{"title":"The Modular Isomorphism Problem – the alternative perspective on counterexamples","authors":"Czesław Bagiński, Kamil Zabielski","doi":"10.1016/j.jpaa.2024.107826","DOIUrl":"10.1016/j.jpaa.2024.107826","url":null,"abstract":"<div><div>As a result of impressive research <span><span>[5]</span></span>, D. García-Lucas, Á. del Río and L. Margolis defined an infinite series of non-isomorphic 2-groups <em>G</em> and <em>H</em>, whose group algebras <span><math><mi>F</mi><mi>G</mi></math></span> and <span><math><mi>F</mi><mi>H</mi></math></span> over the field <span><math><mi>F</mi><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are isomorphic, solving negatively the long-standing Modular Isomorphism Problem (MIP). In this note we give a different perspective on their examples and show that they are special cases of a more general construction. We also show that this type of construction for <span><math><mi>p</mi><mo>></mo><mn>2</mn></math></span> does not provide a similar counterexample to the MIP.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533602","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":"Rouquier dimension versus global dimension","authors":"Greg Stevenson","doi":"10.1016/j.jpaa.2024.107827","DOIUrl":"10.1016/j.jpaa.2024.107827","url":null,"abstract":"<div><div>We give an example of a commutative coherent ring of infinite global dimension such that the category of perfect complexes has finite Rouquier dimension.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444583","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":"Torsion-simple objects in abelian categories","authors":"Sergio Pavon","doi":"10.1016/j.jpaa.2024.107818","DOIUrl":"10.1016/j.jpaa.2024.107818","url":null,"abstract":"<div><div>We introduce the notion of torsion-simple objects in an abelian category: these are the objects which are always either torsion or torsion-free with respect to any torsion pair. We present some general results concerning their properties, and then proceed to investigate the notion in various contexts, such as the category of modules over an artin algebra or a commutative noetherian ring, and the category of quasi-coherent sheaves over the projective line.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426023","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":"Compatible weak factorization systems and model structures","authors":"Zhenxing Di , Liping Li , Li Liang","doi":"10.1016/j.jpaa.2024.107821","DOIUrl":"10.1016/j.jpaa.2024.107821","url":null,"abstract":"<div><div>In this paper, the concept of compatible weak factorization systems in general categories is introduced as a counterpart of compatible complete cotorsion pairs in abelian categories. We describe a method to construct model structures on general categories via two compatible weak factorization systems satisfying certain conditions, and hence, generalize a very useful result by Gillespie for abelian model structures. As particular examples, we show that weak factorization systems associated to some classical model structures (for example, the Kan-Quillen model structure on <span><math><mi>sSet</mi></math></span>) satisfy these conditions.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426022","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":"G-connections on principal bundles over complete G-varieties","authors":"Bivas Khan , Mainak Poddar","doi":"10.1016/j.jpaa.2024.107816","DOIUrl":"10.1016/j.jpaa.2024.107816","url":null,"abstract":"<div><div>Let <em>X</em> be a complete variety over an algebraically closed field <em>k</em> of characteristic zero, equipped with an action of an algebraic group <em>G</em>. Let <em>H</em> be a reductive group. We study the notion of <em>G</em>-connection on a principal <em>H</em>-bundle. We give necessary and sufficient criteria for the existence of <em>G</em>-connections extending the Atiyah-Weil type criterion for holomorphic connections obtained by Azad and Biswas. We also establish a relationship between the existence of <em>G</em>-connection and equivariant structure on a principal <em>H</em>-bundle, under the assumption that <em>G</em> is semisimple and simply connected. These results have been obtained by Biswas et al. when the underlying variety is smooth.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426026","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 note on Ratliff-Rush filtration, reduction number and postulation number of m-primary ideals","authors":"Mousumi Mandal, Shruti Priya","doi":"10.1016/j.jpaa.2024.107822","DOIUrl":"10.1016/j.jpaa.2024.107822","url":null,"abstract":"<div><div>Let <span><math><mo>(</mo><mi>R</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> be a Cohen-Macaulay local ring of dimension <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>, and <em>I</em> an <span><math><mi>m</mi></math></span>-primary ideal. Let <span><math><mi>r</mi><mo>(</mo><mi>I</mi><mo>)</mo></math></span> be the reduction number of <em>I</em>, <span><math><mi>n</mi><mo>(</mo><mi>I</mi><mo>)</mo></math></span> the postulation number and <span><math><mi>ρ</mi><mo>(</mo><mi>I</mi><mo>)</mo></math></span> the stability index of the Ratliff-Rush filtration with respect to <em>I</em>. We prove that for <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>, if <span><math><mi>n</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>=</mo><mi>ρ</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span>, then <span><math><mi>r</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>≤</mo><mi>n</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>+</mo><mn>2</mn></math></span>, and if <span><math><mi>n</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>≠</mo><mi>ρ</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span>, then <span><math><mi>r</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>≥</mo><mi>n</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>+</mo><mn>2</mn></math></span>. For <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>, assuming <em>I</em> is integrally closed, <span><math><mi>depth</mi><mspace></mspace><mi>gr</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>=</mo><mi>d</mi><mo>−</mo><mn>2</mn></math></span>, and <span><math><mi>n</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>=</mo><mo>−</mo><mo>(</mo><mi>d</mi><mo>−</mo><mn>3</mn><mo>)</mo></math></span>, we prove that <span><math><mi>r</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>≥</mo><mi>n</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>+</mo><mi>d</mi></math></span>. Our