Pub 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":"2025-09-12","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 : 2025-09-11DOI: 10.1016/j.jpaa.2025.108089
Thomas Michael Keller , Gavin Pettigrew , Saskia Solotko , Lixin Zheng
For a finite group G, the vertices of the prime graph are the primes that divide , and two vertices p and q are adjacent if and only if there is an element of order pq in G. Prime graphs of solvable groups as well as groups whose noncyclic composition factors have order divisible by exactly three distinct primes have been classified in graph-theoretic terms. In this paper, we begin to develop a general theory on the existence of edges in the prime graph of an arbitrary T-solvable group, that is, a group whose composition factors are cyclic or isomorphic to a fixed nonabelian simple group T. We then apply these results to classify the prime graphs of T-solvable groups for, in a suitable sense, most T such that has exactly four prime divisors. We find that these groups almost always have a 3-colorable prime graph complement containing few possible triangles.
{"title":"Classifying prime graphs of finite groups – a methodical approach","authors":"Thomas Michael Keller , Gavin Pettigrew , Saskia Solotko , Lixin Zheng","doi":"10.1016/j.jpaa.2025.108089","DOIUrl":"10.1016/j.jpaa.2025.108089","url":null,"abstract":"<div><div>For a finite group <em>G</em>, the vertices of the prime graph <span><math><mi>Γ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> are the primes that divide <span><math><mo>|</mo><mi>G</mi><mo>|</mo></math></span>, and two vertices <em>p</em> and <em>q</em> are adjacent if and only if there is an element of order <em>pq</em> in <em>G</em>. Prime graphs of solvable groups as well as groups whose noncyclic composition factors have order divisible by exactly three distinct primes have been classified in graph-theoretic terms. In this paper, we begin to develop a general theory on the existence of edges in the prime graph of an arbitrary <em>T</em>-solvable group, that is, a group whose composition factors are cyclic or isomorphic to a fixed nonabelian simple group <em>T</em>. We then apply these results to classify the prime graphs of <em>T</em>-solvable groups for, in a suitable sense, most <em>T</em> such that <span><math><mo>|</mo><mi>T</mi><mo>|</mo></math></span> has exactly four prime divisors. We find that these groups almost always have a 3-colorable prime graph complement containing few possible triangles.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108089"},"PeriodicalIF":0.8,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145107305","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 : 2025-09-10DOI: 10.1016/j.jpaa.2025.108090
Jun Hu , Huansheng Li , Shuo Li
In this paper, we use the cyclotomic Mackey decomposition and branching rules of the seminormal bases of the semisimple cyclotomic Hecke algebras of type to give a new approach to computing the Schur element of for each ℓ-partition . Our formulae give a simple recursive relation between the Schur element and the Schur element of , where . We give our main results for both the non-degenerate and the degenerate cyclotomic Hecke algebras of type . The formulae of the Schur element that we derived are different superficially from all the known formulae in the literature.
{"title":"New formulae for the Schur elements of the cyclotomic Hecke algebra of type G(ℓ,1,n)","authors":"Jun Hu , Huansheng Li , Shuo Li","doi":"10.1016/j.jpaa.2025.108090","DOIUrl":"10.1016/j.jpaa.2025.108090","url":null,"abstract":"<div><div>In this paper, we use the cyclotomic Mackey decomposition and branching rules of the seminormal bases of the semisimple cyclotomic Hecke algebras of type <span><math><mi>G</mi><mo>(</mo><mi>ℓ</mi><mo>,</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span> to give a new approach to computing the Schur element <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span> of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>ℓ</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> for each <em>ℓ</em>-partition <span><math><mi>λ</mi><mo>∈</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>ℓ</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>. Our formulae give a simple recursive relation between the Schur element <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span> and the Schur element <span><math><msub><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mo>↓</mo><mo>≤</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msub></mrow></msub></math></span> of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>ℓ</mi><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>, where <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mo>↓</mo><mo>≤</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msub><mo>:</mo><mo>=</mo><mi>Shape</mi><mo>(</mo><msub><mrow><mo>(</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>λ</mi></mrow></msup><mo>)</mo></mrow><mrow><mo>↓</mo><mo>≤</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msub><mo>)</mo></math></span>. We give our main results for both the non-degenerate and the degenerate cyclotomic Hecke algebras of type <span><math><mi>G</mi><mo>(</mo><mi>ℓ</mi><mo>,</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span>. The formulae of the Schur element that we derived are different superficially from all the known formulae in the literature.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108090"},"PeriodicalIF":0.8,"publicationDate":"2025-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145061192","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 : 2025-09-10DOI: 10.1016/j.jpaa.2025.108087
