{"title":"Flat covers and injective hulls of persistence modules","authors":"Eero Hyry, Ville Puuska","doi":"10.1016/j.jpaa.2025.107874","DOIUrl":"10.1016/j.jpaa.2025.107874","url":null,"abstract":"<div><div>Motivated by recent progress in topological data analysis, we establish a Matlis duality between injective hulls and flat covers of persistence modules. This extends to a duality between minimal flat and minimal injective resolutions. We utilize the theory of flat cotorsion modules and flat covers developed by Enochs and Xu. By means of this theory we can work with persistence modules which are not tame or even pointwise finite-dimensional.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107874"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163679","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":"On the embedded versions of degree-2 tropical covers of an elliptic curve","authors":"Numan Amin","doi":"10.1016/j.jpaa.2025.107898","DOIUrl":"10.1016/j.jpaa.2025.107898","url":null,"abstract":"<div><div>We study an equivalent problem to the Hurwitz existence problem in the context of tropical algebraic geometry. For this, we introduced the idea of an algebraic realization of a tropical cover <em>c</em> and homeomorphically faithfulness of the embeddings. In this study, our approach is constructive and we constructed algebraic realizations which are homeomorphically faithful for an arbitrary tropical cover of degree 2 and genus 2 of an abstract elliptic curve. On the basis of length conditions, we divide this into two cases: when the lengths in a tropical cover are equal and when the lengths are unequal. To achieve these results, we also progressed in unfolding and generalized a existing technique to unfold a cycle of certain length under certain conditions.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107898"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387714","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":"Relatively endotrivial complexes","authors":"Sam K. Miller","doi":"10.1016/j.jpaa.2025.107867","DOIUrl":"10.1016/j.jpaa.2025.107867","url":null,"abstract":"<div><div>Let <em>G</em> be a finite group and <em>k</em> be a field of characteristic <span><math><mi>p</mi><mo>></mo><mn>0</mn></math></span>. In prior work, we studied endotrivial complexes, the invertible objects of the bounded homotopy category of <em>p</em>-permutation <em>kG</em>-modules <span><math><msup><mrow><mi>K</mi></mrow><mrow><mi>b</mi></mrow></msup><mo>(</mo><mmultiscripts><mrow><mi>triv</mi></mrow><mprescripts></mprescripts><mrow><mi>k</mi><mi>G</mi></mrow><none></none></mmultiscripts><mo>)</mo></math></span>. Using the notion of projectivity relative to a <em>kG</em>-module, we expand on this study by defining notions of “relatively” endotrivial chain complexes, analogous to Lassueur's construction of relatively endotrivial <em>kG</em>-modules. We obtain equivalent characterizations of relative endotriviality and find corresponding local homological data which almost completely determine the isomorphism class of a relatively endotrivial complex. We show this local data must partially satisfy the Borel-Smith conditions, and consider the behavior of restriction to subgroups containing Sylow <em>p</em>-subgroups <em>S</em> of <em>G</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107867"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163717","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":"Artinian Gorenstein algebras of socle degree three which have the weak Lefschetz property","authors":"Andrew R. Kustin","doi":"10.1016/j.jpaa.2025.107878","DOIUrl":"10.1016/j.jpaa.2025.107878","url":null,"abstract":"<div><div>Let <strong><em>k</em></strong> be an arbitrary field and Φ be the Macaulay inverse system for a standard graded Artinian Gorenstein <strong><em>k</em></strong>-algebra <em>A</em> of arbitrary embedding dimension <em>d</em> and socle degree three. Assume that <em>A</em> has the weak Lefschetz property. We identify generators for the defining ideal of <em>A</em> as a quotient of a polynomial ring <em>P</em> over <strong><em>k</em></strong> with <em>d</em> variables and we give an explicit homogeneous resolution, <span><math><mi>X</mi></math></span>, of <em>A</em> by free <em>P</em>-modules. We identify a symmetric bilinear form <em>G</em> which determines how to turn <span><math><mi>X</mi></math></span> into the minimal resolution of <em>A</em>. In particular, when <em>G</em> is identically zero, then <span><math><mi>X</mi></math></span> is already the minimal resolution of <em>A</em>.</div><div>The resolution <span><math><mi>X</mi></math></span> is closely related to the resolution of a Gorenstein algebra with socle degree two. A Gorenstein algebra with socle degree two has a resolution that is as linear as possible.</div><div>The corresponding project has previously been carried out (by the present author, and also by Macias Marques, Veliche, and Weyman), when the embedding dimension <em>d</em> is equal to 4.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107878"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162785","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":"Generators for extensions of valuation rings","authors":"Josnei Novacoski","doi":"10.1016/j.jpaa.2025.107900","DOIUrl":"10.1016/j.jpaa.2025.107900","url":null,"abstract":"<div><div>For a finite valued field extension <span><math><mo>(</mo><mi>L</mi><mo>/</mo><mi>K</mi><mo>,</mo><mi>v</mi><mo>)</mo></math></span> we describe the problem of finding sets of generators for the corresponding extension <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>/</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> of valuation rings. The main tool to obtain such sets is complete sets of (key) polynomials. We show that when the initial index coincides with the ramification index, sequences of key polynomials naturally give rise to sets of generators. We use this to prove Knaf's conjecture for pure extensions.