Pub Date : 2025-11-03DOI: 10.1016/j.jpaa.2025.108123
Ruipeng Zhu
This paper investigates the homological properties of the faithfully flat Hopf Galois extension . It establishes that when B is a noetherian affine PI algebra and A is AS Gorenstein, B inherits the AS Gorenstein property. Furthermore, we demonstrate that injective dimension serves as a monoidal invariant for AS Gorenstein Hopf algebras. Specifically, if two such Hopf algebras have equivalent monoidal categories of comodules, then their injective dimensions are equal.
{"title":"Artin-Schelter Gorenstein property of Hopf Galois extensions","authors":"Ruipeng Zhu","doi":"10.1016/j.jpaa.2025.108123","DOIUrl":"10.1016/j.jpaa.2025.108123","url":null,"abstract":"<div><div>This paper investigates the homological properties of the faithfully flat Hopf Galois extension <span><math><mi>A</mi><mo>⊆</mo><mi>B</mi></math></span>. It establishes that when <em>B</em> is a noetherian affine PI algebra and <em>A</em> is AS Gorenstein, <em>B</em> inherits the AS Gorenstein property. Furthermore, we demonstrate that injective dimension serves as a monoidal invariant for AS Gorenstein Hopf algebras. Specifically, if two such Hopf algebras have equivalent monoidal categories of comodules, then their injective dimensions are equal.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108123"},"PeriodicalIF":0.8,"publicationDate":"2025-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145475353","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 : 2025-11-01DOI: 10.1016/j.jpaa.2025.108118
Ramin Ebrahimi
Let be a skeletally small additive category. Using the canonical equivalence between two different presentations of the free abelian category over , we give a new and simple characterization of definable subcategories of , and in particular definable subcategories of modules over rings. In the end, we give a conceptual proof of Auslander-Gruson-Jensen duality, which makes the duality between definable subcategories of left and right module more transparent.
{"title":"On definable subcategories","authors":"Ramin Ebrahimi","doi":"10.1016/j.jpaa.2025.108118","DOIUrl":"10.1016/j.jpaa.2025.108118","url":null,"abstract":"<div><div>Let <span><math><mi>X</mi></math></span> be a skeletally small additive category. Using the canonical equivalence between two different presentations of the free abelian category over <span><math><mi>X</mi></math></span>, we give a new and simple characterization of definable subcategories of <span><math><mi>Mod</mi><mspace></mspace><mtext>-</mtext><mi>X</mi></math></span>, and in particular definable subcategories of modules over rings. In the end, we give a conceptual proof of Auslander-Gruson-Jensen duality, which makes the duality between definable subcategories of left and right module more transparent.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108118"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416437","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 : 2025-11-01DOI: 10.1016/j.jpaa.2025.108111
Dario Spirito
We generalize the theory of radical factorization from almost Dedekind domain to strongly discrete Prüfer domains; we show that, for a fixed subset X of maximal ideals, the finitely generated ideals with have radical factorization if and only if X contains no critical maximal ideals with respect to X. We use these notions to prove that the group of the invertible ideals of a strongly discrete Prüfer domain is often free: in particular, we show it is free when the spectrum of D is Noetherian or when D is a ring of integer-valued polynomials on a subset over a Dedekind domain.