Li Guo , Yunnan Li , Yunhe Sheng , Rong Tang
The celebrated Milnor-Moore theorem and the more general Cartier-Kostant-Milnor-Moore theorem establish close interconnections among a connected and a pointed cocommutative Hopf algebra, its Lie algebra of primitive elements, and its group of group-like elements. Crossed homomorphisms for Lie algebras, groups and Hopf algebras have been studied extensively, first from a cohomological perspective and then more broadly, with an important case given by difference operators. In this paper, we show that the relationship among the different algebraic structures captured in the Milnor-Moore theorem can be strengthened to include crossed homomorphisms and difference operators, and we also give a graph characterization of Hopf algebra crossed homomorphisms which is compatible with that of the corresponding Lie algebras via the Milnor-Moore theorem. Finally we obtain a Cartier-Kostant-Milnor-Moore type structure theorem for pointed cocommutative difference Hopf algebras. Examples and classifications of difference operators are provided for several Hopf algebras.
{"title":"Crossed homomorphisms and Cartier-Kostant-Milnor-Moore theorem for difference Hopf algebras","authors":"Li Guo , Yunnan Li , Yunhe Sheng , Rong Tang","doi":"10.1016/j.jpaa.2025.108087","DOIUrl":"10.1016/j.jpaa.2025.108087","url":null,"abstract":"<div><div>The celebrated Milnor-Moore theorem and the more general Cartier-Kostant-Milnor-Moore theorem establish close interconnections among a connected and a pointed cocommutative Hopf algebra, its Lie algebra of primitive elements, and its group of group-like elements. Crossed homomorphisms for Lie algebras, groups and Hopf algebras have been studied extensively, first from a cohomological perspective and then more broadly, with an important case given by difference operators. In this paper, we show that the relationship among the different algebraic structures captured in the Milnor-Moore theorem can be strengthened to include crossed homomorphisms and difference operators, and we also give a graph characterization of Hopf algebra crossed homomorphisms which is compatible with that of the corresponding Lie algebras via the Milnor-Moore theorem. Finally we obtain a Cartier-Kostant-Milnor-Moore type structure theorem for pointed cocommutative difference Hopf algebras. Examples and classifications of difference operators are provided for several Hopf algebras.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108087"},"PeriodicalIF":0.8,"publicationDate":"2025-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145050627","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 : 2025-09-09DOI: 10.1016/j.jpaa.2025.108085
Marino Gran , Thomas Letourmy , Leandro Vendramin
The variety of skew braces contains several interesting subcategories as subvarieties, as for instance the varieties of radical rings, of groups and of abelian groups. In this article the methods of non-abelian homological algebra are applied to establish some new Hopf formulae for homology of skew braces, where the coefficient functors are the reflectors from the variety of skew braces to each of the three above-mentioned subvarieties. The corresponding central extensions of skew braces are characterized in purely algebraic terms, leading to some new results, such as an explicit Stallings–Stammbach exact sequence associated with any exact sequence of skew braces, and a new result concerning central series.
{"title":"Hopf formulae for homology of skew braces","authors":"Marino Gran , Thomas Letourmy , Leandro Vendramin","doi":"10.1016/j.jpaa.2025.108085","DOIUrl":"10.1016/j.jpaa.2025.108085","url":null,"abstract":"<div><div>The variety of skew braces contains several interesting subcategories as subvarieties, as for instance the varieties of radical rings, of groups and of abelian groups. In this article the methods of non-abelian homological algebra are applied to establish some new Hopf formulae for homology of skew braces, where the coefficient functors are the reflectors from the variety of skew braces to each of the three above-mentioned subvarieties. The corresponding central extensions of skew braces are characterized in purely algebraic terms, leading to some new results, such as an explicit Stallings–Stammbach exact sequence associated with any exact sequence of skew braces, and a new result concerning central series.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108085"},"PeriodicalIF":0.8,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145050626","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":"2025-09-08","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 : 2025-09-08DOI: 10.1016/j.jpaa.2025.108084
Himanshu Rewri, Surjeet Kour
In this paper, we study the isotropy group of Lotka-Volterra derivations of , i.e., a derivation d of the form . If or , we show that the isotropy group of d is finite. However, for , it is observed that the isotropy group of d need not be finite. Indeed, for , we identify an infinite collection of automorphisms in the isotropy group of d. Moreover, for , we show that the isotropy group of d is isomorphic to the dihedral group of order 2n.
