Pub Date : 2025-11-01Epub Date: 2025-10-24DOI: 10.1016/j.jpaa.2025.108118
Ramin Ebrahimi
Let be a skeletally small additive category. Using the canonical equivalence between two different presentations of the free abelian category over , we give a new and simple characterization of definable subcategories of , and in particular definable subcategories of modules over rings. In the end, we give a conceptual proof of Auslander-Gruson-Jensen duality, which makes the duality between definable subcategories of left and right module more transparent.
{"title":"On definable subcategories","authors":"Ramin Ebrahimi","doi":"10.1016/j.jpaa.2025.108118","DOIUrl":"10.1016/j.jpaa.2025.108118","url":null,"abstract":"<div><div>Let <span><math><mi>X</mi></math></span> be a skeletally small additive category. Using the canonical equivalence between two different presentations of the free abelian category over <span><math><mi>X</mi></math></span>, we give a new and simple characterization of definable subcategories of <span><math><mi>Mod</mi><mspace></mspace><mtext>-</mtext><mi>X</mi></math></span>, and in particular definable subcategories of modules over rings. In the end, we give a conceptual proof of Auslander-Gruson-Jensen duality, which makes the duality between definable subcategories of left and right module more transparent.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108118"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416437","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 : 2025-11-01Epub Date: 2025-09-17DOI: 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 , defined by the relation , where , with . 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 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 , arising from monomials in the inner part of δ. We also make explicit which finite subgroups of 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.
{"title":"On isotropy groups of quantum plane","authors":"Adriano De Santana , Rene Baltazar , Robson Vinciguerra , Wilian De Araujo","doi":"10.1016/j.jpaa.2025.108095","DOIUrl":"10.1016/j.jpaa.2025.108095","url":null,"abstract":"<div><div>This paper investigates the isotropy groups of derivations on the Quantum Plane <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo></math></span>, defined by the relation <span><math><mi>y</mi><mi>x</mi><mo>=</mo><mi>q</mi><mi>x</mi><mi>y</mi></math></span>, where <span><math><mi>q</mi><mo>∈</mo><msup><mrow><mi>k</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, with <span><math><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>≠</mo><mn>1</mn></math></span>. The main goal is to determine the automorphisms of the Quantum Plane that commutes with a fixed derivation <em>δ</em>. We describe conditions under which the isotropy group <span><math><msub><mrow><mtext>Aut</mtext></mrow><mrow><mi>δ</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is trivial, finite, or infinite, depending on the structure of <em>δ</em> and whether <em>q</em> 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 <span><math><msubsup><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>a</mi></mrow></msubsup><msubsup><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>b</mi></mrow></msubsup><mo>=</mo><mn>1</mn></math></span>, arising from monomials in the inner part of <em>δ</em>. We also make explicit which finite subgroups of <span><math><mi>Aut</mi><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo><mo>)</mo></math></span> are isotropy groups of some derivation: either <em>q</em> root of unity or not. Techniques from algebraic geometry, such as intersection multiplicity, are also employed in the classification of the finite case.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108095"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145109500","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}
The purpose of this paper is twofold. First, we introduce the notions of left-symmetric and left-alternative structures on superspaces in characteristic 2. We describe their main properties and classify them in dimension 2. We show that left-symmetric structures can be queerified if and only if they are left-alternative.
Secondly, we present a method of Lagrangian extension of Lie superalgebras in characteristic 2 with a flat torsion-free connection. We show that any strongly polarized quasi-Frobenius Lie superalgebra can be obtained as a Lagrangian extension. Further, we demonstrate that Lagrangian extensions are classified by a certain cohomology space that we introduce. To illustrate our constructions, all Lagrangian extensions in dimension 4 have been described.
