Pub Date : 2025-03-25DOI: 10.1016/j.jpaa.2025.107961
Benjamín García, Alberto G. Raggi-Cárdenas
Fibered Burnside rings appear as Grothendieck rings of fibered permutation representations of a finite group, generalizing Burnside rings and monomial representation rings. Their species, primitive idempotents and their conductors are of particular interest in representation theory as they encode information related to the structure of the group. In this note, we introduce fiber change maps between fibered Burnside rings, and we present results on their functoriality and naturality with respect to biset operations. We present some advances on the conductors for cyclic fiber groups, and fully determine them in particular cases, covering a wide range of interesting examples.
{"title":"On fibered Burnside rings, fiber change maps and cyclic fiber groups","authors":"Benjamín García, Alberto G. Raggi-Cárdenas","doi":"10.1016/j.jpaa.2025.107961","DOIUrl":"10.1016/j.jpaa.2025.107961","url":null,"abstract":"<div><div>Fibered Burnside rings appear as Grothendieck rings of fibered permutation representations of a finite group, generalizing Burnside rings and monomial representation rings. Their species, primitive idempotents and their conductors are of particular interest in representation theory as they encode information related to the structure of the group. In this note, we introduce fiber change maps between fibered Burnside rings, and we present results on their functoriality and naturality with respect to biset operations. We present some advances on the conductors for cyclic fiber groups, and fully determine them in particular cases, covering a wide range of interesting examples.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 6","pages":"Article 107961"},"PeriodicalIF":0.7,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759663","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 : 2025-03-25DOI: 10.1016/j.jpaa.2025.107958
Jian Cui, Xue-Song Lu, Pu Zhang
In contrast with the Hovey correspondence of abelian model structures from two complete cotorsion pairs, Beligiannis and Reiten give a construction of model structures on abelian categories from one hereditary complete cotorsion pair. The aim of this paper is to extend this result to weakly idempotent complete exact categories, by adding the condition of heredity of the complete cotorsion pair. In fact, even for abelian categories, this condition of heredity should be added. This construction really gives model structures which are not necessarily exact in the sense of Gillespie. The correspondence of Beligiannis and Reiten of weakly projective model structures also holds for weakly idempotent complete exact categories.
{"title":"Model structure from one hereditary complete cotorsion pair","authors":"Jian Cui, Xue-Song Lu, Pu Zhang","doi":"10.1016/j.jpaa.2025.107958","DOIUrl":"10.1016/j.jpaa.2025.107958","url":null,"abstract":"<div><div>In contrast with the Hovey correspondence of abelian model structures from two complete cotorsion pairs, Beligiannis and Reiten give a construction of model structures on abelian categories from one hereditary complete cotorsion pair. The aim of this paper is to extend this result to weakly idempotent complete exact categories, by adding the condition of heredity of the complete cotorsion pair. In fact, even for abelian categories, this condition of heredity should be added. This construction really gives model structures which are not necessarily exact in the sense of Gillespie. The correspondence of Beligiannis and Reiten of weakly projective model structures also holds for weakly idempotent complete exact categories.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 7","pages":"Article 107958"},"PeriodicalIF":0.7,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761147","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 : 2025-03-24DOI: 10.1016/j.jpaa.2025.107959
Sebastian Heinrich
In [12], the notion of a module tensor category was introduced as a braided monoidal central functor from a braided monoidal category to a monoidal category , which is a monoidal functor together with a braided monoidal lift to the Drinfeld center of . This is a categorification of a unital associative algebra A over a commutative ring R via a ring homomorphism into the center of A. In this paper, we want to categorify the characterization of an associative algebra as a (not necessarily unital) ring A together with an R–module structure over a commutative ring R, such that multiplication in A and action of R on A are compatible. In doing so, we introduce the more general notion of non–unital module monoidal categories and obtain 2–categories of non–unital and unital module monoidal categories, their functors and natural transformations. We will show that in the unital case the latter definition is equivalent to the definition in [12] by explicitly writing down an equivalence of 2–categories.
