{"title":"Congruence invariants of matrix mutation","authors":"Ahmet I. Seven, İbrahim Ünal","doi":"10.1016/j.jpaa.2025.107920","DOIUrl":"10.1016/j.jpaa.2025.107920","url":null,"abstract":"<div><div>Motivated by the recent work of R. Casals on binary invariants for matrix mutation, we study the matrix congruence relation on quasi-Cartan matrices. In particular, we obtain a classification and determine normal forms modulo 4. As an application, we obtain new mutation invariants, which include the one obtained by R. Casals.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 3","pages":"Article 107920"},"PeriodicalIF":0.7,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474590","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":"Differential torsion theories on Eilenberg-Moore categories of monads","authors":"Divya Ahuja, Surjeet Kour","doi":"10.1016/j.jpaa.2025.107910","DOIUrl":"10.1016/j.jpaa.2025.107910","url":null,"abstract":"<div><div>Let <span><math><mi>C</mi></math></span> be a Grothendieck category and <em>U</em> be a monad on <span><math><mi>C</mi></math></span> that is exact and preserves colimits. In this article, we prove that every hereditary torsion theory on the Eilenberg-Moore category of modules over a monad <em>U</em> is differential. Furthermore, if <span><math><mi>δ</mi><mo>:</mo><mi>U</mi><mo>⟶</mo><mi>U</mi></math></span> denotes a derivation on a monad <em>U</em>, then we show that every <em>δ</em>-derivation on a <em>U</em>-module <em>M</em> extends uniquely to a <em>δ</em>-derivation on the module of quotients of <em>M</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 3","pages":"Article 107910"},"PeriodicalIF":0.7,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143446072","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":"Derived complete complexes at weakly proregular ideals","authors":"Amnon Yekutieli","doi":"10.1016/j.jpaa.2025.107909","DOIUrl":"10.1016/j.jpaa.2025.107909","url":null,"abstract":"<div><div>Weak proregularity of an ideal in a commutative ring is a subtle generalization of the noetherian property of the ring. Weak proregularity is of special importance for the study of derived completion, and it occurs quite often in non-noetherian rings arising in Hochschild and prismatic cohomologies.</div><div>This paper is about several related topics: adically flat modules, recognizing derived complete complexes, the structure of the category of derived complete complexes, and a derived complete Nakayama theorem – all with respect to a weakly proregular ideal; and the preservation of weak proregularity under completion of the ring.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 3","pages":"Article 107909"},"PeriodicalIF":0.7,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474589","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":"Codimension two torus actions on the affine space","authors":"Alvaro Liendo , Charlie Petitjean","doi":"10.1016/j.jpaa.2025.107911","DOIUrl":"10.1016/j.jpaa.2025.107911","url":null,"abstract":"<div><div>In this paper, we classify smooth, contractible affine varieties equipped with faithful torus actions of complexity two, having a unique fixed point and a two-dimensional algebraic quotient isomorphic to a toric blow-up of a toric surface. These varieties are of particular interest as they represent the simplest candidates for potential counterexamples to the linearization conjecture in affine geometry.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 5","pages":"Article 107911"},"PeriodicalIF":0.7,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529836","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":"Reducibility by polynomial functions","authors":"Riccardo Camerlo , Carla Massaza","doi":"10.1016/j.jpaa.2025.107907","DOIUrl":"10.1016/j.jpaa.2025.107907","url":null,"abstract":"<div><div>We study the preorder <span><math><msub><mrow><mo>≤</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span> (and the associated equivalence relation <span><math><msub><mrow><mo>≡</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span>) on the family of subsets of an algebraically closed field <em>k</em> of characteristic 0, defined by letting <span><math><mi>A</mi><msub><mrow><mo>≤</mo></mrow><mrow><mi>Pol</mi></mrow></msub><mi>B</mi></math></span> iff there exists a polynomial <em>P</em> such that <span><math><mi>A</mi><mo>=</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>B</mi><mo>)</mo></math></span>. We concentrate mainly on the finite subsets of <em>k</em> and prove that the <span><math><msub><mrow><mo>≡</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span>-equivalence classes of sets of a given finite cardinality form an affine algebraic variety; inside these varieties, we compute in particular the dimension of the set of <span><math><msub><mrow><mo>≡</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span>-classes that have less representatives than the generic ones and the dimension of the set of <span><math><msub><mrow><mo>≡</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span>-classes that are comparable with a given <span><math><msub><mrow><mo>≡</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span>-class.</div><div>We also show that, in a specified sense, very many <span><math><msub><mrow><mo>≡</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span>-classes are <span><math><msub><mrow><mo>≤</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span>-maximal (or <span><math><msub><mrow><mo>≤</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span>-maximal under the class of singletons, for <span><math><msub><mrow><mo>≡</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span>-classes of finite sets).