include coalgebras, bialgebras, Hopf algebras, braided vector spaces, Yetter–Drinfel'd modules, etc. We find bases for polynomial identities and calculate codimensions in some important particular cases.
利用 Lawvere 的代数理论和 Mac Lane 的 PROPs 的辫状版本,我们介绍了辫状一元范畴 C 中任意代数结构的多项式同素异形体,以及当 C 在某个域上是线性时它们的同维数。新的情况包括煤系、双系、霍普夫系、编织向量空间、Yetter-Drinfel'd 模块等。我们找到了多项式等式的基数,并计算了一些重要特殊情况下的标度。
{"title":"On a general notion of a polynomial identity and codimensions","authors":"A.S. Gordienko","doi":"10.1016/j.jpaa.2024.107814","DOIUrl":"10.1016/j.jpaa.2024.107814","url":null,"abstract":"<div><div>Using the braided version of Lawvere's algebraic theories and Mac Lane's PROPs, we introduce polynomial identities for arbitrary algebraic structures in a braided monoidal category <span><math><mi>C</mi></math></span> as well as their codimensions in the case when <span><math><mi>C</mi></math></span> is linear over some field. The new cases include coalgebras, bialgebras, Hopf algebras, braided vector spaces, Yetter–Drinfel'd modules, etc. We find bases for polynomial identities and calculate codimensions in some important particular cases.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107814"},"PeriodicalIF":0.7,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/j.jpaa.2024.107815
Martin Kreuzer , Tran N.K. Linh , Le N. Long
<div><div>To study a 0-dimensional scheme <span><math><mi>X</mi></math></span> in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> over a perfect field <em>K</em>, we use the module of Kähler differentials <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> of its homogeneous coordinate ring <em>R</em> and its exterior powers, the higher modules of Kähler differentials <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math></span>. One of our main results is a characterization of weakly curvilinear schemes <span><math><mi>X</mi></math></span> by the Hilbert polynomials of the modules <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math></span> which allows us to check this property algorithmically without computing the primary decomposition of the vanishing ideal of <span><math><mi>X</mi></math></span>. Further main achievements are precise formulas for the Hilbert functions and Hilbert polynomials of the modules <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math></span> for a fat point scheme <span><math><mi>X</mi></math></span> which extend and settle previous partial results and conjectures. Underlying these results is a novel method: we first embed the homogeneous coordinate ring <em>R</em> into its truncated integral closure <span><math><mover><mrow><mi>R</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span>. Then we use the corresponding map from the module of Kähler differentials <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> to <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mover><mrow><mi>R</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>/</mo><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> to find a formula for the Hilbert polynomial <span><math><mrow><mi>HP</mi></mrow><mo>(</mo><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mo>)</mo></math></span> and a sharp bound for the regularity index <span><math><mrow><mi>ri</mi></mrow><mo>(</mo><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mo>)</mo></math></span>. Next we extend this to formulas for the Hilbert polynomials <span><math><mrow><mi>HP</mi></mrow><mo>(</mo><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msubsup><mo>)</mo></math></span> and bounds for the regularity indices of the higher modules of Kähler differentials. As a further application, we characterize uniformity conditions on <span><math><mi>X</mi></math></span> using the Hilbert functions of the Kähler diff
为了研究完全域 K 上 Pn 中的 0 维方案 X,我们使用了其同质坐标环 R 的凯勒微分模块 ΩR/K1 及其外部幂,即凯勒微分的高阶模块 ΩR/Km。我们的主要成果之一是通过模块 ΩR/Km 的希尔伯特多项式描述了弱曲线方案 X 的特性,这使我们无需计算 X 消失理想的主分解就能用算法检查这一特性。其他主要成果是胖点方案 X 的模块 ΩR/Km 的希尔伯特函数和希尔伯特多项式的精确公式,这些公式扩展并解决了之前的部分结果和猜想。这些结果的基础是一种新方法:我们首先将同质坐标环 R 嵌入其截积分闭包 R˜。然后,我们利用从凯勒微分模块 ΩR/K1 到 ΩR˜/K1 的相应映射,找到希尔伯特多项式 HP(ΩR/K1) 的公式和正则指数 ri(ΩR/K1) 的尖锐约束。接下来,我们将其扩展到希尔伯特多项式 HP(ΩR/Km) 的公式和凯勒微分高阶模块的正则指数的边界。作为进一步的应用,我们利用 X 及其子方案的凯勒微分模块的希尔伯特函数来描述 X 的均匀性条件。
