{"title":"Free automorphism groups of K3 surfaces with Picard number 3","authors":"Kenji Hashimoto , Kwangwoo Lee","doi":"10.1016/j.jpaa.2024.107845","DOIUrl":"10.1016/j.jpaa.2024.107845","url":null,"abstract":"<div><div>It is known that the automorphism group of any projective K3 surface is finitely generated. In this paper, we consider a certain kind of K3 surfaces with Picard number 3 whose automorphism groups are isomorphic to congruence subgroups of the modular group <span><math><mi>P</mi><mi>S</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span>. In particular, we show that a free group of arbitrarily large rank appears as the automorphism group of such a K3 surface.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107845"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094022","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":"Minimal and cellular free resolutions over polynomial OI-algebras","authors":"Nathan Fieldsteel , Uwe Nagel","doi":"10.1016/j.jpaa.2024.107856","DOIUrl":"10.1016/j.jpaa.2024.107856","url":null,"abstract":"<div><div>Minimal free resolutions of graded modules over a noetherian polynomial ring have been attractive objects of interest for more than a hundred years. We introduce and study two natural extensions in the setting of graded modules over a polynomial OI-algebra, namely <em>minimal</em> and <em>width-wise minimal</em> free resolutions. A minimal free resolution of an OI-module can be characterized by the fact that the free module in every fixed homological degree, say <em>i</em>, has minimal rank among all free resolutions of the module. We show that any finitely generated graded module over a noetherian polynomial OI-algebra admits a graded minimal free resolution and that it is unique. A width-wise minimal free resolution is a free resolution that provides a minimal free resolution of a module in every width. Such a resolution is necessarily minimal. Its existence is not guaranteed. However, we show that certain monomial OI-ideals do admit width-wise minimal free or, more generally, width-wise minimal flat resolutions. These ideals include families of well-known monomial ideals such as Ferrers ideals and squarefree strongly stable ideals. The arguments rely on the theory of cellular resolutions.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107856"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094024","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}
Daniel Lännström , Patrik Lundström , Johan Öinert , Stefan Wagner
{"title":"Prime group graded rings with applications to partial crossed products and Leavitt path algebras","authors":"Daniel Lännström , Patrik Lundström , Johan Öinert , Stefan Wagner","doi":"10.1016/j.jpaa.2024.107842","DOIUrl":"10.1016/j.jpaa.2024.107842","url":null,"abstract":"<div><div>We generalize a classical result by Passman on primeness of unital strongly group graded rings to the class of nearly epsilon-strongly group graded rings which are not necessarily unital. Using this result, we obtain (i) a characterization of prime <em>s</em>-unital strongly group graded rings, and, in particular, of infinite matrix rings and of group rings over <em>s</em>-unital rings, thereby generalizing a well-known result by Connell; (ii) characterizations of prime <em>s</em>-unital partial skew group rings and of prime unital partial crossed products; (iii) a generalization of the well-known characterizations of prime Leavitt path algebras, by Larki and by Abrams-Bell-Rangaswamy.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107842"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094010","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":"The second homology group of the commutative case of Kontsevich's symplectic derivation Lie algebra","authors":"Shuichi Harako","doi":"10.1016/j.jpaa.2024.107841","DOIUrl":"10.1016/j.jpaa.2024.107841","url":null,"abstract":"<div><div>In 1993, Kontsevich introduced the symplectic derivation Lie algebras related to various geometric objects including moduli spaces of graphs and of Riemann surfaces, graph homologies, Hamiltonian vector fields, etc. Each of them is a graded algebra, so that its Chevalley-Eilenberg chain complex has another <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow></msub></math></span>-grading, called weight, than the usual homological degree. We focus on one of the Lie algebras <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span>, called the “commutative case”, and its positive weight part <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>g</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>⊂</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span>. The symplectic invariant homology of <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>g</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> is closely related to the commutative graph homology, hence some computational results are obtained from the viewpoint of graph homology theory. On the other hand, the details of the entire homology group <span><math><msub><mrow><mi>H</mi></mrow><mrow><mo>•</mo></mrow></msub><mo>(</mo><msubsup><mrow><mi>c</mi></mrow><mrow><mi>g</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>)</mo></math></span> are not completely known. We determine <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msubsup><mrow><mi>c</mi></mrow><mrow><mi>g</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>)</mo></math></span> by decomposing it by weight and using the classical representation theory of the symplectic groups.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107841"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094012","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":"Group-graded twisted Calabi–Yau algebras","authors":"Yasmeen S. Baki","doi":"10.1016/j.jpaa.2024.107849","DOIUrl":"10.1016/j.jpaa.2024.107849","url":null,"abstract":"<div><div>Historically, the study of graded (twisted or otherwise) Calabi–Yau algebras has meant the study of such algebras under an <span><math><mi>N</mi></math></span>-grading. In this paper, we propose a suitable definition for a twisted <em>G</em>-graded Calabi–Yau algebra, for <em>G</em> an arbitrary abelian group. Building on the work of Reyes and Rogalski, we show that a <em>G</em>-graded algebra is twisted Calabi–Yau if and only if it is <em>G</em>-graded twisted Calabi–Yau. In the second half of the paper, we prove that localizations of twisted Calabi–Yau algebras at elements which form both left and right denominator sets remain twisted Calabi–Yau. As such, we obtain a large class of <span><math><mi>Z</mi></math></span>-graded twisted Calabi–Yau algebras arising as localizations of Artin–Schelter regular algebras. Throughout the paper, we survey a number of concrete examples of <em>G</em>-graded twisted Calabi–Yau algebras, including the Weyl algebras, families of generalized Weyl algebras, and universal enveloping algebras of finite dimensional Lie algebras.