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107900"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143376711","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":"LULU is syntomic","authors":"E. Javier Elizondo , Paulo Lima-Filho","doi":"10.1016/j.jpaa.2025.107880","DOIUrl":"10.1016/j.jpaa.2025.107880","url":null,"abstract":"<div><div>Let <em>G</em> be a Chevalley group over a field <span><math><mi>k</mi></math></span>. Fix a maximal torus <span><math><mi>T</mi></math></span> in <em>G</em>, along with opposite Borel subgroups <em>B</em> and <span><math><msup><mrow><mi>B</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span> satisfying <span><math><mi>T</mi><mo>=</mo><mi>B</mi><mo>∩</mo><msup><mrow><mi>B</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span>, and denote by <span><math><mi>U</mi><mo>:</mo><mo>=</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>(</mo><mi>B</mi><mo>)</mo></math></span> and <span><math><mi>L</mi><mo>:</mo><mo>=</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>(</mo><msup><mrow><mi>B</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>)</mo></math></span> their respective unipotent radicals. We prove that the multiplication map <span><math><mi>μ</mi><mo>:</mo><mi>L</mi><mo>×</mo><mi>U</mi><mo>×</mo><mi>L</mi><mo>×</mo><mi>U</mi><mo>⟶</mo><mi>G</mi></math></span> is syntomic and faithfully flat over any base field <span><math><mi>k</mi></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107880"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162788","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 Z2-graded dimensions of the free Jordan superalgebra J(D1|D2)","authors":"Shikui Shang","doi":"10.1016/j.jpaa.2025.107879","DOIUrl":"10.1016/j.jpaa.2025.107879","url":null,"abstract":"<div><div>Let <em>k</em> be a field of characteristic 0. For a superspace <span><math><mi>V</mi><mo>=</mo><msub><mrow><mi>V</mi></mrow><mrow><mover><mrow><mn>0</mn></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub><mo>⊕</mo><msub><mrow><mi>V</mi></mrow><mrow><mover><mrow><mn>1</mn></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub></math></span> over <em>k</em>, we call the vector <span><math><mo>(</mo><msub><mrow><mi>dim</mi></mrow><mrow><mi>k</mi></mrow></msub><mo></mo><msub><mrow><mi>V</mi></mrow><mrow><mover><mrow><mn>0</mn></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub><mo>,</mo><msub><mrow><mi>dim</mi></mrow><mrow><mi>k</mi></mrow></msub><mo></mo><msub><mrow><mi>V</mi></mrow><mrow><mover><mrow><mn>1</mn></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub><mo>)</mo></math></span> the (<span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-)graded dimension of <em>V</em>. Let <span><math><mi>J</mi><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> be the free Jordan superalgebra generated by <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> even generators and <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> odd generators. In this paper, we study the graded dimensions of the <em>n</em>-components of <span><math><mi>J</mi><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> and find the connection between them and the homology of Tits-Allison-Gao Lie superalgebra of <span><math><mi>J</mi><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> following the method given by I. Kashuba and O. Mathieu in <span><span>[15]</span></span>, where they deal with the free Jordan algebra. And, four related conjectures on the free Jordan superalgebras and related Lie superalgebras are proposed in this article.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107879"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163716","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":"Strong congruence spaces and dimension in F1-geometry","authors":"Manoel Jarra","doi":"10.1016/j.jpaa.2025.107862","DOIUrl":"10.1016/j.jpaa.2025.107862","url":null,"abstract":"<div><div>We introduce strong congruence spaces, which are topological spaces that provide a useful concept of dimension for monoid schemes. We study their properties and show that, given a toric monoid scheme over an algebraically closed basis, its strong congruence space and the complex toric variety associated to its fan have the same dimension.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107862"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143378688","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":"Low degree rational curves on quasi-polarized K3 surfaces","authors":"Sławomir Rams , Matthias Schütt","doi":"10.1016/j.jpaa.2025.107904","DOIUrl":"10.1016/j.jpaa.2025.107904","url":null,"abstract":"<div><div>We prove that there are at most <span><math><mo>(</mo><mn>24</mn><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> irreducible rational curves of positive low-degree on high-degree models of K3 surfaces with at most Du Val singularities, where <span><math><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is the number of exceptional divisors on the minimal resolution. We also provide several existence results in the above setting (i.e. for rational curves on quasi-polarized K3 surfaces), which imply that for many values of <span><math><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> our bound cannot be improved.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107904"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143394580","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":"Deformations of quasi-categories in modules","authors":"Violeta Borges Marques, Wendy Lowen, Arne Mertens","doi":"10.1016/j.jpaa.2025.107866","DOIUrl":"10.1016/j.jpaa.2025.107866","url":null,"abstract":"<div><div>The framework of templicial objects was put forth in <span><span>[30]</span></span> in order to develop higher categorical concepts in the presence of enrichment. In particular, quasi-categories in modules constitute a subclass of templicial modules which may be considered as a kind of “weak dg-categories (concentrated in homologically positive degrees)” according to <span><span>[29]</span></span>. The main goal of the present paper is to initiate the deformation theory of templicial modules. In particular, we show that quasi-categories in modules are preserved under levelwise flat infinitesimal deformation.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107866"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163055","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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