Pub Date : 2025-02-01DOI: 10.1016/j.jpaa.2025.107864
Philip Hackney, Justin Lynd
We give a new characterization of partial groups as a subcategory of symmetric (simplicial) sets. This subcategory has an explicit reflection, which permits one to compute colimits in the category of partial groups. We also introduce the notion of a partial groupoid, which encompasses both groupoids and partial groups.
{"title":"Partial groups as symmetric simplicial sets","authors":"Philip Hackney, Justin Lynd","doi":"10.1016/j.jpaa.2025.107864","DOIUrl":"10.1016/j.jpaa.2025.107864","url":null,"abstract":"<div><div>We give a new characterization of partial groups as a subcategory of symmetric (simplicial) sets. This subcategory has an explicit reflection, which permits one to compute colimits in the category of partial groups. We also introduce the notion of a partial groupoid, which encompasses both groupoids and partial groups.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107864"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163606","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}
Pub Date : 2025-02-01DOI: 10.1016/j.jpaa.2025.107871
Antonio Ioppolo , Daniela La Mattina
Let A be a finite dimensional algebra endowed with a superautomorphism over a field of characteristic zero. In this paper we study the asymptotic behavior of the sequence of φ-codimensions , . More precisely, we shall prove that always exists and it is an integer related in an explicit way to the dimension of a suitable semisimple subalgebra of A. This result gives a positive answer to a conjecture of Amitsur in this setting. In the final part of the paper we characterize the algebras whose exponential growth is bounded by 2.
{"title":"Codimension growth of algebras with superautomorphism","authors":"Antonio Ioppolo , Daniela La Mattina","doi":"10.1016/j.jpaa.2025.107871","DOIUrl":"10.1016/j.jpaa.2025.107871","url":null,"abstract":"<div><div>Let <em>A</em> be a finite dimensional algebra endowed with a superautomorphism over a field of characteristic zero. In this paper we study the asymptotic behavior of the sequence of <em>φ</em>-codimensions <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>φ</mi></mrow></msubsup><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, <span><math><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo></math></span>. More precisely, we shall prove that <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></msub><mo></mo><mroot><mrow><msubsup><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>φ</mi></mrow></msubsup><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></mroot></math></span> always exists and it is an integer related in an explicit way to the dimension of a suitable semisimple subalgebra of <em>A</em>. This result gives a positive answer to a conjecture of Amitsur in this setting. In the final part of the paper we characterize the algebras whose exponential growth is bounded by 2.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107871"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163665","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}
Pub Date : 2025-02-01DOI: 10.1016/j.jpaa.2025.107874
Eero Hyry, Ville Puuska
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.
{"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}
Pub Date : 2025-02-01DOI: 10.1016/j.jpaa.2025.107898
Numan Amin
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 c 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.
{"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}
Pub Date : 2025-02-01DOI: 10.1016/j.jpaa.2025.107867
Sam K. Miller
Let G be a finite group and k be a field of characteristic . In prior work, we studied endotrivial complexes, the invertible objects of the bounded homotopy category of p-permutation kG-modules . Using the notion of projectivity relative to a kG-module, we expand on this study by defining notions of “relatively” endotrivial chain complexes, analogous to Lassueur's construction of relatively endotrivial kG-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 p-subgroups S of G.
{"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}
Pub Date : 2025-02-01DOI: 10.1016/j.jpaa.2025.107878
Andrew R. Kustin
Let k be an arbitrary field and Φ be the Macaulay inverse system for a standard graded Artinian Gorenstein k-algebra A of arbitrary embedding dimension d and socle degree three. Assume that A has the weak Lefschetz property. We identify generators for the defining ideal of A as a quotient of a polynomial ring P over k with d variables and we give an explicit homogeneous resolution, , of A by free P-modules. We identify a symmetric bilinear form G which determines how to turn into the minimal resolution of A. In particular, when G is identically zero, then is already the minimal resolution of A.
The resolution 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.
The corresponding project has previously been carried out (by the present author, and also by Macias Marques, Veliche, and Weyman), when the embedding dimension d is equal to 4.
{"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}
Pub Date : 2025-02-01DOI: 10.1016/j.jpaa.2025.107900
Josnei Novacoski
For a finite valued field extension we describe the problem of finding sets of generators for the corresponding extension 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.
{"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}
Pub Date : 2025-02-01DOI: 10.1016/j.jpaa.2025.107880
E. Javier Elizondo , Paulo Lima-Filho
Let G be a Chevalley group over a field . Fix a maximal torus in G, along with opposite Borel subgroups B and satisfying , and denote by and their respective unipotent radicals. We prove that the multiplication map is syntomic and faithfully flat over any base field .
{"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}
Pub Date : 2025-02-01DOI: 10.1016/j.jpaa.2025.107879
Shikui Shang
Let k be a field of characteristic 0. For a superspace over k, we call the vector the (-)graded dimension of V. Let be the free Jordan superalgebra generated by even generators and odd generators. In this paper, we study the graded dimensions of the n-components of and find the connection between them and the homology of Tits-Allison-Gao Lie superalgebra of following the method given by I. Kashuba and O. Mathieu in [15], 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.
{"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}
Pub Date : 2025-02-01DOI: 10.1016/j.jpaa.2025.107862
Manoel Jarra
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.
{"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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号