</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 3","pages":"Article 107907"},"PeriodicalIF":0.7,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143446073","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":"Relative Koszul coresolutions and relative Betti numbers","authors":"Hideto Asashiba","doi":"10.1016/j.jpaa.2025.107905","DOIUrl":"10.1016/j.jpaa.2025.107905","url":null,"abstract":"<div><div>Let <em>G</em> be a finitely generated right <em>A</em>-module for a finite-dimensional algebra <em>A</em> over a field <span><math><mi>k</mi></math></span>, and <span><math><mi>I</mi></math></span> the additive closure of <em>G</em>. We will define an <span><math><mi>I</mi></math></span>-relative Koszul coresolution <figure><img></figure> of an indecomposable direct summand <em>V</em> of <em>G</em>, and show that for a finitely generated <em>A</em>-module <em>M</em>, the <span><math><mi>I</mi></math></span>-relative <em>i</em>-th Betti number for <em>M</em> at <em>V</em> is given as the <span><math><mi>k</mi></math></span>-dimension of the <em>i</em>-th homology of the <span><math><mi>I</mi></math></span>-relative Koszul complex <figure><img></figure> of <em>M</em> at <em>V</em> for all <span><math><mi>i</mi><mo>≥</mo><mn>0</mn></math></span>. This is applied to investigate the minimal interval resolution/coresolution of a persistence module <em>M</em>, e.g., to check the interval decomposability of <em>M</em>, and to compute the interval approximation of <em>M</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 3","pages":"Article 107905"},"PeriodicalIF":0.7,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143429062","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":"Minimal resolutions of lattice ideals","authors":"Yupeng Li , Ezra Miller , Erika Ordog","doi":"10.1016/j.jpaa.2025.107901","DOIUrl":"10.1016/j.jpaa.2025.107901","url":null,"abstract":"<div><div>A canonical minimal free resolution of an arbitrary co-artinian lattice ideal over the polynomial ring is constructed over any field whose characteristic is 0 or any but finitely many positive primes. The differential has a closed-form combinatorial description as a sum over lattice paths in <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> of weights that come from sequences of faces in simplicial complexes indexed by lattice points. Over a field of any characteristic, a non-canonical but simpler resolution is constructed by selecting choices of higher-dimensional analogues of spanning trees along lattice paths. These constructions generalize sylvan resolutions for monomial ideals by lifting them equivariantly to lattice modules.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 3","pages":"Article 107901"},"PeriodicalIF":0.7,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143437093","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":"On pyramidal groups of prime power degree","authors":"Xiaofang Gao , Martino Garonzi","doi":"10.1016/j.jpaa.2025.107868","DOIUrl":"10.1016/j.jpaa.2025.107868","url":null,"abstract":"<div><div>A Kirkman Triple System Γ is called <em>m</em>-pyramidal if there exists a subgroup <em>G</em> of the automorphism group of Γ that fixes <em>m</em> points and acts regularly on the other points. Such group <em>G</em> admits a unique conjugacy class <em>C</em> of involutions (elements of order 2) and <span><math><mo>|</mo><mi>C</mi><mo>|</mo><mo>=</mo><mi>m</mi></math></span>. We call groups with this property <em>m</em>-pyramidal. We prove that, if <em>m</em> is an odd prime power <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>, with <span><math><mi>p</mi><mo>≠</mo><mn>7</mn></math></span>, then every <em>m</em>-pyramidal group is solvable if and only if either <span><math><mi>m</mi><mo>=</mo><mn>9</mn></math></span> or <em>k</em> is odd. The primitive permutation groups play an important role in the proof. We also determine the orders of the <em>m</em>-pyramidal groups when <em>m</em> is a prime number.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107868"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163678","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":"Categories of sets with infinite addition","authors":"Pablo Andrés-Martínez , Chris Heunen","doi":"10.1016/j.jpaa.2025.107872","DOIUrl":"10.1016/j.jpaa.2025.107872","url":null,"abstract":"<div><div>We consider sets with infinite addition, called Σ-monoids, and contribute to their literature in three ways. First, our definition subsumes those from previous works and allows us to relate them in terms of adjuctions between their categories. In particular, we discuss Σ-monoids with additive inverses. Second, we show that every Hausdorff commutative monoid is a Σ-monoid, and that there is a free Hausdorff commutative monoid for each Σ-monoid. Third, we prove that Σ-monoids have well-defined tensor products, unlike topological abelian groups.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107872"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163681","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":"Steenrod closed parameter ideals in the mod-2 cohomology of A4 and SO(3)","authors":"Henrik Rüping , Marc Stephan , Ergün Yalçın","doi":"10.1016/j.jpaa.2025.107899","DOIUrl":"10.1016/j.jpaa.2025.107899","url":null,"abstract":"<div><div>In this paper, we classify the parameter ideals in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>B</mi><msub><mrow><mi>A</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>;</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> and in the Dickson algebra <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>B</mi><mi>SO</mi><mo>(</mo><mn>3</mn><mo>)</mo><mo>;</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> that are closed under Steenrod operations. Consequently, we obtain restrictions on the dimensions <span><math><mi>n</mi><mo>,</mo><mi>m</mi></math></span> for which <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> (and <span><math><mi>SO</mi><mo>(</mo><mn>3</mn><mo>)</mo></math></span>) can act freely on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107899"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387715","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}
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