Journal of Algebra最新文献

{"title":"On squarefree powers of simplicial trees","authors":"Elshani Kamberi,&nbsp;Francesco Navarra,&nbsp;Ayesha Asloob Qureshi","doi":"10.1016/j.jalgebra.2026.01.008","DOIUrl":"10.1016/j.jalgebra.2026.01.008","url":null,"abstract":"<div><div>In this article, we study the squarefree powers of facet ideals associated with simplicial trees. Specifically, we examine the linearity of their minimal free resolution and their regularity. Additionally, we investigate when the first syzygy module of squarefree powers of facet ideal of a simplicial tree is generated by linear relations. Finally, we provide a combinatorial formula for the regularity of the squarefree powers of <em>t</em>-path ideals of path graphs.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 240-276"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025652","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}

{"title":"The spectrum of local dualisable modular representations","authors":"Dave Benson ,&nbsp;Srikanth B. Iyengar ,&nbsp;Henning Krause ,&nbsp;Julia Pevtsova","doi":"10.1016/j.jalgebra.2026.01.004","DOIUrl":"10.1016/j.jalgebra.2026.01.004","url":null,"abstract":"<div><div>For a point <span><math><mi>p</mi></math></span> in the spectrum of the cohomology ring of a finite group <em>G</em> over a field <em>k</em>, we calculate the spectrum for the subcategory of dualisable objects inside the tensor triangulated category of <span><math><mi>p</mi></math></span>-local and <span><math><mi>p</mi></math></span>-torsion objects in the (big) stable module category of the group algebra <em>kG</em>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 78-97"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146001823","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}

{"title":"Prismatic Kunz's theorem","authors":"Ryo Ishizuka ,&nbsp;Kei Nakazato","doi":"10.1016/j.jalgebra.2026.01.010","DOIUrl":"10.1016/j.jalgebra.2026.01.010","url":null,"abstract":"<div><div>In this paper, we prove “prismatic Kunz's theorem” which states that a complete Noetherian local ring <em>R</em> of residue characteristic <em>p</em> is a regular local ring if and only if the Frobenius lift on a prismatic complex of (a derived enhancement of) <em>R</em> over a specific prism <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>I</mi><mo>)</mo></math></span> is faithfully flat. This generalizes classical Kunz's theorem from the perspective of extending the “Frobenius map” to mixed characteristic rings. Our approach involves studying the deformation problem of the “regularity” of prisms and demonstrating the faithful flatness of the structure map of the prismatic complex.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 732-769"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146075494","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}

{"title":"Hyperplane arrangements and Vinberg's θ-groups","authors":"Filippo Ambrosio ,&nbsp;Andrea Santi","doi":"10.1016/j.jalgebra.2025.12.023","DOIUrl":"10.1016/j.jalgebra.2025.12.023","url":null,"abstract":"<div><div>Let <span><math><mi>g</mi><mo>=</mo><msub><mrow><mo>⨁</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>Z</mi><mo>/</mo><mi>m</mi><mi>Z</mi></mrow></msub><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> be a periodically graded semisimple complex Lie algebra. In this note, we give a uniform proof of the recent result by W. de Graaf and H. V. Lê that the hyperplane arrangement determined by the restrictions of the roots of <span><math><mi>g</mi></math></span> to a Cartan subspace <span><math><mi>c</mi><mo>⊂</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> coincides with the hyperplane arrangement of (complex) reflections of the little Weyl group of <span><math><mi>g</mi><mo>=</mo><msub><mrow><mo>⨁</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>Z</mi><mo>/</mo><mi>m</mi><mi>Z</mi></mrow></msub><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 770-789"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146075559","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}

{"title":"Set-theoretic complete intersection for smooth surfaces in a smooth affine algebra","authors":"Lisa Mandal,&nbsp;Md. Ali Zinna","doi":"10.1016/j.jalgebra.2025.12.026","DOIUrl":"10.1016/j.jalgebra.2025.12.026","url":null,"abstract":"<div><div>In this article we prove the following results:</div><div>(1) A smooth surface in a smooth affine <span><math><msub><mrow><mover><mrow><mi>F</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>p</mi></mrow></msub></math></span>-algebra with trivial conormal bundle is a set theoretic complete intersection, if its class in the Grothendieck group is torsion.</div><div>(2) A smooth hypersurface in an affine variety of dimension <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span> over <span><math><msub><mrow><mover><mrow><mi>F</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>p</mi></mrow></msub></math></span> can be described set theoretically by <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> equations.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 36-53"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146001820","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}

{"title":"Virtual first Betti number of GGS groups","authors":"Andrew Ng","doi":"10.1016/j.jalgebra.2026.01.002","DOIUrl":"10.1016/j.jalgebra.2026.01.002","url":null,"abstract":"<div><div>We observe a criterion for groups to have vanishing virtual first Betti number and use it to give infinitely many examples of torsion-free, finitely generated, residually finite groups which aren't virtually diffuse. This answers a question raised by Kionke and Raimbault.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 54-61"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146001821","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}

