Journal of Pure and Applied Algebra最新文献

{"title":"On polynomial invariant rings in modular invariant theory","authors":"Manoj Kummini ,&nbsp;Mandira Mondal","doi":"10.1016/j.jpaa.2024.107758","DOIUrl":"10.1016/j.jpaa.2024.107758","url":null,"abstract":"<div><p>Let <span><math><mi>k</mi></math></span> be a field of characteristic <span><math><mi>p</mi><mo>&gt;</mo><mn>0</mn></math></span>, <em>V</em> a finite-dimensional <span><math><mi>k</mi></math></span>-vector-space, and <em>G</em> a finite <em>p</em>-group acting <span><math><mi>k</mi></math></span>-linearly on <em>V</em>. Let <span><math><mi>S</mi><mo>=</mo><mi>Sym</mi><mspace></mspace><msup><mrow><mi>V</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. Confirming a conjecture of Shank-Wehlau-Broer, we show that if <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>G</mi></mrow></msup></math></span> is a direct summand of <em>S</em>, then <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>G</mi></mrow></msup></math></span> is a polynomial ring, in the following cases:</p><ul><li><span>(a)</span><span><p><span><math><mi>k</mi><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>dim</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>⁡</mo><mi>V</mi><mo>=</mo><mn>4</mn></math></span>; or</p></span></li><li><span>(b)</span><span><p><span><math><mo>|</mo><mi>G</mi><mo>|</mo><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>.</p></span></li></ul> In order to prove the above result, we also show that if <span><math><msub><mrow><mi>dim</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>⁡</mo><msup><mrow><mi>V</mi></mrow><mrow><mi>G</mi></mrow></msup><mo>≥</mo><msub><mrow><mi>dim</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>⁡</mo><mi>V</mi><mo>−</mo><mn>2</mn></math></span>, then the Hilbert ideal <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>S</mi></mrow></msub></math></span> is a complete intersection.</div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551431","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 3-preprojective algebras of type A˜","authors":"Darius Dramburg ,&nbsp;Oleksandra Gasanova","doi":"10.1016/j.jpaa.2024.107760","DOIUrl":"https://doi.org/10.1016/j.jpaa.2024.107760","url":null,"abstract":"<div><p>Let <span><math><mi>G</mi><mo>≤</mo><msub><mrow><mi>SL</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> act on <span><math><mi>R</mi><mo>=</mo><mi>C</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><mo>+</mo><mn>1</mn></mrow></msub><mo>]</mo></math></span> by change of variables. Then, the skew-group algebra <span><math><mi>R</mi><mo>⁎</mo><mi>G</mi></math></span> is bimodule <span><math><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-Calabi-Yau. In certain circumstances, this algebra admits a locally finite-dimensional grading of Gorenstein parameter 1, in which case it is the <span><math><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-preprojective algebra of its <em>n</em>-representation infinite degree 0 piece, as defined in <span>[10]</span>. If the group <em>G</em> is abelian, the <span><math><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-preprojective algebra is said to be of type <span><math><mover><mrow><mi>A</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span>. For a given group <em>G</em>, it is not obvious whether <span><math><mi>R</mi><mo>⁎</mo><mi>G</mi></math></span> admits such a grading making it into an <span><math><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-preprojective algebra. We study the case when <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span> and <em>G</em> is abelian. We give an explicit classification of groups such that <span><math><mi>R</mi><mo>⁎</mo><mi>G</mi></math></span> is 3-preprojective by constructing such gradings. This is possible as long as <em>G</em> is not a subgroup of <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> and not <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>×</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. For a fixed <em>G</em>, the algebra <span><math><mi>R</mi><mo>⁎</mo><mi>G</mi></math></span> admits different 3-preprojective gradings, so we associate a type to a grading and classify all types. Then we show that gradings of the same type are related by a certain kind of mutation. This gives a classification of 2-representation infinite algebras of type <span><math><mover><mrow><mi>A</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span>. The involved quivers are those arising from hexagonal dimer models on the torus, and the gradings we consider correspond to perfect matchings on the dimer, or equivalently to periodic lozenge tilings of the plane. Consequently, we classify these tilings up to flips, which correspond to the mutation we consider.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022404924001579/pdfft?md5=0a6792a213bba8d7f8d057c5a015caf7&pid=1-s2.0-S0022404924001579-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141539785","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}