{"title":"Left-symmetric superalgebras and Lagrangian extensions of Lie superalgebras in characteristic 2","authors":"Saïd Benayadi , Sofiane Bouarroudj , Quentin Ehret","doi":"10.1016/j.jpaa.2025.108086","DOIUrl":"10.1016/j.jpaa.2025.108086","url":null,"abstract":"<div><div>The purpose of this paper is twofold. First, we introduce the notions of left-symmetric and left-alternative structures on superspaces in characteristic 2. We describe their main properties and classify them in dimension 2. We show that left-symmetric structures can be queerified if and only if they are left-alternative.</div><div>Secondly, we present a method of Lagrangian extension of Lie superalgebras in characteristic 2 with a flat torsion-free connection. We show that any strongly polarized quasi-Frobenius Lie superalgebra can be obtained as a Lagrangian extension. Further, we demonstrate that Lagrangian extensions are classified by a certain cohomology space that we introduce. To illustrate our constructions, all Lagrangian extensions in dimension 4 have been described.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108086"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145107307","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 : 2025-11-01Epub Date: 2025-10-20DOI: 10.1016/j.jpaa.2025.108109
Thomas Bouchet
Let K be a field of characteristic 0. We present an explicit algorithm that, given the invariants of a generic homogeneous polynomial f under the linear action of or , returns a polynomial differing from f only by a linear change of variables with coefficients in a finite extension of K. Our approach uses the theory of covariants and the Veronese embeddings to characterize the linear equivalence class of a homogeneous polynomial through equations whose coefficients are invariants. As applications, we derive explicit formulas for reconstructing of a generic non-hyperelliptic curve of genus 4 from its invariants, as well as reconstructing generic non-hyperelliptic curves of genus 3 from their Dixmier-Ohno invariants. Formulas for the reconstruction of cubic surfaces from their Salmon-Clebsch invariants. In all of these cases, the coefficients of the reconstructed object lie in its field of moduli.
{"title":"Reconstruction of hypersurfaces from their invariants","authors":"Thomas Bouchet","doi":"10.1016/j.jpaa.2025.108109","DOIUrl":"10.1016/j.jpaa.2025.108109","url":null,"abstract":"<div><div>Let <em>K</em> be a field of characteristic 0. We present an explicit algorithm that, given the invariants of a generic homogeneous polynomial <em>f</em> under the linear action of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> or <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, returns a polynomial differing from <em>f</em> only by a linear change of variables with coefficients in a finite extension of <em>K</em>. Our approach uses the theory of covariants and the Veronese embeddings to characterize the linear equivalence class of a homogeneous polynomial through equations whose coefficients are invariants. As applications, we derive explicit formulas for reconstructing of a generic non-hyperelliptic curve of genus 4 from its invariants, as well as reconstructing generic non-hyperelliptic curves of genus 3 from their Dixmier-Ohno invariants. Formulas for the reconstruction of cubic surfaces from their Salmon-Clebsch invariants. In all of these cases, the coefficients of the reconstructed object lie in its field of moduli.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108109"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145363615","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 : 2025-11-01Epub Date: 2025-10-20DOI: 10.1016/j.jpaa.2025.108114
Andrea Lucchini , Patricia Medina Capilla
Let be the smallest cardinality of a generating set of a finite group G. We give a complete classification of the finite groups with the property that, whenever , for any there exists such that . We also prove that for every finite group G and every maximal subgroup M of G, there exists a generating set for G of minimal size in which at least elements belong to M. We conjecture that the stronger statement holds, that there exists a generating set of size in which only one element does not belong to M, and we prove this conjecture for some suitable choices of M.
{"title":"Finite groups with the minimal generating set exchange property","authors":"Andrea Lucchini , Patricia Medina Capilla","doi":"10.1016/j.jpaa.2025.108114","DOIUrl":"10.1016/j.jpaa.2025.108114","url":null,"abstract":"<div><div>Let <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the smallest cardinality of a generating set of a finite group <em>G</em>. We give a complete classification of the finite groups with the property that, whenever <span><math><mo>〈</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mo>〉</mo><mo>=</mo><mo>〈</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mo>〉</mo><mo>=</mo><mi>G</mi></math></span>, for any <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> there exists <span><math><mn>1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> such that <span><math><mo>〈</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mo>〉</mo><mo>=</mo><mi>G</mi></math></span>. We also prove that for every finite group <em>G</em> and every maximal subgroup <em>M</em> of <em>G</em>, there exists a generating set for <em>G</em> of minimal size in which at least <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>2</mn></math></span> elements belong to <em>M</em>. We conjecture that the stronger statement holds, that there exists a generating set of size <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> in which only one element does not belong to <em>M</em>, and we prove this conjecture for some suitable choices of <em>M</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108114"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145363619","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 : 2025-11-01Epub Date: 2025-09-19DOI: 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 . The degree of the extension field generated over 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 is 2, 3, or 4, the Picard number ρ is necessarily large. Moreover, for an abelian variety X of dimension g with , we establish a structure-type result: X must be isogenous to , where E is an elliptic curve without complex multiplication. In this case, the Picard number satisfies . As a byproduct, we obtain that if is odd, then .
