Journal of Algebra最新文献

{"title":"Hypertranscendence and linear difference equations, the exponential case","authors":"Thomas Dreyfus","doi":"10.1016/j.jalgebra.2025.08.025","DOIUrl":"10.1016/j.jalgebra.2025.08.025","url":null,"abstract":"<div><div>In this paper we study meromorphic solutions of linear shift difference equations with coefficients in <span><math><mi>C</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> involving the operator <span><math><mi>ρ</mi><mo>:</mo><mi>y</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>↦</mo><mi>y</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>h</mi><mo>)</mo></math></span>, for some <span><math><mi>h</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. We prove that if <em>f</em> is a solution of an algebraic differential equation, then <em>f</em> belongs to a ring that is generated by periodic functions and exponentials. Our proof is based on the parametrized difference Galois theory initiated by Hardouin and Singer.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 775-792"},"PeriodicalIF":0.8,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145045367","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 restricted model for the bounded derived category of gentle algebras","authors":"Esha Gupta","doi":"10.1016/j.jalgebra.2025.08.018","DOIUrl":"10.1016/j.jalgebra.2025.08.018","url":null,"abstract":"<div><div>We present a restricted model for the bounded derived category of gentle algebras that encodes the indecomposable objects and positive extensions between them. The model is then used to count the number of <em>d</em>-term silting objects for linearly oriented <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, recovering the result that they are counted by the Pfaff-Fuss-Catalan numbers.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 635-649"},"PeriodicalIF":0.8,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145019380","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":"Numerical homological regularities over positively graded algebras","authors":"Quanshui Wu,&nbsp;Bojuan Yi","doi":"10.1016/j.jalgebra.2025.08.016","DOIUrl":"10.1016/j.jalgebra.2025.08.016","url":null,"abstract":"<div><div>We study numerical regularities for complexes over noncommutative noetherian locally finite <span><math><mi>N</mi></math></span>-graded algebras <em>A</em> such as CM-regularity, Tor-regularity (Ext-regularity) and Ex-regularity, which are the supremum or infimum degrees of some associated canonical complexes. We introduce their companions—lowercase named regularities, which are defined by taking the infimum or supremum degrees of the respective canonical associated complexes. We show that for any right bounded complex <em>X</em> with finitely generated cohomologies, the supremum degree of <span><math><mi>R</mi><msub><mrow><munder><mrow><mi>Hom</mi></mrow><mo>_</mo></munder></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> coincides with the opposite of the infimum degree of <em>X</em> if <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is semisimple. If <em>A</em> has a balanced dualizing complex and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is semisimple, we prove that the CM-regularity of <em>X</em> coincides with the supremum degree of <span><math><mi>R</mi><msub><mrow><munder><mrow><mi>Hom</mi></mrow><mo>_</mo></munder></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mi>X</mi><mo>)</mo></math></span> for any left bounded complex <em>X</em> with finitely generated cohomologies.</div><div>Several inequalities concerning the numerical regularities and the supremum or infimum degrees of derived Hom or derived tensor complexes are given for noncommutative noetherian locally finite <span><math><mi>N</mi></math></span>-graded algebras. Some of these are generalizations of Jørgensen's results on the inequalities between the CM-regularity and Tor-regularity, some are new even in the connected graded case. Conditions are given under which the inequalities become equalities by establishing two technical lemmas.</div><div>Following Kirkman, Won and Zhang, we also use the numerical AS-regularity (resp. little AS-regularity) to study Artin-Schelter regular property (finite-dimensional property) for noetherian <span><math><mi>N</mi></math></span>-graded algebras. We prove that the numerical AS-regularity of <em>A</em> is zero if and only if that <em>A</em> is an <span><math><mi>N</mi></math></span>-graded AS-regular algebra under some mild conditions, which generalizes a result of Dong-Wu and a result of Kirkman-Won-Zhang. If <em>A</em> has a balanced dualizing complex and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is semisimple, we prove that the little AS-regularity of <em>A</em> is zero if and only if <em>A</em> is finite-dimensional.