Journal of Pure and Applied Algebra最新文献

{"title":"Δ-locally nilpotent algebras, their ideal structure and simplicity criteria","authors":"V.V. Bavula","doi":"10.1016/j.jpaa.2024.107861","DOIUrl":"10.1016/j.jpaa.2024.107861","url":null,"abstract":"<div><div>The class of Δ-locally nilpotent algebras introduced in the paper is a wide generalization of the algebras of differential operators on commutative algebras. Examples include all the rings <span><math><mi>D</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> of differential operators on commutative algebras in arbitrary characteristic, all subalgebras of <span><math><mi>D</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> that contain the algebra <em>A</em>, the universal enveloping algebras of nilpotent, solvable and semi-simple Lie algebras, the Poisson universal enveloping algebra of an arbitrary Poisson algebra, iterated Ore extensions <span><math><mi>A</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>;</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span>, certain generalized Weyl algebras, and others.</div><div>In <span><span>[8]</span></span>, simplicity criteria are given for the algebras differential operators on commutative algebras. To find the simplicity criterion was a long standing problem from 60'th. The aim of the paper is to describe the ideal structure of Δ-locally nilpotent algebras and as a corollary to give simplicity criteria for them. These results are generalizations of the results of <span><span>[8]</span></span>. Examples are considered.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107861"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163664","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":"Classification of G2-orbits for pairs of octonions","authors":"Artem Lopatin ,&nbsp;Alexandr N. Zubkov","doi":"10.1016/j.jpaa.2025.107875","DOIUrl":"10.1016/j.jpaa.2025.107875","url":null,"abstract":"<div><div>Over an algebraically closed field, we described a minimal set of representatives for <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-orbits on the set <span><math><msup><mrow><mi>O</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> of pairs of octonions.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107875"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163666","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":"Semi-galois Categories I: The classical Eilenberg variety theory","authors":"Takeo Uramoto","doi":"10.1016/j.jpaa.2025.107863","DOIUrl":"10.1016/j.jpaa.2025.107863","url":null,"abstract":"<div><div>This paper is an extended version of our proceedings paper <span><span>[49]</span></span> announced at LICS'16; in order to complement it, this version is written from a different viewpoint including topos-theoretic aspects of <span><span>[49]</span></span> that were not discussed there. Technically, this paper introduces and studies the class of <em>semi-galois categories</em>, which extend galois categories and are dual to profinite monoids in the same way as galois categories are dual to profinite groups; the study on this class of categories is aimed at providing an axiomatic reformulation of <em>Eilenberg's theory of varieties of regular languages</em>—a branch in formal language theory that has been developed since the mid 1960s and particularly concerns systematic classification of regular languages, finite monoids and deterministic finite automata. In this paper, detailed proofs of our central results announced at LICS'16 are presented, together with topos-theoretic considerations. The main results include (I) a proof of the duality theorem between profinite monoids and semi-galois categories, extending the duality theorem between profinite groups and galois categories; based on these results on semi-galois categories we then discuss (II) a reinterpretation of Eilenberg's theory from a viewpoint of the duality theorem; in relation with this reinterpretation of the theory, (III) we also give a purely topos-theoretic characterization of classifying topoi <span><math><mi>B</mi><mi>M</mi></math></span> of profinite monoids <em>M</em> among general coherent topoi, which is a topos-theoretic application of (I). This characterization states that a topos <span><math><mi>E</mi></math></span> is equivalent to the classifying topos <span><math><mi>B</mi><mi>M</mi></math></span> of some profinite monoid <em>M</em> if and only if <span><math><mi>E</mi></math></span> is (i) coherent, (ii) noetherian, and (iii) has a surjective coherent point <span><math><mi>p</mi><mo>:</mo><mrow><mi>Sets</mi></mrow><mo>→</mo><mi>E</mi></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107863"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163676","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":"Toric splittings","authors":"Anargyros Katsabekis,&nbsp;Apostolos Thoma","doi":"10.1016/j.jpaa.2025.107870","DOIUrl":"10.1016/j.jpaa.2025.107870","url":null,"abstract":"<div><div>The toric ideal <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span> is splittable if it has a toric splitting; namely, if there exist toric ideals <span><math><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math></span> such that <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math></span> and <span><math><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>≠</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span> for all <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mn>2</mn></math></span>. We provide a necessary and sufficient condition for a toric ideal to be splittable in terms of <em>A</em>, and we apply it to prove or disprove that certain classes of toric ideals are splittable.