{"title":"Radical factorization in higher dimension","authors":"Dario Spirito","doi":"10.1016/j.jpaa.2025.108111","DOIUrl":"10.1016/j.jpaa.2025.108111","url":null,"abstract":"<div><div>We generalize the theory of radical factorization from almost Dedekind domain to strongly discrete Prüfer domains; we show that, for a fixed subset <em>X</em> of maximal ideals, the finitely generated ideals with <span><math><mi>V</mi><mo>(</mo><mi>I</mi><mo>)</mo><mo>⊆</mo><mi>X</mi></math></span> have radical factorization if and only if <em>X</em> contains no critical maximal ideals with respect to <em>X</em>. We use these notions to prove that the group <span><math><mrow><mi>Inv</mi></mrow><mo>(</mo><mi>D</mi><mo>)</mo></math></span> of the invertible ideals of a strongly discrete Prüfer domain is often free: in particular, we show it is free when the spectrum of <em>D</em> is Noetherian or when <em>D</em> is a ring of integer-valued polynomials on a subset over a Dedekind domain.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108111"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145424588","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 : 2025-11-01DOI: 10.1016/j.jpaa.2025.108112
Neil Epstein
The Fundamental Theorem of Algebra can be thought of as a statement about the real numbers as a space, considered as an algebraic set over the real numbers as a field. This paper introduces what it means for an algebraic set or affine variety over a field to be fundamental, in a way that encompasses the Fundamental Theorem of Algebra as a special case. The related concept of local fundamentality is introduced and its behavior developed. On the algebraic side, the notions of locally, geometrically, and generically unit-additive rings are introduced, thus complementing unit-additivity as previously defined by the author and Jay Shapiro. A number of results are extended from the previous joint paper from unit-additivity to local unit-additivity. It is shown that an affine variety is (locally) fundamental if and only if its coordinate ring is (locally) unit-additive. To do so, a theorem is proved showing that there are many equivalent definitions of local unit-additivity. Illustrative examples are sprinkled throughout.
{"title":"Fundamental algebraic sets and locally unit-additive rings","authors":"Neil Epstein","doi":"10.1016/j.jpaa.2025.108112","DOIUrl":"10.1016/j.jpaa.2025.108112","url":null,"abstract":"<div><div>The Fundamental Theorem of Algebra can be thought of as a statement about the real numbers as a space, considered as an algebraic set over the real numbers as a field. This paper introduces what it means for an algebraic set or affine variety over a field to be <em>fundamental</em>, in a way that encompasses the Fundamental Theorem of Algebra as a special case. The related concept of <em>local</em> fundamentality is introduced and its behavior developed. On the algebraic side, the notions of <em>locally</em>, <em>geometrically</em>, and <em>generically unit-additive</em> rings are introduced, thus complementing <em>unit-additivity</em> as previously defined by the author and Jay Shapiro. A number of results are extended from the previous joint paper from unit-additivity to local unit-additivity. It is shown that an affine variety is (locally) fundamental if and only if its coordinate ring is (locally) unit-additive. To do so, a theorem is proved showing that there are many equivalent definitions of local unit-additivity. Illustrative examples are sprinkled throughout.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108112"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416440","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 : 2025-11-01DOI: 10.1016/j.jpaa.2025.108113
Ritesh Kumar Pandey, Sachin S. Sharma
In this paper, we extend the notion of Weyl modules for twisted toroidal Lie algebra . We prove that the level one global Weyl modules of are isomorphic to the tensor product of the level one representation of twisted affine Lie algebras and certain lattice vertex algebras. As a byproduct, we calculate the graded character of the level one local Weyl modules of .
{"title":"Weyl modules for twisted toroidal Lie algebras","authors":"Ritesh Kumar Pandey, Sachin S. Sharma","doi":"10.1016/j.jpaa.2025.108113","DOIUrl":"10.1016/j.jpaa.2025.108113","url":null,"abstract":"<div><div>In this paper, we extend the notion of Weyl modules for twisted toroidal Lie algebra <span><math><mi>T</mi><mo>(</mo><mi>μ</mi><mo>)</mo></math></span>. We prove that the level one global Weyl modules of <span><math><mi>T</mi><mo>(</mo><mi>μ</mi><mo>)</mo></math></span> are isomorphic to the tensor product of the level one representation of twisted affine Lie algebras and certain lattice vertex algebras. As a byproduct, we calculate the graded character of the level one local Weyl modules of <span><math><mi>T</mi><mo>(</mo><mi>μ</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108113"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416438","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 : 2025-11-01DOI: 10.1016/j.jpaa.2025.108115
Teena Gerhardt , Maximilien Péroux , W. Hermann B. Soré
We establish comparison maps between the classical algebraic K-theory of algebras over a field and its analogue , an algebraic K-theory for coalgebras over a field. The comparison maps are compatible with the Hattori–Stallings (co)traces. We identify conditions on the algebras or coalgebras under which the comparison maps are equivalences. Notably, the algebraic K-theory of the power series ring is equivalent to the -theory of the divided power coalgebra. We also establish comparison maps between the G-theory of finite dimensional representations of an algebra and its analogue for coalgebras. In particular, we show that the Swan theory of a group is equivalent to the -theory of the representative functions coalgebra, reframing the classical character of a group as a trace in coHochschild homology.