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 677-748"},"PeriodicalIF":0.8,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145045406","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 index in d-exact categories","authors":"Francesca Fedele ,&nbsp;Peter Jørgensen ,&nbsp;Amit Shah","doi":"10.1016/j.jalgebra.2025.08.021","DOIUrl":"10.1016/j.jalgebra.2025.08.021","url":null,"abstract":"<div><div>Starting from its original definition in module categories with respect to projective modules, the index has played an important role in various aspects of homological algebra, categorification of cluster algebras and <em>K</em>-theory. In the last few years, the notion of index has been generalised to several different contexts in (higher) homological algebra, typically with respect to a (higher) cluster-tilting subcategory <span><math><mi>X</mi></math></span> of the relevant ambient category <span><math><mi>C</mi></math></span>. The recent tools of extriangulated and higher-exangulated categories have permitted some conditions on the subcategory <span><math><mi>X</mi></math></span> to be relaxed. In this paper, we introduce the index with respect to a generating, contravariantly finite subcategory of a <em>d</em>-exact category that has <em>d</em>-kernels. We show that our index has the important property of being additive on <em>d</em>-exact sequences up to an error term.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 814-835"},"PeriodicalIF":0.8,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145105586","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":"Towards the nerves of steel conjecture","authors":"Logan Hyslop","doi":"10.1016/j.jalgebra.2025.08.028","DOIUrl":"10.1016/j.jalgebra.2025.08.028","url":null,"abstract":"<div><div>Given a local ⊗-triangulated category, and a fiber sequence <span><math><mi>y</mi><mover><mrow><mo>→</mo></mrow><mrow><mi>g</mi></mrow></mover><mn>1</mn><mover><mrow><mo>→</mo></mrow><mrow><mi>f</mi></mrow></mover><mi>x</mi></math></span>, one may ask if there is always a nonzero object <em>z</em> such that either <span><math><mi>z</mi><mo>⊗</mo><mi>f</mi></math></span> or <span><math><mi>z</mi><mo>⊗</mo><mi>g</mi></math></span> is ⊗-nilpotent. The claim that this property holds for all local ⊗-triangulated categories is equivalent to Balmer's “nerves of steel conjecture” <span><span>[7, Remark 5.15]</span></span>. In the present paper, we will see how this property can fail if the category we start with is not rigid, discuss a large class of categories where the property holds, and ultimately prove that the nerves of steel conjecture is equivalent to a stronger form of this property.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 544-565"},"PeriodicalIF":0.8,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145026719","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 arithmetic of power monoids","authors":"Laura Cossu ,&nbsp;Salvatore Tringali","doi":"10.1016/j.jalgebra.2025.08.026","DOIUrl":"10.1016/j.jalgebra.2025.08.026","url":null,"abstract":"<div><div>Given a monoid <em>H</em> (written multiplicatively), the family <span><math><msub><mrow><mi>P</mi></mrow><mrow><mrow><mi>fin</mi></mrow><mo>,</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo></math></span> of all non-empty finite subsets of <em>H</em> containing the identity element is itself a monoid, called the reduced finitary power monoid of <em>H</em>, under the operation of setwise multiplication induced by <em>H</em>.</div><div>We investigate the arithmetic of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mrow><mi>fin</mi></mrow><mo>,</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo></math></span> from the perspective of minimal factorizations into irreducibles, paying particular attention to the potential presence of non-trivial idempotents. Among other results, we provide necessary and sufficient conditions on <em>H</em> for <span><math><msub><mrow><mi>P</mi></mrow><mrow><mrow><mi>fin</mi></mrow><mo>,</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo></math></span> to admit unique minimal factorizations.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 793-813"},"PeriodicalIF":0.8,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145060210","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":"Space pairs of the Grassmann algebra: Unexpected pairs with polynomial growth of the codimensions","authors":"Alan Guimarães ,&nbsp;David Levi da Silva Macêdo","doi":"10.1016/j.jalgebra.2025.08.024","DOIUrl":"10.1016/j.jalgebra.2025.08.024","url":null,"abstract":"<div><div>Let <em>K</em> be a field of characteristic zero and <em>E</em> the infinite dimensional Grassmann algebra over <em>K</em>. We determine the weak polynomial identities for pairs <span><math><mo>(</mo><mi>E</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span>, where <em>S</em> is a subspace of <em>E</em> such that <span><math><mi>S</mi><mo>∪</mo><mo>{</mo><mn>1</mn><mo>}</mo></math></span> generates <em>E</em>. Depending on the structure of <em>S</em>, we divide this analysis into four distinct cases. When <span><math><mo>(</mo><mi>E</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span> belongs to the first two types, we show that the codimension sequence is <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>E</mi><mo>,</mo><mi>S</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span> for all <em>n</em>. On the other hand, for certain pairs of the third type, we prove that <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>E</mi><mo>,</mo><mi>S</mi><mo>)</mo><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, i.e., in this case, the codimension sequence coincides with the ordinary (associative) case. We further investigate certain classes of associative pairs <span><math><mo>(</mo><mi>E</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span>, exhibiting polynomial growth, particularly those whose codimension sequences correspond to Young diagrams with unbounded numbers of boxes below the first row. These pairs show that an usual characterization of polynomial growth of the codimensions does not hold in the case of associative-space pairs.