Journal of Pure and Applied Algebra最新文献

{"title":"Extensions representing Nori-Srinivas obstruction","authors":"Yukihide Takayama","doi":"10.1016/j.jpaa.2024.107783","DOIUrl":"10.1016/j.jpaa.2024.107783","url":null,"abstract":"<div><p>Let <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> be a pair of a smooth variety <em>X</em> over an algebraically closed field <em>k</em> of characteristic <span><math><mi>p</mi><mo>&gt;</mo><mn>0</mn></math></span> and its Frobenius morphism <em>F</em>. Given a Frobenius <span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo></math></span>-lifting <span><math><mo>(</mo><mover><mrow><mi>X</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>,</mo><mover><mrow><mi>F</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>)</mo></math></span> of the pair <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>, Nori and Srinivas <span><span>[9]</span></span> determined the obstruction <span><math><mi>o</mi><mi>b</mi><msub><mrow><mi>s</mi></mrow><mrow><mover><mrow><mi>X</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>,</mo><mover><mrow><mi>F</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub><mo>∈</mo><mi>Ext</mi><mo>(</mo><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>X</mi><mo>/</mo><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mo>,</mo><mi>B</mi><msub><mrow><mi>F</mi></mrow><mrow><mo>⁎</mo></mrow></msub><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>X</mi><mo>/</mo><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mo>)</mo></math></span> to Frobenius <span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo></math></span>-lifting of <span><math><mo>(</mo><mover><mrow><mi>X</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>,</mo><mover><mrow><mi>F</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>)</mo></math></span> in terms of Čech cohomology. The extension representing <span><math><mi>o</mi><mi>b</mi><msub><mrow><mi>s</mi></mrow><mrow><mover><mrow><mi>X</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>,</mo><mover><mrow><mi>F</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub></math></span> has been only known for <span><math><mi>n</mi><mo>=</mo><mn>1</mn></math></span>, which uses the Cartier operator. In this paper, we interpret <span><math><mi>o</mi><mi>b</mi><msub><mrow><mi>s</mi></mrow><mrow><mover><mrow><mi>X</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>,</mo><mover><mrow><mi>F</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub></math></span> in terms of Kato's version of de Rham-Witt Cartier operator <span><span>[8]</span></span> and determine the extension representing <span><math><mi>o</mi><mi>b</mi><msub><mrow><mi>s</mi></mrow><mrow><mover><mrow><mi>X</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>,</mo><mover><mrow><mi>F</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141782342","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":"The nucleus of a compact Lie group, and support of singularity categories","authors":"Thomas Peirce","doi":"10.1016/j.jpaa.2024.107780","DOIUrl":"10.1016/j.jpaa.2024.107780","url":null,"abstract":"<div><p>In this paper we adapt the notion of the nucleus defined by Benson, Carlson, and Robinson to compact Lie groups in non-modular characteristic. We show that it describes the singularities of the projective scheme of the cohomology of its classifying space. A notion of support for singularity categories of ring spectra (in the sense of Greenlees and Stevenson) is established, and is shown to be precisely the nucleus in this case, consistent with a conjecture of Benson and Greenlees for finite groups.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022404924001774/pdfft?md5=1caa8c3af6bce11b853017fe930a9f81&pid=1-s2.0-S0022404924001774-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141782343","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":"The minimal primes of localizations of rings","authors":"V.V. Bavula","doi":"10.1016/j.jpaa.2024.107776","DOIUrl":"10.1016/j.jpaa.2024.107776","url":null,"abstract":"<div><p>The set of minimal primes of a ring is a very important set as far the spectrum of a ring is concerned as every prime contains a minimal prime. So, knowing the minimal primes is the first (important and difficult) step in describing the spectrum. In the algebraic geometry, the minimal primes of the algebra of regular functions on an algebraic variety determine/correspond to the irreducible components of the variety. The aim of the paper is to obtain several descriptions of the set of minimal prime ideals of localizations of rings under several natural assumptions. In particular, the following cases are considered: a localization of a semiprime ring with finite set of minimal primes; a localization of a prime rich ring where the localization respects the ideal structure of primes and primeness of certain minimal primes; a localization of a ring at a left denominator set generated by normal elements, and others. As an application, for a semiprime ring with finitely many minimal primes, a description of the minimal primes of its largest left/right quotient ring is obtained.