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107870"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163677","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 the derived category of a toric stack bundle","authors":"Qian Chao ,&nbsp;Jiun-Cheng Chen ,&nbsp;Hsian-Hua Tseng","doi":"10.1016/j.jpaa.2025.107882","DOIUrl":"10.1016/j.jpaa.2025.107882","url":null,"abstract":"<div><div>We establish some properties of the derived category of torus-equivariant coherent sheaves on a split toric stack bundle. Our main result is a semi-orthogonal decomposition of such a category.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107882"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163680","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":"Exact weights and path metrics for triangulated categories and the derived category of persistence modules","authors":"Peter Bubenik ,&nbsp;José A. Vélez-Marulanda","doi":"10.1016/j.jpaa.2025.107903","DOIUrl":"10.1016/j.jpaa.2025.107903","url":null,"abstract":"<div><div>We define exact weights on a pretriangulated category to be nonnegative functions on objects satisfying a subadditivity condition with respect to distinguished triangles. Such weights induce a metric on objects in the category, which we call a path metric. Our exact weights generalize the rank functions of J. Chuang and A. Lazarev for triangulated categories and are analogous to the exact weights for an exact category given by the first author and J. Scott and D. Stanley. We show that (co)homological functors from a triangulated category to an abelian category with an additive weight induce an exact weight on the triangulated category. We prove that triangle equivalences induce an isometry for the path metrics induced by cohomological functors. In the perfectly generated or compactly generated case, we use Brown representability to express the exact weight on the triangulated category. We give three characterizations of exactness for a weight on a pretriangulated category and show that they are equivalent. We also define Wasserstein distances for triangulated categories. Finally, we apply our work to derived categories of persistence modules and to representations of continuous quivers of type <span><math><mi>A</mi></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107903"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143394581","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":"Schur powers of the cokernel of a graded morphism","authors":"Jan O. Kleppe ,&nbsp;Rosa M. Miró-Roig","doi":"10.1016/j.jpaa.2025.107902","DOIUrl":"10.1016/j.jpaa.2025.107902","url":null,"abstract":"<div><div>Let <span><math><mi>φ</mi><mo>:</mo><mi>F</mi><mo>⟶</mo><mi>G</mi></math></span> be a graded morphism between free <em>R</em>-modules of rank <em>t</em> and <span><math><mi>t</mi><mo>+</mo><mi>c</mi><mo>−</mo><mn>1</mn></math></span>, respectively, and let <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>(</mo><mi>φ</mi><mo>)</mo></math></span> be the ideal generated by the <span><math><mi>j</mi><mo>×</mo><mi>j</mi></math></span> minors of a matrix representing <em>φ</em>. In this paper: (1) We show that the canonical module of <span><math><mi>R</mi><mo>/</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>(</mo><mi>φ</mi><mo>)</mo></math></span> is up to twist equal to a suitable Schur power <span><math><msup><mrow><mi>Σ</mi></mrow><mrow><mi>I</mi></mrow></msup><mi>M</mi></math></span> of <span><math><mi>M</mi><mo>=</mo><mi>coker</mi><mo>(</mo><msup><mrow><mi>φ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></math></span>; thus equal to <span><math><msup><mrow><mo>∧</mo></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>j</mi></mrow></msup><mi>M</mi></math></span> if <span><math><mi>c</mi><mo>=</mo><mn>2</mn></math></span> in which case we find a minimal free <em>R</em>-resolution of <span><math><msup><mrow><mo>∧</mo></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>j</mi></mrow></msup><mi>M</mi></math></span> for any <em>j</em>, (2) For <span><math><mi>c</mi><mo>=</mo><mn>3</mn></math></span>, we construct a free <em>R</em>-resolution of <span><math><msup><mrow><mo>∧</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>M</mi></math></span> which starts almost minimally (i.e. the first three terms are minimal up to a precise summand), and (3) For <span><math><mi>c</mi><mo>≥</mo><mn>4</mn></math></span>, we construct under a certain depth condition the first