{"title":"Differential theory of zero-dimensional schemes","authors":"Martin Kreuzer , Tran N.K. Linh , Le N. Long","doi":"10.1016/j.jpaa.2024.107815","DOIUrl":"10.1016/j.jpaa.2024.107815","url":null,"abstract":"<div><div>To study a 0-dimensional scheme <span><math><mi>X</mi></math></span> in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> over a perfect field <em>K</em>, we use the module of Kähler differentials <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> of its homogeneous coordinate ring <em>R</em> and its exterior powers, the higher modules of Kähler differentials <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math></span>. One of our main results is a characterization of weakly curvilinear schemes <span><math><mi>X</mi></math></span> by the Hilbert polynomials of the modules <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math></span> which allows us to check this property algorithmically without computing the primary decomposition of the vanishing ideal of <span><math><mi>X</mi></math></span>. Further main achievements are precise formulas for the Hilbert functions and Hilbert polynomials of the modules <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math></span> for a fat point scheme <span><math><mi>X</mi></math></span> which extend and settle previous partial results and conjectures. Underlying these results is a novel method: we first embed the homogeneous coordinate ring <em>R</em> into its truncated integral closure <span><math><mover><mrow><mi>R</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span>. Then we use the corresponding map from the module of Kähler differentials <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> to <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mover><mrow><mi>R</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>/</mo><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> to find a formula for the Hilbert polynomial <span><math><mrow><mi>HP</mi></mrow><mo>(</mo><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mo>)</mo></math></span> and a sharp bound for the regularity index <span><math><mrow><mi>ri</mi></mrow><mo>(</mo><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mo>)</mo></math></span>. Next we extend this to formulas for the Hilbert polynomials <span><math><mrow><mi>HP</mi></mrow><mo>(</mo><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msubsup><mo>)</mo></math></span> and bounds for the regularity indices of the higher modules of Kähler differentials. As a further application, we characterize uniformity conditions on <span><math><mi>X</mi></math></span> using the Hilbert functions of the Kähler diff","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107815"},"PeriodicalIF":0.7,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-30DOI: 10.1016/j.jpaa.2024.107813
Sam Evens , Yu Li
For a semisimple algebraic group G of adjoint type with Lie algebra over the complex numbers, we establish a bijection between the set of closed orbits of the group acting on the variety of Lagrangian subalgebras of and the set of abelian ideals of a fixed Borel subalgebra of . In particular, the number of such orbits equals by Peterson's theorem on abelian ideals.
对于具有复数上的李代数 g 的邻接型半简代数群 G,我们建立了作用于 g⋉g⁎ 的各种拉格朗日子代数上的群 G⋉g⁎ 的闭轨道集与 g 的固定伯尔子代数的无边际理想集之间的双射关系。特别是,根据彼得森的无边际理想定理,这样的轨道数等于 2rkg。
{"title":"Abelian ideals and the variety of Lagrangian subalgebras","authors":"Sam Evens , Yu Li","doi":"10.1016/j.jpaa.2024.107813","DOIUrl":"10.1016/j.jpaa.2024.107813","url":null,"abstract":"<div><div>For a semisimple algebraic group <em>G</em> of adjoint type with Lie algebra <span><math><mi>g</mi></math></span> over the complex numbers, we establish a bijection between the set of closed orbits of the group <span><math><mi>G</mi><mo>⋉</mo><msup><mrow><mi>g</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> acting on the variety of Lagrangian subalgebras of <span><math><mi>g</mi><mo>⋉</mo><msup><mrow><mi>g</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> and the set of abelian ideals of a fixed Borel subalgebra of <span><math><mi>g</mi></math></span>. In particular, the number of such orbits equals <span><math><msup><mrow><mn>2</mn></mrow><mrow><mtext>rk</mtext><mi>g</mi></mrow></msup></math></span> by Peterson's theorem on abelian ideals.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107813"},"PeriodicalIF":0.7,"publicationDate":"2024-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-27DOI: 10.1016/j.jpaa.2024.107807
Carlos Améndola , Francesco Galuppi , Ángel David Ríos Ortiz , Pierpaola Santarsiero , Tim Seynnaeve
The signature of a path is a sequence of tensors whose entries are iterated integrals, playing a key role in stochastic analysis and applications. The set of all signature tensors at a particular level gives rise to the universal signature variety. We show that the parametrization of this variety induces a natural decomposition of the tensor space via representation theory, and connect this to the study of path invariants. We also reveal certain constraints that apply to the rank and symmetry of a signature tensor.
