Pub Date : 2025-12-01DOI: 10.1016/j.jpaa.2025.108133
Ihechukwu Chinyere , Martin Edjvet , Gerald Williams
We consider the cyclically presented groups defined by cyclic presentations with 2m generators whose relators are the 2m positive length three relators . We show that they are hyperbolic if and only if . This completes the classification of the hyperbolic cyclically presented groups with positive length three relators.
{"title":"All hyperbolic cyclically presented groups with positive length three relators","authors":"Ihechukwu Chinyere , Martin Edjvet , Gerald Williams","doi":"10.1016/j.jpaa.2025.108133","DOIUrl":"10.1016/j.jpaa.2025.108133","url":null,"abstract":"<div><div>We consider the cyclically presented groups defined by cyclic presentations with 2<em>m</em> generators <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> whose relators are the 2<em>m</em> positive length three relators <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>. We show that they are hyperbolic if and only if <span><math><mi>m</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>9</mn><mo>}</mo></math></span>. This completes the classification of the hyperbolic cyclically presented groups with positive length three relators.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108133"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145693704","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 : 2025-12-01DOI: 10.1016/j.jpaa.2025.108136
John G. Miller
This paper studies how persistence categories and triangulated persistence categories behave with respect to taking idempotent completions. In particular we study when the idempotent completion (Karoubi envelope) of categories admitting persistence refinement also admits such a refinement. In doing so, we introduce notions of persistence semi-categories and persistent presheaves and explore their properties.
{"title":"Idempotent completion of persistence categories","authors":"John G. Miller","doi":"10.1016/j.jpaa.2025.108136","DOIUrl":"10.1016/j.jpaa.2025.108136","url":null,"abstract":"<div><div>This paper studies how persistence categories and triangulated persistence categories behave with respect to taking idempotent completions. In particular we study when the idempotent completion (Karoubi envelope) of categories admitting persistence refinement also admits such a refinement. In doing so, we introduce notions of persistence semi-categories and persistent presheaves and explore their properties.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108136"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623946","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 : 2025-12-01DOI: 10.1016/j.jpaa.2025.108138
Matthew B. Day , Trevor Nakamura
We consider normal subgroups N of the braid group such that the quotient is an extension of the symmetric group by an abelian group. We show that, if , then there are exactly 8 commensurability classes of such subgroups. We define a Specht subgroup to be a subgroup of this form that is maximal in its commensurability class. We give descriptions of the Specht subgroups in terms of winding numbers and in terms of infinite generating sets. The quotient of the pure braid group by a Specht subgroup is a module over the symmetric group. We show that the modules arising this way are closely related to Specht modules for the partitions and , working over the integers. We compute the second cohomology of the symmetric group with coefficients in both of these Specht modules, working over an arbitrary commutative ring. Finally, we determine which of the extensions of the symmetric group arising from Specht subgroups are split extensions.
{"title":"Quotients of the braid group that are extensions of the symmetric group","authors":"Matthew B. Day , Trevor Nakamura","doi":"10.1016/j.jpaa.2025.108138","DOIUrl":"10.1016/j.jpaa.2025.108138","url":null,"abstract":"<div><div>We consider normal subgroups <em>N</em> of the braid group <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> such that the quotient <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>/</mo><mi>N</mi></math></span> is an extension of the symmetric group by an abelian group. We show that, if <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>, then there are exactly 8 commensurability classes of such subgroups. We define a <em>Specht subgroup</em> to be a subgroup of this form that is maximal in its commensurability class. We give descriptions of the Specht subgroups in terms of winding numbers and in terms of infinite generating sets. The quotient of the pure braid group by a Specht subgroup is a module over the symmetric group. We show that the modules arising this way are closely related to Specht modules for the partitions <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, working over the integers. We compute the second cohomology of the symmetric group with coefficients in both of these Specht modules, working over an arbitrary commutative ring. Finally, we determine which of the extensions of the symmetric group arising from Specht subgroups are split extensions.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108138"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145747650","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 : 2025-12-01DOI: 10.1016/j.jpaa.2025.108141
Xiao-Wu Chen , Ren Wang
Let be a field of characteristic p and G be a cyclic p-group which acts on a finite acyclic quiver Q. The folding process associates a Cartan triple to the action. We establish a Morita equivalence between the skew group algebra of the preprojective algebra of Q and the generalized preprojective algebra associated to the Cartan triple in the sense of Geiss, Leclerc and Schröer. The Morita equivalence induces an isomorphism between certain ideal monoids of these preprojective algebras, which is compatible with the embedding of Weyl groups appearing in the folding process.