three terms of a free <em>R</em>-resolution of <span><math><msup><mrow><mo>∧</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>M</mi></math></span> which are minimal up to a precise summand. As a byproduct we answer the first open case of a question posed by Buchsbaum and Eisenbud in <span><span>[2, pg. 299]</span></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107902"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143420767","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":"Matrix invertible extensions over commutative rings. Part I: General theory","authors":"Grigore Călugăreanu ,&nbsp;Horia F. Pop ,&nbsp;Adrian Vasiu","doi":"10.1016/j.jpaa.2024.107852","DOIUrl":"10.1016/j.jpaa.2024.107852","url":null,"abstract":"<div><div>A unimodular <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrix with entries in a commutative <em>R</em> is called extendable (resp. simply extendable) if it extends to an invertible <span><math><mn>3</mn><mo>×</mo><mn>3</mn></math></span> matrix (resp. invertible <span><math><mn>3</mn><mo>×</mo><mn>3</mn></math></span> matrix whose <span><math><mo>(</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></span> entry is 0). We obtain necessary and sufficient conditions for a unimodular <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrix to be extendable (resp. simply extendable) and use them to study the class <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> (resp. <span><math><mi>S</mi><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>) of rings <em>R</em> with the property that all unimodular <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrices with entries in <em>R</em> are extendable (resp. simply extendable). We also study the larger class <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> of rings <em>R</em> with the property that all unimodular <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrices of determinant 0 and with entries in <em>R</em> are (simply) extendable (e.g., rings with trivial Picard groups or pre-Schreier domains). Among Dedekind domains, polynomial rings over <span><math><mi>Z</mi></math></span> and Hermite rings, only the EDRs belong to the class <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> or <span><math><mi>S</mi><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. If <em>R</em> has stable range at most 2 (e.g., <em>R</em> is a Hermite ring or <span><math><mi>dim</mi><mo>⁡</mo><mo>(</mo><mi>R</mi><mo>)</mo><mo>≤</mo><mn>1</mn></math></span>), then <em>R</em> is an <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> ring iff it is an <span><math><mi>S</mi><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> ring.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107852"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094014","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":"Projective surfaces not as Hadamard products and the dimensions of the Hadamard joins","authors":"Edoardo Ballico","doi":"10.1016/j.jpaa.2024.107853","DOIUrl":"10.1016/j.jpaa.2024.107853","url":null,"abstract":"<div><div>We study the dimensions of Hadamard products of <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span> varieties if we allow to modify <span><math><mi>k</mi><mo>−</mo><mn>1</mn></math></span> of them by the action of a general projective linear transformation. We also prove that the join of a variety not contained in a coordinate hyperplane with a “nice” curve always has the expected dimension.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107853"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094017","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":"Non-finitely generated monoids corresponding to finitely generated homogeneous subalgebras","authors":"Akihiro Higashitani,&nbsp;Koichiro Tani","doi":"10.1016/j.jpaa.2024.107854","DOIUrl":"10.1016/j.jpaa.2024.107854","url":null,"abstract":"<div><div>The goal of this paper is to study the possible monoids appearing as the associated monoids of the initial algebra of a finitely generated homogeneous <span><math><mi>k</mi></math></span>-subalgebra of a polynomial ring <span><math><mi>k</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span>. Clearly, any affine monoid can be realized since the initial algebra of the affine monoid <span><math><mi>k</mi></math></span>-algebra is itself. On the other hand, the initial algebra of a finitely generated homogeneous <span><math><mi>k</mi></math></span>-algebra is not necessarily finitely generated. In this paper, we provide a new family of non-finitely generated monoids which can be realized as the initial algebras of finitely generated homogeneous <span><math><mi>k</mi></math></span>-algebras. Moreover, we also provide an example of a non-finitely generated monoid which cannot be realized as the initial algebra of any finitely generated homogeneous <span><math><mi>k</mi></math></span>-algebra.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107854"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094023","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}