</p><p>For a semiprime ring <em>R</em> with finitely many minimal primes <span><math><mi>min</mi><mo>⁡</mo><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, criteria are given for the map<span><span><span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>R</mi><mo>,</mo><mi>min</mi></mrow></msub><mo>:</mo><mi>min</mi><mo>⁡</mo><mo>(</mo><mi>R</mi><mo>)</mo><mo>→</mo><mi>min</mi><mo>⁡</mo><mo>(</mo><mi>Z</mi><mo>(</mo><mi>R</mi><mo>)</mo><mo>)</mo><mo>,</mo><mspace></mspace><mspace></mspace><mi>p</mi><mo>↦</mo><mi>p</mi><mo>∩</mo><mi>Z</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span></span></span> being a well-defined map or surjective where <span><math><mi>Z</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> is the centre of <em>R</em>.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022404924001737/pdfft?md5=1e0b674103716acf2964ce8202dc8825&pid=1-s2.0-S0022404924001737-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141881799","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":"Weak dimension of power series rings over valuation rings","authors":"Adam Jones","doi":"10.1016/j.jpaa.2024.107778","DOIUrl":"10.1016/j.jpaa.2024.107778","url":null,"abstract":"<div><p>We examine the power series ring <span><math><mi>R</mi><mo>[</mo><mo>[</mo><mi>X</mi><mo>]</mo><mo>]</mo></math></span> over a valuation ring <em>R</em> of rank 1, with proper, dense value group. We give a counterexample to Hilbert's syzygy theorem for <span><math><mi>R</mi><mo>[</mo><mo>[</mo><mi>X</mi><mo>]</mo><mo>]</mo></math></span>, i.e. an <span><math><mi>R</mi><mo>[</mo><mo>[</mo><mi>X</mi><mo>]</mo><mo>]</mo></math></span>-module <em>C</em> that is flat over <em>R</em> and has flat dimension at least 2 over <span><math><mi>R</mi><mo>[</mo><mo>[</mo><mi>X</mi><mo>]</mo><mo>]</mo></math></span>, contradicting a previously published result. The key ingredient in our construction is an exploration of the valuation theory of <span><math><mi>R</mi><mo>[</mo><mo>[</mo><mi>X</mi><mo>]</mo><mo>]</mo></math></span>. We also use this theory to give a new proof that <span><math><mi>R</mi><mo>[</mo><mo>[</mo><mi>X</mi><mo>]</mo><mo>]</mo></math></span> is not a coherent ring, a fact which is essential in our construction of the module <em>C</em>.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022404924001750/pdfft?md5=7d7a61796914e797af61b233ad5207c2&pid=1-s2.0-S0022404924001750-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141782344","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":"Presentations of Schur and Specht modules in characteristic zero","authors":"Mihalis Maliakas ,&nbsp;Maria Metzaki ,&nbsp;Dimitra-Dionysia Stergiopoulou","doi":"10.1016/j.jpaa.2024.107774","DOIUrl":"10.1016/j.jpaa.2024.107774","url":null,"abstract":"<div><p>New presentations of Specht modules of symmetric groups over fields of characteristic zero have been obtained by Brauner, Friedmann, Hanlon, Stanley and Wachs. These involve generators that are column tabloids and relations that are Garnir relations with maximal number of exchanges between consecutive columns or symmetrization of Garnir relations with minimal number of exchanges between consecutive columns. In this paper, we examine Garnir relations and their symmetrization with any number of exchanges. In both cases, we provide sufficient arithmetic conditions so that the corresponding quotient is a Specht module. In particular, in the first case this yields new presentations of Specht modules if the parts of the conjugate partition that correspond to maximal number of exchanges greater than 1 are distinct. These results generalize the presentations mentioned above and offer an answer to a question of Friedmann, Hanlon and Wachs. Our approach is via representations of the general linear group.