{"title":"Coalgebraic K-theory","authors":"Teena Gerhardt , Maximilien Péroux , W. Hermann B. Soré","doi":"10.1016/j.jpaa.2025.108115","DOIUrl":"10.1016/j.jpaa.2025.108115","url":null,"abstract":"<div><div>We establish comparison maps between the classical algebraic <em>K</em>-theory of algebras over a field and its analogue <span><math><msup><mrow><mi>K</mi></mrow><mrow><mi>c</mi></mrow></msup></math></span>, an algebraic <em>K</em>-theory for coalgebras over a field. The comparison maps are compatible with the Hattori–Stallings (co)traces. We identify conditions on the algebras or coalgebras under which the comparison maps are equivalences. Notably, the algebraic <em>K</em>-theory of the power series ring is equivalent to the <span><math><msup><mrow><mi>K</mi></mrow><mrow><mi>c</mi></mrow></msup></math></span>-theory of the divided power coalgebra. We also establish comparison maps between the <em>G</em>-theory of finite dimensional representations of an algebra and its analogue <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>c</mi></mrow></msup></math></span> for coalgebras. In particular, we show that the Swan theory of a group is equivalent to the <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>c</mi></mrow></msup></math></span>-theory of the representative functions coalgebra, reframing the classical character of a group as a trace in coHochschild homology.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108115"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416436","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 : 2025-11-01DOI: 10.1016/j.jpaa.2025.108117
J. Miguel Calderón
In this article, we describe the lattice of ideals of some Green biset functors. We consider Green biset functors which satisfy that each evaluation is a finite-dimensional split semisimple commutative algebra and use the idempotents in these evaluations to characterize any ideal of these Green biset functors. For this we will give the definition of MC-group, this definition generalizes that of a B-group, given for the Burnside functor.
Given a Green biset functor A, with the above hypotheses, the set of all MC-groups of A has a structure of a poset and we prove that there exists an isomorphism of lattices between the set of ideals of A and the set of upward closed subsets of the MC-groups of A.
{"title":"Ideals of some Green biset functors","authors":"J. Miguel Calderón","doi":"10.1016/j.jpaa.2025.108117","DOIUrl":"10.1016/j.jpaa.2025.108117","url":null,"abstract":"<div><div>In this article, we describe the lattice of ideals of some Green biset functors. We consider Green biset functors which satisfy that each evaluation is a finite-dimensional split semisimple commutative algebra and use the idempotents in these evaluations to characterize any ideal of these Green biset functors. For this we will give the definition of <em>MC</em>-group, this definition generalizes that of a <em>B</em>-group, given for the Burnside functor.</div><div>Given a Green biset functor <em>A</em>, with the above hypotheses, the set of all <em>MC</em>-groups of <em>A</em> has a structure of a poset and we prove that there exists an isomorphism of lattices between the set of ideals of <em>A</em> and the set of upward closed subsets of the <em>MC</em>-groups of <em>A</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108117"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416439","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 : 2025-10-30DOI: 10.1016/j.jpaa.2025.108120
Fulin Chen , Xin Huang , Fei Kong , Shaobin Tan
In this paper, we associate the quantum toroidal algebra of type with quantum vertex algebra through equivariant ϕ-coordinated quasi modules. More precisely, for every , by deforming the universal affine vertex algebra of , we construct an ħ-adic quantum -vertex algebra . Then we prove that the category of restricted -modules of level ℓ is canonically isomorphic to that of equivariant ϕ-coordinated quasi -modules.