{"title":"Rationality patterns","authors":"Takuma Hayashi","doi":"10.1016/j.jalgebra.2026.01.013","DOIUrl":"10.1016/j.jalgebra.2026.01.013","url":null,"abstract":"<div><div>In this paper, we establish general categorical frameworks that extend Loewy's classification scheme for finite-dimensional real irreducible representations of groups and Borel–Tits' criterion for the existence of rational forms of representations of <span><math><mover><mrow><mi>F</mi></mrow><mrow><mo>¯</mo></mrow></mover><msub><mrow><mo>⊗</mo></mrow><mrow><mi>F</mi></mrow></msub><mi>G</mi></math></span> for a connected reductive algebraic group <em>G</em> over a field <em>F</em> of characteristic zero and its algebraic closure <span><math><mover><mrow><mi>F</mi></mrow><mrow><mo>¯</mo></mrow></mover></math></span>. We also discuss applications of these general formalisms to the theory of Harish-Chandra modules, specifically to classify irreducible Harish-Chandra modules over fields <em>F</em> of characteristic zero and to identify smaller fields of definition of irreducible Harish-Chandra modules over <span><math><mover><mrow><mi>F</mi></mrow><mrow><mo>¯</mo></mrow></mover></math></span>, particularly in the case of cohomological irreducible essentially unitarizable modules.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 157-212"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025650","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}

{"title":"Sylow subgroups for distinct primes and intersection of nilpotent subgroups","authors":"Francesca Lisi,&nbsp;Luca Sabatini","doi":"10.1016/j.jalgebra.2025.12.028","DOIUrl":"10.1016/j.jalgebra.2025.12.028","url":null,"abstract":"<div><div>Let <em>G</em> be a finite group and let <span><math><msubsup><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> be Sylow subgroups for distinct primes <span><math><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We conjecture that there exists <span><math><mi>x</mi><mo>∈</mo><mi>G</mi></math></span> such that <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∩</mo><msubsup><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>x</mi></mrow></msubsup></math></span> is inclusion-minimal in <span><math><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∩</mo><msubsup><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>g</mi></mrow></msubsup><mo>:</mo><mi>g</mi><mo>∈</mo><mi>G</mi><mo>}</mo></math></span> for all <em>i</em>. As a first step in this direction, we show that a finite group cannot be covered by (proper) Sylow normalizers for distinct primes. Then we settle the conjecture in two opposite situations: symmetric and alternating groups of large degree and metanilpotent groups of odd order. Applications concerning the intersections of nilpotent subgroups are discussed.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 824-837"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146170546","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}

{"title":"Presentations for the ghost algebra and the label algebra","authors":"Madeline Nurcombe","doi":"10.1016/j.jalgebra.2025.12.027","DOIUrl":"10.1016/j.jalgebra.2025.12.027","url":null,"abstract":"<div><div>The ghost algebra is a two-boundary extension of the Temperley–Lieb algebra, constructed recently via a diagrammatic presentation. The existing two-boundary Temperley–Lieb algebra has a basis of two-boundary string diagrams, where the number of strings connected to each boundary must be even. The ghost algebra is similar, but allows this number to be odd, using bookkeeping dots called <em>ghosts</em> to assign a consistent parity to each string endpoint on each boundary. Equivalently, one can discard the ghosts and label each string endpoint with its parity; the resulting algebra is readily generalised to allow any number of possible labels, instead of just odd or even. We call the generalisation the <em>label algebra</em>, and establish a non-diagrammatic presentation for it. A similar presentation for the ghost algebra follows from this.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 611-682"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146075491","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}

{"title":"On the solutions of the generalized Fermat equation over totally real number fields","authors":"Satyabrat Sahoo","doi":"10.1016/j.jalgebra.2026.01.006","DOIUrl":"10.1016/j.jalgebra.2026.01.006","url":null,"abstract":"<div><div>Let <em>K</em> be a totally real number field and <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> be the ring of integers of <em>K</em>. In this article, we study the asymptotic solutions of the generalized Fermat equation, namely <span><math><mi>A</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mrow><mi>y</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>C</mi><msup><mrow><mi>z</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><mn>0</mn></math></span> over <em>K</em> with prime exponent <em>p</em>, where <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>∈</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span> with <em>ABC</em> is even. For certain class of fields <em>K</em>, we prove that the equation <span><math><mi>A</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mrow><mi>y</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>C</mi><msup><mrow><mi>z</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><mn>0</mn></math></span> has no asymptotic solution <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo><mo>∈</mo><msubsup><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msubsup></math></span> with <span><math><mn>2</mn><mo>|</mo><mi>a</mi><mi>b</mi><mi>c</mi></math></span>. Then, under some assumptions on <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi></math></span>, we also prove that <span><math><mi>A</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mrow><mi>y</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>C</mi><msup><mrow><mi>z</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><mn>0</mn></math></span> has no asymptotic solution in <span><math><msup><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. Finally, we give several purely local criteria of <em>K</em> such that <span><math><mi>A</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mrow><mi>y</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>C</mi><msup><mrow><mi>z</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><mn>0</mn></math></span> has no asymptotic solutions in <span><math><msup><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, and calculate the density of such fields <em>K</em> when <em>K</em> is a real quadratic field.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 690-709"},"PeriodicalIF":0.8,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146075493","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}