{"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":"229 1","pages":"Article 107821"},"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":"229 1","pages":"Article 107816"},"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":"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":"229 1","pages":"Article 107817"},"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 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":"229 1","pages":"Article 107822"},"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":"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":"229 1","pages":"Article 107820"},"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":"229 1","pages":"Article 107814"},"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":"229 1","pages":"Article 107815"},"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}
{"title":"Abelian ideals and the variety of Lagrangian subalgebras","authors":"Sam Evens , Yu Li","doi":"10.1016/j.jpaa.2024.107813","DOIUrl":"10.1016/j.jpaa.2024.107813","url":null,"abstract":"<div><div>For a semisimple algebraic group <em>G</em> of adjoint type with Lie algebra <span><math><mi>g</mi></math></span> over the complex numbers, we establish a bijection between the set of closed orbits of the group <span><math><mi>G</mi><mo>⋉</mo><msup><mrow><mi>g</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> acting on the variety of Lagrangian subalgebras of <span><math><mi>g</mi><mo>⋉</mo><msup><mrow><mi>g</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> and the set of abelian ideals of a fixed Borel subalgebra of <span><math><mi>g</mi></math></span>. In particular, the number of such orbits equals <span><math><msup><mrow><mn>2</mn></mrow><mrow><mtext>rk</mtext><mi>g</mi></mrow></msup></math></span> by Peterson's theorem on abelian ideals.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107813"},"PeriodicalIF":0.7,"publicationDate":"2024-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425562","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}
Carlos Améndola , Francesco Galuppi , Ángel David Ríos Ortiz , Pierpaola Santarsiero , Tim Seynnaeve
{"title":"Decomposing tensor spaces via path signatures","authors":"Carlos Améndola , Francesco Galuppi , Ángel David Ríos Ortiz , Pierpaola Santarsiero , Tim Seynnaeve","doi":"10.1016/j.jpaa.2024.107807","DOIUrl":"10.1016/j.jpaa.2024.107807","url":null,"abstract":"<div><div>The signature of a path is a sequence of tensors whose entries are iterated integrals, playing a key role in stochastic analysis and applications. The set of all signature tensors at a particular level gives rise to the universal signature variety. We show that the parametrization of this variety induces a natural decomposition of the tensor space via representation theory, and connect this to the study of path invariants. We also reveal certain constraints that apply to the rank and symmetry of a signature tensor.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107807"},"PeriodicalIF":0.7,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425561","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}
Luca Chiantini , Pietro De Poi , Łucja Farnik , Giuseppe Favacchio , Brian Harbourne , Giovanna Ilardi , Juan Migliore , Tomasz Szemberg , Justyna Szpond
{"title":"Geproci sets on skew lines in P3 with two transversals","authors":"Luca Chiantini , Pietro De Poi , Łucja Farnik , Giuseppe Favacchio , Brian Harbourne , Giovanna Ilardi , Juan Migliore , Tomasz Szemberg , Justyna Szpond","doi":"10.1016/j.jpaa.2024.107809","DOIUrl":"10.1016/j.jpaa.2024.107809","url":null,"abstract":"<div><div>The purpose of this work is to pursue classification of geproci sets. Specifically we classify <span><math><mo>[</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>]</mo></math></span>-geproci sets <em>Z</em> which consist of <span><math><mi>m</mi><mo>=</mo><mn>4</mn></math></span> points on each of <em>n</em> skew lines, assuming the skew lines have two transversals in common. We show in this case that <span><math><mi>n</mi><mo>≤</mo><mn>6</mn></math></span>. Moreover we show that all geproci sets of this type and with no points on the transversals are contained in the <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> configuration. We conjecture that a similar result is true for an arbitrary number <em>m</em> of points on each skew line, replacing containment in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> by containment in a half grid obtained by the so-called <em>standard construction</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107809"},"PeriodicalIF":0.7,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425558","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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