{"title":"Isotropy group of Lotka-Volterra derivations","authors":"Himanshu Rewri, Surjeet Kour","doi":"10.1016/j.jpaa.2025.108084","DOIUrl":"10.1016/j.jpaa.2025.108084","url":null,"abstract":"<div><div>In this paper, we study the isotropy group of Lotka-Volterra derivations of <span><math><mi>K</mi><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>n</mi></mrow></msub><mo>]</mo></math></span>, i.e., a derivation <em>d</em> of the form <span><math><mi>d</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span>. If <span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span> or <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>, we show that the isotropy group of <em>d</em> is finite. However, for <span><math><mi>n</mi><mo>=</mo><mn>4</mn></math></span>, it is observed that the isotropy group of <em>d</em> need not be finite. Indeed, for <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mo>−</mo><mn>1</mn></math></span>, we identify an infinite collection of automorphisms in the isotropy group of <em>d</em>. Moreover, for <span><math><mi>n</mi><mo>≥</mo><mn>3</mn><mo>,</mo><mspace></mspace><mspace></mspace><mtext>and</mtext><mspace></mspace><mspace></mspace><msub><mrow><mi>C</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn>1</mn></math></span>, we show that the isotropy group of <em>d</em> is isomorphic to the dihedral group of order 2<em>n</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108084"},"PeriodicalIF":0.8,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145107308","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 : 2025-09-05DOI: 10.1016/j.jpaa.2025.108083
Kürşat Sözer , Alexis Virelizier
Given a crossed module χ, we introduce Hopf χ-(co)algebras which generalize Hopf algebras and Hopf group-(co)algebras. We interpret them as Hopf algebras in some symmetric monoidal category. We prove that their categories of representations are monoidal and χ-graded (meaning that both objects and morphisms have degrees which are related via χ).
{"title":"Hopf crossed module (co)algebras","authors":"Kürşat Sözer , Alexis Virelizier","doi":"10.1016/j.jpaa.2025.108083","DOIUrl":"10.1016/j.jpaa.2025.108083","url":null,"abstract":"<div><div>Given a crossed module <em>χ</em>, we introduce Hopf <em>χ</em>-(co)algebras which generalize Hopf algebras and Hopf group-(co)algebras. We interpret them as Hopf algebras in some symmetric monoidal category. We prove that their categories of representations are monoidal and <em>χ</em>-graded (meaning that both objects and morphisms have degrees which are related via <em>χ</em>).</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108083"},"PeriodicalIF":0.8,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145027711","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 : 2025-09-03DOI: 10.1016/j.jpaa.2025.108082
Antonino Ficarra , Somayeh Moradi
In this paper, we investigate the componentwise linearity and the Castelnuovo-Mumford regularity of symbolic powers of polymatroidal ideals. For a polymatroidal ideal I, we conjecture that every symbolic power is componentwise linear and for all . We prove that for all when I has no embedded associated primes, for instance if I is a matroidal ideal. Moreover, we establish a criterion on the symbolic Rees algebra of a monomial ideal of minimal intersection type which guarantees that every symbolic power has linear quotients and hence, is componentwise linear for all . By applying our criterion to squarefree Veronese ideals and certain matching-matroidal ideals, we verify both conjectures for these families. We establish the Conforti-Cornuéjols conjecture for any matroidal ideal, and we show that a matroidal ideal is packed if and only if it is the product of monomial prime ideals with pairwise disjoint supports. Furthermore, we identify several classes of non-squarefree polymatroidal ideals for which the ordinary and symbolic powers coincide. Hence, we confirm our conjectures for transversal polymatroidal ideals and principal Borel ideals. Finally, we verify our conjectures for all polymatroidal ideals either generated in small degrees or in a small number of variables.