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145010260","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":"Symmetric ideals and invariant Hilbert schemes","authors":"Sebastian Debus ,&nbsp;Andreas Kretschmer","doi":"10.1016/j.jalgebra.2025.08.023","DOIUrl":"10.1016/j.jalgebra.2025.08.023","url":null,"abstract":"<div><div>A symmetric ideal is an ideal in a polynomial ring which is stable under all permutations of the variables. In this paper we initiate a global study of zero-dimensional symmetric ideals. By this we mean a geometric study of the invariant Hilbert schemes <span><math><msubsup><mrow><mi>Hilb</mi></mrow><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msubsup><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> parametrizing symmetric subschemes of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> whose coordinate rings, as <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>-modules, are isomorphic to a given representation <em>ρ</em>. In the case that <span><math><mi>ρ</mi><mo>=</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>λ</mi></mrow></msup></math></span> is a permutation module corresponding to certain special types of partitions <em>λ</em> of <em>n</em>, we prove that <span><math><msubsup><mrow><mi>Hilb</mi></mrow><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msubsup><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> is irreducible or even smooth. We also prove irreducibility whenever <span><math><mi>dim</mi><mo>⁡</mo><mi>ρ</mi><mo>≤</mo><mn>2</mn><mi>n</mi></math></span> and the invariant Hilbert scheme is non-empty. In this same range, we classify all homogeneous symmetric ideals and decide which of these define singular points of <span><math><msubsup><mrow><mi>Hilb</mi></mrow><mrow><mi>ρ</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msubsup><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span>. A central tool is the combinatorial theory of higher Specht polynomials.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 595-634"},"PeriodicalIF":0.8,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145026721","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":"Hom-orthogonal modules and brick-Brauer-Thrall conjectures","authors":"Kaveh Mousavand ,&nbsp;Charles Paquette","doi":"10.1016/j.jalgebra.2025.08.027","DOIUrl":"10.1016/j.jalgebra.2025.08.027","url":null,"abstract":"<div><div>For finite dimensional algebras over algebraically closed fields, we study the sets of pairwise Hom-orthogonal modules and obtain new results on some open conjectures on the behaviour of bricks and several related problems, which we generally refer to as brick-Brauer-Thrall (bBT) conjectures. Using some algebraic and geometric tools, and in terms of the notion of Hom-orthogonality, we find necessary and sufficient conditions for the existence of infinite families of bricks of the same dimension. This sheds new light on the bBT conjectures and we prove some of them for new families of algebras. Our results imply some interesting algebraic and geometric characterizations of brick-finite algebras as conceptual generalizations of local algebras. We also verify the bBT conjectures for any algebra whose Auslander-Reiten quiver has a generalized standard component, which particularly extends some results of Chindris-Kinser-Weyman on the algebras with preprojective components.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 650-676"},"PeriodicalIF":0.8,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145026722","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 computational study of certain Weyl modules for type G2 in characteristic 2","authors":"Stephen Doty","doi":"10.1016/j.jalgebra.2025.08.022","DOIUrl":"10.1016/j.jalgebra.2025.08.022","url":null,"abstract":"<div><div>Using the <span>WeylModules</span> <span>GAP</span> Package, we compute structural information about certain Weyl modules for type <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> in characteristic 2. This gives counterexamples to two conjectures stated by S. Donkin in 1990. It also illustrates capabilities of the package, which can in principle be applied to Weyl modules for any simple, simply-connected algebraic group in any characteristic, subject of course to time and space limitations of computational resources.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 503-543"},"PeriodicalIF":0.8,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145019381","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}