{"title":"Large automorphism groups of bordered tori","authors":"E. Bujalance ,&nbsp;F.J. Cirre ,&nbsp;J.M. Gamboa","doi":"10.1016/j.jpaa.2024.107757","DOIUrl":"10.1016/j.jpaa.2024.107757","url":null,"abstract":"<div><p>We study large groups of automorphisms of compact orientable bordered Klein surfaces of topological genus one. Here, <em>large</em> means that the order of the group is greater than or equal to <span><math><mn>4</mn><mo>(</mo><mi>g</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>, where <span><math><mi>g</mi><mo>≥</mo><mn>2</mn></math></span> is the algebraic genus of the surface. We find all such groups, providing presentations by means of generators and relations of them. We also determine which of these groups act as the full automorphism group of some bordered torus.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022404924001543/pdfft?md5=7c2f0e2211052948517b0149c23295e8&pid=1-s2.0-S0022404924001543-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551429","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}

{"title":"Isotropy indices of Pfister multiples in characteristic 2","authors":"Nico Lorenz ,&nbsp;Kristýna Zemková","doi":"10.1016/j.jpaa.2024.107759","DOIUrl":"10.1016/j.jpaa.2024.107759","url":null,"abstract":"<div><p>Let <em>F</em> be a field of characteristic 2, <em>π</em> an <em>n</em>-fold bilinear Pfister form over <em>F</em> and <em>φ</em> an arbitrary quadratic form over <em>F</em>. In this note, we investigate Witt index, defect, total isotropy index and higher isotropy indices of <em>φ</em> and <span><math><mi>π</mi><mo>⊗</mo><mi>φ</mi></math></span> and prove relations among the indices of these two forms over certain field extensions.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022404924001567/pdfft?md5=4a793c50fe87144e70295cee7055ed3c&pid=1-s2.0-S0022404924001567-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509314","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}

{"title":"A note on Deligne's formula","authors":"Peter Schenzel","doi":"10.1016/j.jpaa.2024.107754","DOIUrl":"https://doi.org/10.1016/j.jpaa.2024.107754","url":null,"abstract":"<div><p>Let <em>R</em> denote a Noetherian ring and an ideal <span><math><mi>J</mi><mo>⊂</mo><mi>R</mi></math></span> with <span><math><mi>U</mi><mo>=</mo><mi>Spec</mi><mspace></mspace><mi>R</mi><mo>∖</mo><mi>V</mi><mo>(</mo><mi>J</mi><mo>)</mo></math></span>. For an <em>R</em>-module <em>M</em> there is an isomorphism <span><math><mi>Γ</mi><mo>(</mo><mi>U</mi><mo>,</mo><mover><mrow><mi>M</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>)</mo><mo>≅</mo><munder><mi>lim</mi><mo>→</mo></munder><msub><mrow><mi>Hom</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>(</mo><msup><mrow><mi>J</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>M</mi><mo>)</mo></math></span> known as Deligne's formula (see <span>[8, p. 217]</span> and Deligne's Appendix in <span>[7]</span>). We extend the isomorphism for any <em>R</em>-module <em>M</em> in the non-Noetherian case of <em>R</em> and <span><math><mi>J</mi><mo>=</mo><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>k</mi></mrow></msub><mo>)</mo></math></span> a certain finitely generated ideal. Moreover, we recall a corresponding sheaf construction.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022404924001518/pdfft?md5=a8edd94b07a7a3b04be5dc0efdc259af&pid=1-s2.0-S0022404924001518-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141479559","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}