{"title":"Symbolic powers of polymatroidal ideals","authors":"Antonino Ficarra , Somayeh Moradi","doi":"10.1016/j.jpaa.2025.108082","DOIUrl":"10.1016/j.jpaa.2025.108082","url":null,"abstract":"<div><div>In this paper, we investigate the componentwise linearity and the Castelnuovo-Mumford regularity of symbolic powers of polymatroidal ideals. For a polymatroidal ideal <em>I</em>, we conjecture that every symbolic power <span><math><msup><mrow><mi>I</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> is componentwise linear and<span><span><span><math><mtext>reg</mtext><mspace></mspace><msup><mrow><mi>I</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup><mo>=</mo><mtext>reg</mtext><mspace></mspace><msup><mrow><mi>I</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span></span></span> for all <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span>. We prove that <span><math><mtext>reg</mtext><mspace></mspace><msup><mrow><mi>I</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup><mo>≥</mo><mtext>reg</mtext><mspace></mspace><msup><mrow><mi>I</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> for all <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span> when <em>I</em> has no embedded associated primes, for instance if <em>I</em> is a matroidal ideal. Moreover, we establish a criterion on the symbolic Rees algebra <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo></math></span> of a monomial ideal of minimal intersection type which guarantees that every symbolic power <span><math><msup><mrow><mi>I</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> has linear quotients and hence, is componentwise linear for all <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span>. By applying our criterion to squarefree Veronese ideals and certain matching-matroidal ideals, we verify both conjectures for these families. We establish the Conforti-Cornuéjols conjecture for any matroidal ideal, and we show that a matroidal ideal is packed if and only if it is the product of monomial prime ideals with pairwise disjoint supports. Furthermore, we identify several classes of non-squarefree polymatroidal ideals for which the ordinary and symbolic powers coincide. Hence, we confirm our conjectures for transversal polymatroidal ideals and principal Borel ideals. Finally, we verify our conjectures for all polymatroidal ideals either generated in small degrees or in a small number of variables.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 10","pages":"Article 108082"},"PeriodicalIF":0.8,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144996518","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 : 2025-09-03DOI: 10.1016/j.jpaa.2025.108078
Dave Murphy
We classify thick subcategories in a Paquette-Yıldırım completion of a discrete cluster category of Dynkin type . To do this we introduce the notion of homologically connected objects, and the hc (=homologically connected) decomposition of an object into homologically connected objects in a Hom-finite, Krull-Schmidt triangulated category. We show that any object in a has a hc decomposition, and that the hc decomposition determines the thick closure of an object. Moreover, we use this result to classify the classical generators of as homologically connected objects satisfying a maximality condition.
Every homologically connected object has an invariant, known as the homological length, and we show that in this homological length is an upper bound for the generation time of a classical generator. This allows us to provide an upper bound for the Orlov spectrum of , as well as giving the Rouquier dimension.
{"title":"Bounding the Orlov spectrum for a completion of discrete cluster categories","authors":"Dave Murphy","doi":"10.1016/j.jpaa.2025.108078","DOIUrl":"10.1016/j.jpaa.2025.108078","url":null,"abstract":"<div><div>We classify thick subcategories in a Paquette-Yıldırım completion <span><math><mover><mrow><mi>C</mi></mrow><mo>‾</mo></mover></math></span> of a discrete cluster category of Dynkin type <span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>. To do this we introduce the notion of homologically connected objects, and the hc (=homologically connected) decomposition of an object into homologically connected objects in a Hom-finite, Krull-Schmidt triangulated category. We show that any object in a <span><math><mover><mrow><mi>C</mi></mrow><mo>‾</mo></mover></math></span> has a hc decomposition, and that the hc decomposition determines the thick closure of an object. Moreover, we use this result to classify the classical generators of <span><math><mover><mrow><mi>C</mi></mrow><mo>‾</mo></mover></math></span> as homologically connected objects satisfying a maximality condition.</div><div>Every homologically connected object has an invariant, known as the homological length, and we show that in <span><math><mover><mrow><mi>C</mi></mrow><mo>‾</mo></mover></math></span> this homological length is an upper bound for the generation time of a classical generator. This allows us to provide an upper bound for the Orlov spectrum of <span><math><mover><mrow><mi>C</mi></mrow><mo>‾</mo></mover></math></span>, as well as giving the Rouquier dimension.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 10","pages":"Article 108078"},"PeriodicalIF":0.8,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144988166","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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