在[12]中，模块张量范畴的概念被引入为从一个辫状单环范畴 V 到一个单环范畴 T 的辫状单环中心函子 F:V⟶T，它是一个单环函子 F:V⟶T 连同到 T 的 Drinfeld 中心的辫状单环提升 FZ:V⟶Z(T)。在本文中，我们希望把关联代数的特征归类为一个（不一定是单整的）环 A 和一个交换环 R 上的 R 模块结构，从而使 A 中的乘法和 R 对 A 的作用是相容的。在此过程中，我们引入了非空模单范畴这一更一般的概念，并得到了非空模单范畴和空模单范畴的 2 个范畴、它们的函数和自然变换。我们将通过明确写出 2 维类的等价性来证明，在非ital 的情况下，后一个定义等价于 [12] 中的定义。
{"title":"Module monoidal categories as categorification of associative algebras","authors":"Sebastian Heinrich","doi":"10.1016/j.jpaa.2025.107959","DOIUrl":"10.1016/j.jpaa.2025.107959","url":null,"abstract":"<div><div>In <span><span>[12]</span></span>, the notion of a module tensor category was introduced as a braided monoidal central functor <span><math><mi>F</mi><mo>:</mo><mi>V</mi><mo>⟶</mo><mi>T</mi></math></span> from a braided monoidal category <span><math><mi>V</mi></math></span> to a monoidal category <span><math><mi>T</mi></math></span>, which is a monoidal functor <span><math><mi>F</mi><mo>:</mo><mi>V</mi><mo>⟶</mo><mi>T</mi></math></span> together with a braided monoidal lift <span><math><msup><mrow><mi>F</mi></mrow><mrow><mi>Z</mi></mrow></msup><mo>:</mo><mi>V</mi><mo>⟶</mo><mi>Z</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> to the Drinfeld center of <span><math><mi>T</mi></math></span>. This is a categorification of a unital associative algebra <em>A</em> over a commutative ring <em>R</em> via a ring homomorphism <span><math><mi>f</mi><mo>:</mo><mi>R</mi><mo>⟶</mo><mi>Z</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> into the center of <em>A</em>. In this paper, we want to categorify the characterization of an associative algebra as a (not necessarily unital) ring <em>A</em> together with an <em>R</em>–module structure over a commutative ring <em>R</em>, such that multiplication in <em>A</em> and action of <em>R</em> on <em>A</em> are compatible. In doing so, we introduce the more general notion of <em>non–unital module monoidal categories</em> and obtain 2–categories of non–unital and unital module monoidal categories, their functors and natural transformations. We will show that in the unital case the latter definition is equivalent to the definition in <span><span>[12]</span></span> by explicitly writing down an equivalence of 2–categories.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 7","pages":"Article 107959"},"PeriodicalIF":0.7,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143738875","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 : 2025-03-24DOI: 10.1016/j.jpaa.2025.107960
Panagiotis Kostas, Chrysostomos Psaroudakis
In this paper we investigate injective generation for graded rings. We first examine the relation between injective generation and graded injective generation for graded rings. We then reduce the study of injective generation for graded rings to the study of injective generation for certain Morita context rings and we provide sufficient conditions for injective generation of the latter. We then provide necessary and sufficient conditions so that injectives generate for tensor rings and for trivial extension rings. We provide two proofs for the class of tensor rings, one uses covering theory and the other uses the framework of cleft extensions of module categories. We finally prove injective generation for twisted tensor products of finite dimensional algebras.