{"title":"Idempotents in nilpotent quotients and triangulated categories","authors":"Teimuraz Pirashvili","doi":"10.1016/j.jpaa.2024.107755","DOIUrl":"https://doi.org/10.1016/j.jpaa.2024.107755","url":null,"abstract":"<div><p>We will prove that if <span><math><mi>I</mi></math></span> is a nilpotent ideal of an additive category <span><math><mi>A</mi></math></span>, then an idempotent <em>e</em> of the category <span><math><mi>A</mi></math></span> splits iff its image splits in <span><math><mi>A</mi><mo>/</mo><mi>I</mi></math></span>. Based on this fact, we give a short proof of a crucial proposition of Le and Chen.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141434279","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":"Multiplicative isomorphisms and derivations on axial algebras","authors":"Bruno L.M. Ferreira ,&nbsp;Douglas de Araujo Smigly ,&nbsp;Elisabete Barreiro","doi":"10.1016/j.jpaa.2024.107753","DOIUrl":"10.1016/j.jpaa.2024.107753","url":null,"abstract":"<div><p>In this paper, we show that the multiplicative derivations on <span><math><mi>J</mi><mo>(</mo><mi>α</mi><mo>)</mo></math></span>-axial algebras, with <span><math><mi>α</mi><mo>≠</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, are additive under suitable conditions, which nowadays are called Martindale-type conditions. Besides, with proper assumptions, we proceed to study the additivity of multiplicative isomorphisms and derivations in the context of <span><math><mi>M</mi><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span>-axial algebras, except for multiplicative derivations when <span><math><mi>β</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. In this case, we mention a research question at the end.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141411796","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":"Diagramatics for cyclic pointed fusion categories","authors":"Agustina Czenky","doi":"10.1016/j.jpaa.2024.107752","DOIUrl":"https://doi.org/10.1016/j.jpaa.2024.107752","url":null,"abstract":"<div><p>We give a parametrization of cyclic pointed categories associated to the cyclic group of order <em>n</em> in terms of <em>n</em>-th roots of unity. We also provide a diagramatic description of these categories by generators and relations, and use it to characterize their 2-group of automorphisms.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141322869","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":"A geometrization of Stanley–Reisner theory","authors":"Fernando Sancho de Salas,&nbsp;Alejandro Torres Sancho","doi":"10.1016/j.jpaa.2024.107743","DOIUrl":"https://doi.org/10.1016/j.jpaa.2024.107743","url":null,"abstract":"<div><p>We give a geometric interpretation of the Stanley–Reisner correspondence, extend it to schemes, and interpret it in terms of the field of one element.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141313827","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 functorial equivalence classes of blocks of finite groups","authors":"Deniz Yılmaz","doi":"10.1016/j.jpaa.2024.107744","DOIUrl":"https://doi.org/10.1016/j.jpaa.2024.107744","url":null,"abstract":"<div><p>Let <em>k</em> be an algebraically closed field of characteristic <span><math><mi>p</mi><mo>&gt;</mo><mn>0</mn></math></span> and let <span><math><mi>F</mi></math></span> be an algebraically closed field of characteristic 0. Recently, together with Bouc, we introduced the notion of functorial equivalences between blocks of finite groups and proved that given a <em>p</em>-group <em>D</em>, there is only a finite number of pairs <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span> of a finite group <em>G</em> and a block <em>b</em> of <em>kG</em> with defect groups isomorphic to <em>D</em>, up to functorial equivalence over <span><math><mi>F</mi></math></span>. In this paper, we classify the functorial equivalence classes over <span><math><mi>F</mi></math></span> of blocks with cyclic defect groups and 2-blocks of defects 2 and 3. In particular, we prove that for all these blocks, the functorial equivalence classes depend only on the fusion system of the block.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141313826","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}