{"title":"On the Picard number and the extension degree of period matrices of complex tori","authors":"Robert Auffarth , Jorge Duque Franco","doi":"10.1016/j.jpaa.2025.108097","DOIUrl":"10.1016/j.jpaa.2025.108097","url":null,"abstract":"<div><div>The rank <em>ρ</em> of the Néron-Severi group of a complex torus <em>X</em> of dimension <em>g</em> satisfies <span><math><mn>0</mn><mo>≤</mo><mi>ρ</mi><mo>≤</mo><msup><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></math></span>. The degree <span><math><mi>d</mi></math></span> of the extension field generated over <span><math><mi>Q</mi></math></span> by the entries of a period matrix of <em>X</em> imposes constraints on its Picard number <em>ρ</em> and, consequently, on the structure of <em>X</em>. In this paper, we show that when <span><math><mi>d</mi></math></span> is 2, 3, or 4, the Picard number <em>ρ</em> is necessarily large. Moreover, for an abelian variety <em>X</em> of dimension <em>g</em> with <span><math><mi>d</mi><mo>=</mo><mn>3</mn></math></span>, we establish a structure-type result: <em>X</em> must be isogenous to <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>g</mi></mrow></msup></math></span>, where <em>E</em> is an elliptic curve without complex multiplication. In this case, the Picard number satisfies <span><math><mi>ρ</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>g</mi><mo>(</mo><mi>g</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. As a byproduct, we obtain that if <span><math><mi>d</mi></math></span> is odd, then <span><math><mi>ρ</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mi>g</mi><mo>(</mo><mi>g</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108097"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145159547","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 : 2025-11-01Epub Date: 2025-10-06DOI: 10.1016/j.jpaa.2025.108107
Suprajo Das , Saipriya Dubey , Sudeshna Roy , Jugal K. Verma
This article investigates the computational aspects of the ε-multiplicity. Primarily, we show that the ε-multiplicity of a homogeneous ideal I in a two-dimensional standard graded domain of finite type over an algebraically closed field of arbitrary characteristic, is always a rational number. In this situation, we produce a formula for the ε-multiplicity of I in terms of certain mixed multiplicities associated to I. In any dimension, under the assumptions that the saturated Rees algebra of I is finitely generated, we give a different expression of the ε-multiplicity in terms of mixed multiplicities by using the Veronese degree. This enabled us to make various explicit computations of ε-multiplicities. We further write a Macaulay2 algorithm to compute ε-multiplicity (under the Noetherian hypotheses) even when the base ring is not necessarily standard graded.
{"title":"Computing epsilon multiplicities in graded algebras","authors":"Suprajo Das , Saipriya Dubey , Sudeshna Roy , Jugal K. Verma","doi":"10.1016/j.jpaa.2025.108107","DOIUrl":"10.1016/j.jpaa.2025.108107","url":null,"abstract":"<div><div>This article investigates the computational aspects of the <em>ε</em>-multiplicity. Primarily, we show that the <em>ε</em>-multiplicity of a homogeneous ideal <em>I</em> in a two-dimensional standard graded domain of finite type over an algebraically closed field of arbitrary characteristic, is always a rational number. In this situation, we produce a formula for the <em>ε</em>-multiplicity of <em>I</em> in terms of certain mixed multiplicities associated to <em>I</em>. In any dimension, under the assumptions that the saturated Rees algebra of <em>I</em> is finitely generated, we give a different expression of the <em>ε</em>-multiplicity in terms of mixed multiplicities by using the Veronese degree. This enabled us to make various explicit computations of <em>ε</em>-multiplicities. We further write a Macaulay2 algorithm to compute <em>ε</em>-multiplicity (under the Noetherian hypotheses) even when the base ring is not necessarily standard graded.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108107"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268345","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 : 2025-11-01Epub Date: 2025-09-12DOI: 10.1016/j.jpaa.2025.108088
Edwin J. Beggs, James E. Blake
We construct a Leray-Serre spectral sequence for fibre bundles for de Rham sheaf cohomology on noncommutative algebras. The morphisms are bimodules with zero-curvature extendable bimodule connections. This generalises definitions involving differentiable algebra maps to differentiable completely positive maps by using the KSGNS construction and Hilbert -bimodules with bimodule connections. We give examples of noncommutative fibre bundles, involving group algebras, matrix algebras, and the quantum torus.