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141782345","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":"Some applications of Gröbner-Shirshov bases to Lie algebras","authors":"Luis Mendonça","doi":"10.1016/j.jpaa.2024.107773","DOIUrl":"10.1016/j.jpaa.2024.107773","url":null,"abstract":"<div><p>We show that if a countably generated Lie algebra <em>H</em> does not contain isomorphic copies of certain finite-dimensional nilpotent Lie algebras <em>A</em> and <em>B</em> (satisfying some mild conditions), then <em>H</em> embeds into a quotient of <span><math><mi>A</mi><mo>⁎</mo><mi>B</mi></math></span> that is at the same time hopfian and cohopfian. This is a Lie algebraic version of an embedding theorem proved by C. Miller and P. Schupp for groups. We also prove that any finitely presentable Lie algebra is the quotient of a finitely presented, centerless, residually nilpotent and SQ-universal Lie algebra of cohomological dimension at most 2 by an ideal that can be generated by two elements as a Lie subalgebra. This is reminiscent of the Rips construction in group theory. In both results we use the theory of Gröbner-Shirshov bases.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141782346","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":"Reduction map in the higher K-theory of the rings of integers in number fields","authors":"Soumyadip Sahu","doi":"10.1016/j.jpaa.2024.107771","DOIUrl":"10.1016/j.jpaa.2024.107771","url":null,"abstract":"<div><p>This article studies the reduction maps in the higher <em>K</em>-theory of the ring of integers in a number field arising from the canonical reduction maps at nonzero prime ideals. It proves an explicit density estimate for the subset of primes where the images of a fixed collection of elements vanish. Our result applies to a collection of elements possibly having different degrees and suggests that the linearly independent elements of global <em>K</em>-theory exhibit mutually independent reduction patterns. We also relate the reduction map in <em>K</em>-theory to the reduction map in stable cohomology of general linear groups. This connection allows us to examine the pullback of Quillen's <em>e</em>-classes in the cohomology of the stable general linear group over a finite field. During the proof of the main result, we construct the smallest Galois extension which trivializes a Galois cohomology class of degree one, and show that the linear independence of classes results in disjointness of corresponding field extensions.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141782347","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":"Locally nilpotent derivations on A2-fibrations with A1-fibration kernels","authors":"Janaki Raman Babu ,&nbsp;Prosenjit Das ,&nbsp;Animesh Lahiri","doi":"10.1016/j.jpaa.2024.107772","DOIUrl":"10.1016/j.jpaa.2024.107772","url":null,"abstract":"<div><p>In this paper, we give a characterization of locally nilpotent derivations on <span><math><msup><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-fibrations having kernels isomorphic to <span><math><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-fibrations over Noetherian normal domains containing <span><math><mi>Q</mi></math></span>.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141782348","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":"Quasi-parabolic Kazhdan-Lusztig bases and reflection subgroups","authors":"Zachary Carlini,&nbsp;Yaolong Shen","doi":"10.1016/j.jpaa.2024.107777","DOIUrl":"10.1016/j.jpaa.2024.107777","url":null,"abstract":"<div><p>Recently, Wang and the second author constructed a bar involution and canonical basis for a quasi-permutation module of the Hecke algebra associated to a type B Weyl group <em>W</em>, where the basis is parameterized by left cosets of a quasi-parabolic reflection subgroup in <em>W</em>. In this paper we provide an alternative approach to these constructions, and then generalize these constructions to Coxeter groups which contain a product of type B Weyl groups as a parabolic subgroup.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141782403","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":"Profinite properties of algebraically clean graphs of free groups","authors":"Kasia Jankiewicz ,&nbsp;Kevin Schreve","doi":"10.1016/j.jpaa.2024.107775","DOIUrl":"10.1016/j.jpaa.2024.107775","url":null,"abstract":"<div><p>We prove that for every prime <em>p</em> algebraically clean graphs of groups are virtually residually <em>p</em>-finite and cohomologically <em>p</em>-complete. We also prove that they are cohomologically good. We apply this to certain 2-dimensional Artin groups.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022404924001725/pdfft?md5=70a6736e62f29b5f2cba9a6ec821ca5a&pid=1-s2.0-S0022404924001725-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141707406","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}