{"title":"Injective generation for graded rings","authors":"Panagiotis Kostas, Chrysostomos Psaroudakis","doi":"10.1016/j.jpaa.2025.107960","DOIUrl":"10.1016/j.jpaa.2025.107960","url":null,"abstract":"<div><div>In this paper we investigate injective generation for graded rings. We first examine the relation between injective generation and graded injective generation for graded rings. We then reduce the study of injective generation for graded rings to the study of injective generation for certain Morita context rings and we provide sufficient conditions for injective generation of the latter. We then provide necessary and sufficient conditions so that injectives generate for tensor rings and for trivial extension rings. We provide two proofs for the class of tensor rings, one uses covering theory and the other uses the framework of cleft extensions of module categories. We finally prove injective generation for twisted tensor products of finite dimensional algebras.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 7","pages":"Article 107960"},"PeriodicalIF":0.7,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761148","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 : 2025-03-21DOI: 10.1016/j.jpaa.2025.107953
Xiaoxiang Zhou
In this article, we construct affine pavings for quiver partial flag varieties when the quiver is of Dynkin type. To achieve our results, we extend methods from Cerulli-Irelli–Esposito–Franzen–Reineke and Maksimau as well as techniques from Auslander–Reiten theory.
{"title":"Affine pavings of quiver flag varieties","authors":"Xiaoxiang Zhou","doi":"10.1016/j.jpaa.2025.107953","DOIUrl":"10.1016/j.jpaa.2025.107953","url":null,"abstract":"<div><div>In this article, we construct affine pavings for quiver partial flag varieties when the quiver is of Dynkin type. To achieve our results, we extend methods from Cerulli-Irelli–Esposito–Franzen–Reineke and Maksimau as well as techniques from Auslander–Reiten theory.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 6","pages":"Article 107953"},"PeriodicalIF":0.7,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704285","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 : 2025-03-19DOI: 10.1016/j.jpaa.2025.107952
Martin Palmer , Arthur Soulié
We construct a 3-variable enrichment of the Lawrence-Krammer-Bigelow (LKB) representation of the braid groups, which is the limit of a pro-nilpotent tower of representations having the original LKB representation as its bottom layer. We also construct analogous pro-nilpotent towers of representations of surface braid groups and loop braid groups.
{"title":"The pro-nilpotent Lawrence-Krammer-Bigelow representation","authors":"Martin Palmer , Arthur Soulié","doi":"10.1016/j.jpaa.2025.107952","DOIUrl":"10.1016/j.jpaa.2025.107952","url":null,"abstract":"<div><div>We construct a 3-variable enrichment of the Lawrence-Krammer-Bigelow (LKB) representation of the braid groups, which is the limit of a pro-nilpotent tower of representations having the original LKB representation as its bottom layer. We also construct analogous pro-nilpotent towers of representations of surface braid groups and loop braid groups.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 6","pages":"Article 107952"},"PeriodicalIF":0.7,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143738976","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 : 2025-03-19DOI: 10.1016/j.jpaa.2025.107951
Àngel García-Blázquez, Ángel del Río
We prove that if two finite metacyclic groups have isomorphic rational group algebras, then they are isomorphic. This contributes to understand where the line separating positive and negative solutions to the Isomorphism Problem for group algebras lies.
{"title":"The isomorphism problem for rational group algebras of finite metacyclic groups","authors":"Àngel García-Blázquez, Ángel del Río","doi":"10.1016/j.jpaa.2025.107951","DOIUrl":"10.1016/j.jpaa.2025.107951","url":null,"abstract":"<div><div>We prove that if two finite metacyclic groups have isomorphic rational group algebras, then they are isomorphic. This contributes to understand where the line separating positive and negative solutions to the Isomorphism Problem for group algebras lies.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 6","pages":"Article 107951"},"PeriodicalIF":0.7,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683882","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}
A quandle equipped with a good involution is referred to as symmetric. It is known that the cohomology of symmetric quandles gives rise to strong cocycle invariants for classical and surface links, even when they are not necessarily oriented. In this paper, we introduce the category of symmetric quandle modules and prove that these modules completely determine the Beck modules in the category of symmetric quandles. Consequently, this establishes suitable coefficient objects for constructing appropriate (co)homology theories. We develop an extension theory of modules over symmetric quandles, and propose a generalized (co)homology theory for symmetric quandles with coefficients in a homogeneous Beck module, which also recovers the symmetric quandle (co)homology developed by Kamada and Oshiro (2010) [16]. Our constructions also apply to symmetric racks. We conclude by establishing an explicit isomorphism between the second cohomology of a symmetric quandle and the first cohomology of its associated group.