{"title":"Preprojective algebras, skew group algebras and Morita equivalences","authors":"Xiao-Wu Chen , Ren Wang","doi":"10.1016/j.jpaa.2025.108141","DOIUrl":"10.1016/j.jpaa.2025.108141","url":null,"abstract":"<div><div>Let <span><math><mi>K</mi></math></span> be a field of characteristic <em>p</em> and <em>G</em> be a cyclic <em>p</em>-group which acts on a finite acyclic quiver <em>Q</em>. The folding process associates a Cartan triple to the action. We establish a Morita equivalence between the skew group algebra of the preprojective algebra of <em>Q</em> and the generalized preprojective algebra associated to the Cartan triple in the sense of Geiss, Leclerc and Schröer. The Morita equivalence induces an isomorphism between certain ideal monoids of these preprojective algebras, which is compatible with the embedding of Weyl groups appearing in the folding process.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108141"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145693703","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 : 2025-12-01DOI: 10.1016/j.jpaa.2025.108143
Onofrio M. Di Vincenzo , Vincenzo C. Nardozza
Let F be a field of characteristic zero and let E be the Grassmann algebra of an infinite-dimensional F-vector space. We consider a class of solvable nonabelian finite-dimensional Lie algebras acting on E by derivations, and completely describe the differential polynomial identities satisfied by E. The corresponding -cocharacter and differential codimension sequences are computed. Finally, we prove that the differential exponent exists and equals the ordinary exponent of E.
{"title":"Differential identities of the Grassmann algebra","authors":"Onofrio M. Di Vincenzo , Vincenzo C. Nardozza","doi":"10.1016/j.jpaa.2025.108143","DOIUrl":"10.1016/j.jpaa.2025.108143","url":null,"abstract":"<div><div>Let <em>F</em> be a field of characteristic zero and let <em>E</em> be the Grassmann algebra of an infinite-dimensional <em>F</em>-vector space. We consider a class of solvable nonabelian finite-dimensional Lie algebras acting on <em>E</em> by derivations, and completely describe the differential polynomial identities satisfied by <em>E</em>. The corresponding <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>-cocharacter and differential codimension sequences are computed. Finally, we prove that the differential exponent exists and equals the ordinary exponent of <em>E</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108143"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145693705","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 : 2025-12-01DOI: 10.1016/j.jpaa.2025.108144
Cheng Meng
In this paper, we prove that if P is a homogeneous prime ideal inside a standard graded polynomial ring S with , and for , adjoining s general linear forms to the prime ideal changes the -th Hilbert coefficient by 1, then . This criterion also tells us about possible restrictions on the generic initial ideal of a prime ideal inside a polynomial ring.
本文证明了如果P是标准渐变多项式环S内的齐次素理想，且当S≤d时，与该素理想相邻的S种一般线性形式使(d - S)-希尔伯特系数改变1，则深度(S/P)= S - 1。这个判据也告诉我们多项式环内素数理想的一般初始理想的可能约束。
{"title":"Restrictions on Hilbert coefficients give depths of graded domains","authors":"Cheng Meng","doi":"10.1016/j.jpaa.2025.108144","DOIUrl":"10.1016/j.jpaa.2025.108144","url":null,"abstract":"<div><div>In this paper, we prove that if <em>P</em> is a homogeneous prime ideal inside a standard graded polynomial ring <em>S</em> with <span><math><mi>dim</mi><mo></mo><mo>(</mo><mi>S</mi><mo>/</mo><mi>P</mi><mo>)</mo><mo>=</mo><mi>d</mi></math></span>, and for <span><math><mi>s</mi><mo>≤</mo><mi>d</mi></math></span>, adjoining <em>s</em> general linear forms to the prime ideal changes the <span><math><mo>(</mo><mi>d</mi><mo>−</mo><mi>s</mi><mo>)</mo></math></span>-th Hilbert coefficient by 1, then <span><math><mtext>depth</mtext><mo>(</mo><mi>S</mi><mo>/</mo><mi>P</mi><mo>)</mo><mo>=</mo><mi>s</mi><mo>−</mo><mn>1</mn></math></span>. This criterion also tells us about possible restrictions on the generic initial ideal of a prime ideal inside a polynomial ring.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108144"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145693706","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 : 2025-12-01DOI: 10.1016/j.jpaa.2025.108134
Carmelo Antonio Finocchiaro , K. Alan Loper
Let D be a Dedekind domain (not a field) with finite residue fields and let be the ring of integer-valued polynomials over D. We completely classify in topological terms some relevant classes of radical unitary ideals of (and of its overrings). This project strongly extends the classification given in a previous paper and regarding special unitary ideals, precisely the ones lying over a given maximal ideal of D.
{"title":"On radical unitary ideals of rings of integer-valued polynomials","authors":"Carmelo Antonio Finocchiaro , K. Alan Loper","doi":"10.1016/j.jpaa.2025.108134","DOIUrl":"10.1016/j.jpaa.2025.108134","url":null,"abstract":"<div><div>Let <em>D</em> be a Dedekind domain (not a field) with finite residue fields and let <span><math><mi>Int</mi><mo>(</mo><mi>D</mi><mo>)</mo></math></span> be the ring of integer-valued polynomials over <em>D</em>. We completely classify in topological terms some relevant classes of radical unitary ideals of <span><math><mi>Int</mi><mo>(</mo><mi>D</mi><mo>)</mo></math></span> (and of its overrings). This project strongly extends the classification given in a previous paper and regarding special unitary ideals, precisely the ones lying over a given maximal ideal of <em>D</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108134"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623948","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 : 2025-12-01DOI: 10.1016/j.jpaa.2025.108146
Yilin Wu , Jinyi Xu , Guodong Zhou
Several GAGA-type results for singularity categories are presented. Firstly, as an easy consequence of Serre's GAGA theorem, we show that for a complex projective variety, its singularity category is naturally equivalent to that of its analytification.