在非交换代数上构造了de Rham轴上同调的纤维束的Leray-Serre谱序列。态射是具有零曲率可扩展双模连接的双模。利用KSGNS构造和具有双模连接的Hilbert C - C -双模，将涉及可微代数映射的定义推广到可微的完全正映射。我们给出了非交换纤维束的例子，涉及群代数、矩阵代数和量子环面。
{"title":"Noncommutative fibre bundles via bimodules","authors":"Edwin J. Beggs, James E. Blake","doi":"10.1016/j.jpaa.2025.108088","DOIUrl":"10.1016/j.jpaa.2025.108088","url":null,"abstract":"<div><div>We construct a Leray-Serre spectral sequence for fibre bundles for de Rham sheaf cohomology on noncommutative algebras. The morphisms are bimodules with zero-curvature extendable bimodule connections. This generalises definitions involving differentiable algebra maps to differentiable completely positive maps by using the KSGNS construction and Hilbert <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-bimodules with bimodule connections. We give examples of noncommutative fibre bundles, involving group algebras, matrix algebras, and the quantum torus.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108088"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145107306","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 : 2025-11-01Epub Date: 2025-09-29DOI: 10.1016/j.jpaa.2025.108100
Mark Colarusso , Sam Evens
<div><div>Let <span><math><mi>G</mi><mo>=</mo><mi>G</mi><mi>L</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> complex general linear group and let <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be its flag variety. The standard Borel subgroup <em>B</em> of upper triangular matrices acts on the product <span><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></span> diagonally with finitely many orbits. In this paper, we study the <em>B</em>-orbits on the subvarieties <span><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></span>, where <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is the <em>B</em>-orbit on <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> containing the line through the origin in the direction of the <em>i</em>-th standard basis vector of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. For each <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></math></span>, we construct a bijection between <em>B</em>-orbits on <span><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></span> and certain pairs of Schubert cells in <span><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></span>. We also show that this bijection can be used to understand the Richardson-Springer monoid action on such <em>B</em>-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 <span><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></span> and <span><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></span>. In the sequel to this paper, we use the results of this paper to construct a correspondence between <em>B</em>-orbits on <span><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>
{"title":"Orbits on a product of two flags and a line and the Bruhat order, I","authors":"Mark Colarusso , Sam Evens","doi":"10.1016/j.jpaa.2025.108100","DOIUrl":"10.1016/j.jpaa.2025.108100","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi><mo>=</mo><mi>G</mi><mi>L</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> complex general linear group and let <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be its flag variety. The standard Borel subgroup <em>B</em> of upper triangular matrices acts on the product <span><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></span> diagonally with finitely many orbits. In this paper, we study the <em>B</em>-orbits on the subvarieties <span><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></span>, where <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is the <em>B</em>-orbit on <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> containing the line through the origin in the direction of the <em>i</em>-th standard basis vector of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. For each <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></math></span>, we construct a bijection between <em>B</em>-orbits on <span><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></span> and certain pairs of Schubert cells in <span><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></span>. We also show that this bijection can be used to understand the Richardson-Springer monoid action on such <em>B</em>-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 <span><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></span> and <span><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></span>. In the sequel to this paper, we use the results of this paper to construct a correspondence between <em>B</em>-orbits on <span><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>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108100"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145222120","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 : 2025-11-01Epub Date: 2025-10-24DOI: 10.1016/j.jpaa.2025.108111
Dario Spirito
We generalize the theory of radical factorization from almost Dedekind domain to strongly discrete Prüfer domains; we show that, for a fixed subset X of maximal ideals, the finitely generated ideals with have radical factorization if and only if X contains no critical maximal ideals with respect to X. We use these notions to prove that the group of the invertible ideals of a strongly discrete Prüfer domain is often free: in particular, we show it is free when the spectrum of D is Noetherian or when D is a ring of integer-valued polynomials on a subset over a Dedekind domain.
{"title":"Radical factorization in higher dimension","authors":"Dario Spirito","doi":"10.1016/j.jpaa.2025.108111","DOIUrl":"10.1016/j.jpaa.2025.108111","url":null,"abstract":"<div><div>We generalize the theory of radical factorization from almost Dedekind domain to strongly discrete Prüfer domains; we show that, for a fixed subset <em>X</em> of maximal ideals, the finitely generated ideals with <span><math><mi>V</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>⊆</mo><mi>X</mi></math></span> have radical factorization if and only if <em>X</em> contains no critical maximal ideals with respect to <em>X</em>. We use these notions to prove that the group <span><math><mrow><mi>Inv</mi></mrow><mo>(</mo><mi>D</mi><mo>)</mo></math></span> of the invertible ideals of a strongly discrete Prüfer domain is often free: in particular, we show it is free when the spectrum of <em>D</em> is Noetherian or when <em>D</em> is a ring of integer-valued polynomials on a subset over a Dedekind domain.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108111"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145424588","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}
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