{"title":"Quantum vertex algebra associated to quantum toroidal glN","authors":"Fulin Chen , Xin Huang , Fei Kong , Shaobin Tan","doi":"10.1016/j.jpaa.2025.108120","DOIUrl":"10.1016/j.jpaa.2025.108120","url":null,"abstract":"<div><div>In this paper, we associate the quantum toroidal algebra <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> of type <span><math><msub><mrow><mi>gl</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> with quantum vertex algebra through equivariant <em>ϕ</em>-coordinated quasi modules. More precisely, for every <span><math><mi>ℓ</mi><mo>∈</mo><mi>C</mi></math></span>, by deforming the universal affine vertex algebra of <span><math><msub><mrow><mi>sl</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>, we construct an <em>ħ</em>-adic quantum <span><math><mi>Z</mi></math></span>-vertex algebra <span><math><msub><mrow><mi>V</mi></mrow><mrow><msub><mrow><mover><mrow><mi>sl</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mo>∞</mo></mrow></msub><mo>,</mo><mi>ħ</mi></mrow></msub><mo>(</mo><mi>ℓ</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></span>. Then we prove that the category of restricted <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span>-modules of level <em>ℓ</em> is canonically isomorphic to that of equivariant <em>ϕ</em>-coordinated quasi <span><math><msub><mrow><mi>V</mi></mrow><mrow><msub><mrow><mover><mrow><mi>sl</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mo>∞</mo></mrow></msub><mo>,</mo><mi>ħ</mi></mrow></msub><mo>(</mo><mi>ℓ</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></span>-modules.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108120"},"PeriodicalIF":0.8,"publicationDate":"2025-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145475354","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 : 2025-10-27DOI: 10.1016/j.jpaa.2025.108119
Victor Carmona
This work addresses the homotopical analysis of enveloping operads in a general cofibrantly generated symmetric monoidal model category. We show the potential of this analysis by obtaining, in a uniform way, several central results regarding the homotopy theory of operadic algebras.
{"title":"Enveloping operads and applications","authors":"Victor Carmona","doi":"10.1016/j.jpaa.2025.108119","DOIUrl":"10.1016/j.jpaa.2025.108119","url":null,"abstract":"<div><div>This work addresses the homotopical analysis of enveloping operads in a general cofibrantly generated symmetric monoidal model category. We show the potential of this analysis by obtaining, in a uniform way, several central results regarding the homotopy theory of operadic algebras.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108119"},"PeriodicalIF":0.8,"publicationDate":"2025-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145384599","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 : 2025-10-21DOI: 10.1016/j.jpaa.2025.108110
Giacomo Tendas
Working within enriched category theory, we further develop the use of soundness, introduced by Adámek, Borceux, Lack, and Rosický for ordinary categories. In particular we investigate: (1) the theory of locally Φ-presentable -categories for a sound class Φ, (2) the problem of whether every Φ-accessible -category is Ψ-accessible, for given sound classes , and (3) a notion of Φ-ary equational theory whose -categories of models characterize algebras for Φ-ary monads on .
{"title":"More on soundness in the enriched context","authors":"Giacomo Tendas","doi":"10.1016/j.jpaa.2025.108110","DOIUrl":"10.1016/j.jpaa.2025.108110","url":null,"abstract":"<div><div>Working within enriched category theory, we further develop the use of soundness, introduced by Adámek, Borceux, Lack, and Rosický for ordinary categories. In particular we investigate: (1) the theory of locally Φ-presentable <span><math><mi>V</mi></math></span>-categories for a sound class Φ, (2) the problem of whether every Φ-accessible <span><math><mi>V</mi></math></span>-category is Ψ-accessible, for given sound classes <span><math><mi>Φ</mi><mo>⊆</mo><mi>Ψ</mi></math></span>, and (3) a notion of Φ-ary equational theory whose <span><math><mi>V</mi></math></span>-categories of models characterize algebras for Φ-ary monads on <span><math><mi>V</mi></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108110"},"PeriodicalIF":0.8,"publicationDate":"2025-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145363618","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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