Secondly, we introduce the torsion singularity category of a formal scheme. Under Orlov's (ELF) condition, we prove that for the formal completion of a Noetherian scheme along a closed subset, its torsion singularity category is equivalent to the singularity category of the original scheme, with support in the closed subset.
Lastly, using the Artin Approximation Theorem and the result above, we provide an alternative proof of a result of Orlov. Namely, for a Noetherian local ring with an isolated singularity, its singularity category is equivalent (up to direct summands) to that of its Henselization, which in turn is equivalent to that of its completion.
{"title":"GAGA type results for singularity categories","authors":"Yilin Wu , Jinyi Xu , Guodong Zhou","doi":"10.1016/j.jpaa.2025.108146","DOIUrl":"10.1016/j.jpaa.2025.108146","url":null,"abstract":"<div><div>Several GAGA-type results for singularity categories are presented. Firstly, as an easy consequence of Serre's GAGA theorem, we show that for a complex projective variety, its singularity category is naturally equivalent to that of its analytification.</div><div>Secondly, we introduce the torsion singularity category of a formal scheme. Under Orlov's (ELF) condition, we prove that for the formal completion of a Noetherian scheme along a closed subset, its torsion singularity category is equivalent to the singularity category of the original scheme, with support in the closed subset.</div><div>Lastly, using the Artin Approximation Theorem and the result above, we provide an alternative proof of a result of Orlov. Namely, for a Noetherian local ring with an isolated singularity, its singularity category is equivalent (up to direct summands) to that of its Henselization, which in turn is equivalent to that of its completion.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108146"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145747651","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 : 2025-12-01DOI: 10.1016/j.jpaa.2025.108142
A.A. Ambily, H. Sugilesh
We prove Horrocks' theorem for the odd elementary orthogonal group, which gives a decomposition of an orthogonal matrix with entries from a polynomial ring , over a commutative ring R in which 2 is invertible, as a product of an orthogonal matrix with entries in R and an elementary orthogonal matrix with entries from .
{"title":"Horrocks' theorem for odd orthogonal groups","authors":"A.A. Ambily, H. Sugilesh","doi":"10.1016/j.jpaa.2025.108142","DOIUrl":"10.1016/j.jpaa.2025.108142","url":null,"abstract":"<div><div>We prove Horrocks' theorem for the odd elementary orthogonal group, which gives a decomposition of an orthogonal matrix with entries from a polynomial ring <span><math><mi>R</mi><mo>[</mo><mi>X</mi><mo>]</mo></math></span>, over a commutative ring <em>R</em> in which 2 is invertible, as a product of an orthogonal matrix with entries in <em>R</em> and an elementary orthogonal matrix with entries from <span><math><mi>R</mi><mo>[</mo><mi>X</mi><mo>]</mo></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108142"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145747652","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 : 2025-12-01DOI: 10.1016/j.jpaa.2025.108139
Jiří Adámek
Classical varieties were characterized by Lawvere as the categories with effective congruences and a varietal generator: an abstractly finite, regular generator which is regularly projective (its hom-functor preserves regular epimorphisms). We characterize varieties of quantitative algebras of Mardare, Panangaden and Plotkin analogously as metric-enriched categories. We introduce the concept of a subcongruence (a metric-enriched analogue of a congruence) and the corresponding subregular epimorphisms, obtained via colimits of subcongruences. Varieties of quantitative algebras are precisely the metric-enriched categories with effective subcongruences and a subvarietal generator: an abstractly finite, subregular generator which is subregularly projective (its hom-functor preserves subregular epimorphisms).
{"title":"Which categories are varieties of quantitative algebras?","authors":"Jiří Adámek","doi":"10.1016/j.jpaa.2025.108139","DOIUrl":"10.1016/j.jpaa.2025.108139","url":null,"abstract":"<div><div>Classical varieties were characterized by Lawvere as the categories with effective congruences and a varietal generator: an abstractly finite, regular generator which is regularly projective (its hom-functor preserves regular epimorphisms). We characterize varieties of quantitative algebras of Mardare, Panangaden and Plotkin analogously as metric-enriched categories. We introduce the concept of a subcongruence (a metric-enriched analogue of a congruence) and the corresponding subregular epimorphisms, obtained via colimits of subcongruences. Varieties of quantitative algebras are precisely the metric-enriched categories with effective subcongruences and a subvarietal generator: an abstractly finite, subregular generator which is subregularly projective (its hom-functor preserves subregular epimorphisms).</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108139"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145747653","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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