{"title":"Generalized (co)homology of symmetric quandles over homogeneous Beck modules","authors":"Biswadeep Karmakar, Deepanshi Saraf, Mahender Singh","doi":"10.1016/j.jpaa.2025.107956","DOIUrl":"10.1016/j.jpaa.2025.107956","url":null,"abstract":"<div><div>A quandle equipped with a good involution is referred to as symmetric. It is known that the cohomology of symmetric quandles gives rise to strong cocycle invariants for classical and surface links, even when they are not necessarily oriented. In this paper, we introduce the category of symmetric quandle modules and prove that these modules completely determine the Beck modules in the category of symmetric quandles. Consequently, this establishes suitable coefficient objects for constructing appropriate (co)homology theories. We develop an extension theory of modules over symmetric quandles, and propose a generalized (co)homology theory for symmetric quandles with coefficients in a homogeneous Beck module, which also recovers the symmetric quandle (co)homology developed by Kamada and Oshiro (2010) <span><span>[16]</span></span>. Our constructions also apply to symmetric racks. We conclude by establishing an explicit isomorphism between the second cohomology of a symmetric quandle and the first cohomology of its associated group.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 6","pages":"Article 107956"},"PeriodicalIF":0.7,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683880","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 : 2025-03-19DOI: 10.1016/j.jpaa.2025.107957
Nguyen Bin , Vicente Lorenzo
Examples of algebraic surfaces of general type with maximal Picard number are not abundant in the literature. Moreover, most known examples either possess low invariants, lie near the Noether line or are somewhat scattered. A notable exception is Persson's sequence of double covers of the projective plane with maximal Picard number, whose invariants converge to the Severi line . This note is devoted to the construction of infinitely many new sequences of surfaces of general type with maximal Picard number whose invariants converge to the Severi line.
{"title":"Infinitely many new sequences of surfaces of general type with maximal Picard number converging to the Severi line","authors":"Nguyen Bin , Vicente Lorenzo","doi":"10.1016/j.jpaa.2025.107957","DOIUrl":"10.1016/j.jpaa.2025.107957","url":null,"abstract":"<div><div>Examples of algebraic surfaces of general type with maximal Picard number are not abundant in the literature. Moreover, most known examples either possess low invariants, lie near the Noether line <span><math><msup><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>2</mn><mi>χ</mi><mo>−</mo><mn>6</mn></math></span> or are somewhat scattered. A notable exception is Persson's sequence of double covers of the projective plane with maximal Picard number, whose invariants converge to the Severi line <span><math><msup><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>4</mn><mi>χ</mi></math></span>. This note is devoted to the construction of infinitely many new sequences of surfaces of general type with maximal Picard number whose invariants converge to the Severi line.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 6","pages":"Article 107957"},"PeriodicalIF":0.7,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143696108","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 : 2025-03-19DOI: 10.1016/j.jpaa.2025.107949
Phùng Hô Hai , Hop D. Nguyen , João Pedro dos Santos
A very useful result concerning flatness in Algebraic Geometry is EGA's “fiber” criterion. We propose similar fiber criteria to verify flatness of a module while avoiding “finiteness” assumptions. Motivated by a Tannakian viewpoint (where the category of representations comes to the front), we derive applications to the theory of affine and flat group schemes.
{"title":"Fiber criteria for flatness and homomorphisms of flat affine group schemes","authors":"Phùng Hô Hai , Hop D. Nguyen , João Pedro dos Santos","doi":"10.1016/j.jpaa.2025.107949","DOIUrl":"10.1016/j.jpaa.2025.107949","url":null,"abstract":"<div><div>A very useful result concerning flatness in Algebraic Geometry is EGA's “fiber” criterion. We propose similar fiber criteria to verify flatness of a module while avoiding “finiteness” assumptions. Motivated by a Tannakian viewpoint (where the category of representations comes to the front), we derive applications to the theory of affine and flat group schemes.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 6","pages":"Article 107949"},"PeriodicalIF":0.7,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683881","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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