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107849"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094016","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":"Kernels of linear maps","authors":"Arno van den Essen , Jan Schoone","doi":"10.1016/j.jpaa.2024.107847","DOIUrl":"10.1016/j.jpaa.2024.107847","url":null,"abstract":"<div><div>The theorem of Duistermaat and Van der Kallen from 1998 proved the first case of the Mathieu conjecture. Using the theory of Mathieu-Zhao spaces, we can reformulate this theorem as Ker <em>L</em> is a Mathieu-Zhao space where <em>L</em> is the linear map <span><math><mi>L</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></mrow></msub><mo>,</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>]</mo><mo>→</mo><mi>C</mi><mo>,</mo><mspace></mspace><mi>f</mi><mo>↦</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. In this paper, we generalize this result (for <span><math><mi>n</mi><mo>=</mo><mn>1</mn></math></span>) to all non-trivial linear maps <span><math><mi>L</mi><mo>:</mo><mi>C</mi><mo>[</mo><mi>X</mi><mo>,</mo><msup><mrow><mi>X</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>]</mo><mo>→</mo><mi>C</mi></math></span> such that <span><math><mo>{</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>|</mo><mo>|</mo><mi>n</mi><mo>|</mo><mo>≥</mo><mi>N</mi><mo>}</mo><mo>⊂</mo><mi>Ker</mi><mspace></mspace><mi>L</mi></math></span> for some <span><math><mi>N</mi><mo>≥</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107847"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094015","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":"Topology and monoid representations I: Foundations","authors":"Benjamin Steinberg","doi":"10.1016/j.jpaa.2024.107848","DOIUrl":"10.1016/j.jpaa.2024.107848","url":null,"abstract":"<div><div>This paper aims to use topological methods to compute Ext between an irreducible representation of a finite monoid inflated from its group completion and one inflated from its group of units, or more generally coinduced from a maximal subgroup, via a spectral sequence that collapses on the <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-page over fields of good characteristic. As an application, we determine the global dimension of the algebra of the monoid of all affine transformations of a vector space over a finite field. We provide a topological characterization of when a monoid homomorphism induces a homological epimorphism of monoid algebras and apply it to semidirect products. Topology is used to construct projective resolutions of modules inflated from the group completion for sufficiently nice monoids. A sequel paper will use these results to study the representation theory Hsiao's monoid of ordered <em>G</em>-partitions (connected to the Mantaci-Reutenauer descent algebra for the wreath product <span><math><mi>G</mi><mo>≀</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>).</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107848"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094018","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":"An inductive model structure for strict ∞-categories","authors":"Simon Henry , Félix Loubaton","doi":"10.1016/j.jpaa.2024.107859","DOIUrl":"10.1016/j.jpaa.2024.107859","url":null,"abstract":"<div><div>We construct a left semi-model category of “marked strict ∞-categories” for which the fibrant objects are those whose marked arrows satisfy natural closure properties and are invertible up to higher marked arrows. The canonical model structure on strict ∞-categories can be recovered as a left Bousfield localization of this model structure. We show that an appropriate extension of the Street nerve to the marked setting produces a Quillen adjunction between our model category and the Verity model structure for complicial sets, generalizing previous results by the second named author. Finally, we use this model structure to study, in the setting of strict ∞-categories, the idea that, because they are two different “truncation functors” taking an <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mi>n</mi><mo>)</mo></math></span> to an <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-category, there are two non-equivalent definitions for the <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-category of <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>-categories as a limit of the <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-categories of <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mi>n</mi><mo>)</mo></math></span>-categories. We show that in fact there seem to be at least three non-equivalent ways of constructing an <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-category of <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>-categories.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107859"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099040","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":"Rings of almost everywhere defined functions","authors":"Matthias Schötz","doi":"10.1016/j.jpaa.2024.107851","DOIUrl":"10.1016/j.jpaa.2024.107851","url":null,"abstract":"<div><div>The following representation theorem is proven: A partially ordered commutative ring <span><math><mi>R</mi></math></span> is a subring of a ring of almost everywhere defined continuous real-valued functions on a compact Hausdorff space <em>X</em> if and only if <span><math><mi>R</mi></math></span> is archimedean and localizable. Here we assume that the positive cone of <span><math><mi>R</mi></math></span> is closed under multiplication and stable under multiplication with squares, but actually one of these assumptions implies the other. An almost everywhere defined function on <em>X</em> is one that is defined on a dense open subset of <em>X</em>. A partially ordered commutative ring <span><math><mi>R</mi></math></span> is archimedean if the underlying additive partially ordered abelian group is archimedean, and <span><math><mi>R</mi></math></span> is localizable essentially if its order is compatible with the construction of a localization with sufficiently large, positive denominators. As applications we discuss the <em>σ</em>-bounded case, lattice-ordered commutative rings (<em>f</em>-rings), partially ordered fields, and commutative operator algebras.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107851"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094021","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":"Fano manifolds whose Chern characters satisfy some positivity conditions","authors":"Taku Suzuki","doi":"10.1016/j.jpaa.2025.107865","DOIUrl":"10.1016/j.jpaa.2025.107865","url":null,"abstract":"<div><div>In this paper, we investigate Fano manifolds whose Chern characters satisfy some positivity conditions. We prove that such manifolds admit long chains of higher order minimal families of rational curves and are covered by higher rational varieties.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107865"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099042","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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