main result generalizes a result by Marley on the relation between the Hilbert-Samuel function and the Hilbert-Samuel polynomial by relaxing the condition on the depth of the associated graded ring to the good behavior of the Ratliff-Rush filtration with respect to <em>I</em> mod a superficial sequence. From this result, it follows that for Cohen-Macaulay local rings of dimension <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>, if <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo></math></span> for some <span><math><mi>k</mi><mo>≥</mo><mi>ρ</mi><mo>(</mo><mi>I</mi><mo>)</mo></math></span>, then <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for all <span><math><mi>n</mi><mo>≥</mo><mi>k</mi></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142441733","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 algebraic approach to Harder-Narasimhan filtrations","authors":"Hipolito Treffinger","doi":"10.1016/j.jpaa.2024.107817","DOIUrl":"10.1016/j.jpaa.2024.107817","url":null,"abstract":"<div><div>In this article we study chains of torsion classes in an abelian category <span><math><mi>A</mi></math></span>. We prove that chains of torsion classes satisfying mild technical conditions induce a Harder-Narasimhan filtration for every non-zero object <em>M</em> in <span><math><mi>A</mi></math></span>, generalising a well-known property of stability conditions. We also characterise the slicings of <span><math><mi>A</mi></math></span> in terms of chains of torsion classes. We finish the paper by showing that chains of torsion classes induce wall-crossing formulas in the completed Hall algebra of the category.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426024","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 note on an effective bound for the gonality conjecture","authors":"Alexander S. Duncan , Wenbo Niu , Jinhyung Park","doi":"10.1016/j.jpaa.2024.107820","DOIUrl":"10.1016/j.jpaa.2024.107820","url":null,"abstract":"<div><div>The gonality conjecture, proved by Ein–Lazarsfeld, asserts that the gonality of a nonsingular projective curve of genus <em>g</em> can be detected from its syzygies in the embedding given by a line bundle of sufficiently large degree. An effective result obtained by Rathmann says that any line bundle of degree at least <span><math><mn>4</mn><mi>g</mi><mo>−</mo><mn>3</mn></math></span> would work in the gonality theorem. In this note, we develop a new method to improve the degree bound to <span><math><mn>4</mn><mi>g</mi><mo>−</mo><mn>4</mn></math></span> with two exceptional cases.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426025","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 a general notion of a polynomial identity and codimensions","authors":"A.S. Gordienko","doi":"10.1016/j.jpaa.2024.107814","DOIUrl":"10.1016/j.jpaa.2024.107814","url":null,"abstract":"<div><div>Using the braided version of Lawvere's algebraic theories and Mac Lane's PROPs, we introduce polynomial identities for arbitrary algebraic structures in a braided monoidal category <span><math><mi>C</mi></math></span> as well as their codimensions in the case when <span><math><mi>C</mi></math></span> is linear over some field. The new cases include coalgebras, bialgebras, Hopf algebras, braided vector spaces, Yetter–Drinfel'd modules, etc. We find bases for polynomial identities and calculate codimensions in some important particular cases.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425132","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":"Differential theory of zero-dimensional schemes","authors":"Martin Kreuzer , Tran N.K. Linh , Le N. Long","doi":"10.1016/j.jpaa.2024.107815","DOIUrl":"10.1016/j.jpaa.2024.107815","url":null,"abstract":"<div><div>To study a 0-dimensional scheme <span><math><mi>X</mi></math></span> in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> over a perfect field <em>K</em>, we use the module of Kähler differentials <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> of its homogeneous coordinate ring <em>R</em> and its exterior powers, the higher modules of Kähler differentials <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math></span>. One of our main results is a characterization of weakly curvilinear schemes <span><math><mi>X</mi></math></span> by the Hilbert polynomials of the modules <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math></span> which allows us to check this property algorithmically without computing the primary decomposition of the vanishing ideal of <span><math><mi>X</mi></math></span>. Further main achievements are precise formulas for the Hilbert functions and Hilbert polynomials of the modules <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math></span> for a fat point scheme <span><math><mi>X</mi></math></span> which extend and settle previous partial results and conjectures. Underlying these results is a novel method: we first embed the homogeneous coordinate ring <em>R</em> into its truncated integral closure <span><math><mover><mrow><mi>R</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span>. Then we use the corresponding map from the module of Kähler differentials <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> to <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mover><mrow><mi>R</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>/</mo><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> to find a formula for the Hilbert polynomial <span><math><mrow><mi>HP</mi></mrow><mo>(</mo><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mo>)</mo></math></span> and a sharp bound for the regularity index <span><math><mrow><mi>ri</mi></mrow><mo>(</mo><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mo>)</mo></math></span>. Next we extend this to formulas for the Hilbert polynomials <span><math><mrow><mi>HP</mi></mrow><mo>(</mo><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>R</mi><mo>/</mo><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msubsup><mo>)</mo></math></span> and bounds for the regularity indices of the higher modules of Kähler differentials. As a further application, we characterize uniformity conditions on <span><math><mi>X</mi></math></span> using the Hilbert functions of the Kähler diff","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426021","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}
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