{"title":"Decomposing tensor spaces via path signatures","authors":"Carlos Améndola , Francesco Galuppi , Ángel David Ríos Ortiz , Pierpaola Santarsiero , Tim Seynnaeve","doi":"10.1016/j.jpaa.2024.107807","DOIUrl":"10.1016/j.jpaa.2024.107807","url":null,"abstract":"<div><div>The signature of a path is a sequence of tensors whose entries are iterated integrals, playing a key role in stochastic analysis and applications. The set of all signature tensors at a particular level gives rise to the universal signature variety. We show that the parametrization of this variety induces a natural decomposition of the tensor space via representation theory, and connect this to the study of path invariants. We also reveal certain constraints that apply to the rank and symmetry of a signature tensor.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107807"},"PeriodicalIF":0.7,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425561","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-27DOI: 10.1016/j.jpaa.2024.107809
Luca Chiantini , Pietro De Poi , Łucja Farnik , Giuseppe Favacchio , Brian Harbourne , Giovanna Ilardi , Juan Migliore , Tomasz Szemberg , Justyna Szpond
The purpose of this work is to pursue classification of geproci sets. Specifically we classify -geproci sets Z which consist of points on each of n skew lines, assuming the skew lines have two transversals in common. We show in this case that . Moreover we show that all geproci sets of this type and with no points on the transversals are contained in the configuration. We conjecture that a similar result is true for an arbitrary number m of points on each skew line, replacing containment in by containment in a half grid obtained by the so-called standard construction.
这项工作的目的是研究 geproci 集的分类。具体地说,我们对[m,n]-geproci 集 Z 进行分类,它由 n 条斜线上的 m=4 个点组成,假定这些斜线有两条共同的横线。在这种情况下,我们证明 n≤6.此外,我们还证明了所有这种类型且横轴上没有点的开普西集都包含在 F4 配置中。我们猜想,对于每条斜线上任意数目的 m 点,用所谓标准构造得到的半网格中的包含来代替 F4 中的包含,也会得到类似的结果。
{"title":"Geproci sets on skew lines in P3 with two transversals","authors":"Luca Chiantini , Pietro De Poi , Łucja Farnik , Giuseppe Favacchio , Brian Harbourne , Giovanna Ilardi , Juan Migliore , Tomasz Szemberg , Justyna Szpond","doi":"10.1016/j.jpaa.2024.107809","DOIUrl":"10.1016/j.jpaa.2024.107809","url":null,"abstract":"<div><div>The purpose of this work is to pursue classification of geproci sets. Specifically we classify <span><math><mo>[</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>]</mo></math></span>-geproci sets <em>Z</em> which consist of <span><math><mi>m</mi><mo>=</mo><mn>4</mn></math></span> points on each of <em>n</em> skew lines, assuming the skew lines have two transversals in common. We show in this case that <span><math><mi>n</mi><mo>≤</mo><mn>6</mn></math></span>. Moreover we show that all geproci sets of this type and with no points on the transversals are contained in the <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> configuration. We conjecture that a similar result is true for an arbitrary number <em>m</em> of points on each skew line, replacing containment in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> by containment in a half grid obtained by the so-called <em>standard construction</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107809"},"PeriodicalIF":0.7,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-27DOI: 10.1016/j.jpaa.2024.107811
Konrad Schmüdgen
A variant of the Archimedean Positivstellensatz is proved which is based on Archimedean semirings or quadratic modules of generating subalgebras. It allows one to obtain representations of strictly positive polynomials on compact semi-algebraic sets by means of smaller sets of squares or polynomials. A large number of examples is developed in detail.
{"title":"A note on the Archimedean Positivstellensatz in real algebraic geometry","authors":"Konrad Schmüdgen","doi":"10.1016/j.jpaa.2024.107811","DOIUrl":"10.1016/j.jpaa.2024.107811","url":null,"abstract":"<div><div>A variant of the Archimedean Positivstellensatz is proved which is based on Archimedean semirings or quadratic modules of generating subalgebras. It allows one to obtain representations of strictly positive polynomials on compact semi-algebraic sets by means of smaller sets of squares or polynomials. A large number of examples is developed in detail.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107811"},"PeriodicalIF":0.7,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}