Journal of Algebra最新文献

{"title":"Representation of partially ordered sets over Von Neumann regular algebras. More prime, non-primitive regular rings","authors":"Giuseppe Baccella","doi":"10.1016/j.jalgebra.2025.08.005","DOIUrl":"10.1016/j.jalgebra.2025.08.005","url":null,"abstract":"<div><div>Given an <em>infinite</em> cardinal <strong>ℵ</strong>, let <span><math><mi>P</mi><mi>o</mi><msub><mrow><mi>s</mi></mrow><mrow><mi>ℵ</mi></mrow></msub></math></span> be the category whose objects are all partially ordered sets <em>I</em>, having a finite cofinal subset and such that both <em>I</em> and the set of all maximal chains of <em>I</em> have at most cardinality <strong>ℵ</strong>, while the morphisms are all maps which are order imbeddings of the domain as an upper subset of the codomain. Given a field <span><math><mi>K</mi></math></span>, for every object <em>I</em> of <span><math><mi>P</mi><mi>o</mi><msub><mrow><mi>s</mi></mrow><mrow><mi>ℵ</mi></mrow></msub></math></span> we find a unital, locally matricial and hence unit-regular <span><math><mi>K</mi></math></span>-algebra <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>ℵ</mi><mo>,</mo><mi>K</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo></math></span> such that the lattice of all its ideals is order isomorphic to the lattice of all lower subsets of <em>I</em>. We show that the Grothendieck group <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>ℵ</mi><mo>,</mo><mi>K</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>)</mo></math></span>, with its natural partial order, is order isomorphic to the restricted Hahn power of <span><math><mi>Z</mi></math></span> by <em>I</em>; this gives a contribution to solve the <em>Realization Problem for Dimension Groups with order-unit</em>. We also show that the algebra <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>ℵ</mi><mo>,</mo><mi>K</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo></math></span> has the following features: (a) <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>ℵ</mi><mo>,</mo><mi>K</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo></math></span> is prime if and only if <em>I</em> is lower directed; (b) <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>ℵ</mi><mo>,</mo><mi>K</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo></math></span> is primitive if and only if <em>I</em> has a coinitial chain; (c) <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>ℵ</mi><mo>,</mo><mi>K</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo></math></span> is semiartinian if and only if <em>I</em> is artinian, in which case <em>I</em> is order isomorphic to the primitive spectrum of <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>ℵ</mi><mo>,</mo><mi>K</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo></math></span>. Finally we show that the assignment <span><math><mi>I</mi><mo>↦</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>ℵ</mi><mo>,</mo><mi>K</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo></math></span> extends to a pair of functors from <span><math><mi>P</mi><mi>o</mi><msub><mrow><mi>s</mi></mrow><mrow><mi>ℵ</mi></mrow></msub></math></span> to the category of <span><math><mi>K</mi></math></span>-algebras, one covariant and the other contravariant.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 38-100"},"PeriodicalIF":0.8,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144886414","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":"Tame fields, graded rings and finite complete sequences of key polynomials","authors":"Silva de Souza C.H.","doi":"10.1016/j.jalgebra.2025.07.046","DOIUrl":"10.1016/j.jalgebra.2025.07.046","url":null,"abstract":"<div><div>In this paper, we present a criterion for <span><math><mo>(</mo><mi>K</mi><mo>,</mo><mi>v</mi><mo>)</mo></math></span> to be henselian and defectless in terms of finite complete sequences of key polynomials. For this, we use the theory of Mac Lane-Vaquié chains and abstract key polynomials. We then prove that a valued field <span><math><mo>(</mo><mi>K</mi><mo>,</mo><mi>v</mi><mo>)</mo></math></span> is tame if and only if <em>vK</em> is <em>p</em>-divisible, <em>Kv</em> is perfect and every simple algebraic extension of <em>K</em> admits a finite complete sequence of key polynomials. The properties <em>vK p</em>-divisible and <em>Kv</em> perfect are described by the Frobenius endomorphism on the associated graded ring. We also make considerations on simply defectless and algebraically maximal valued fields and purely inertial and purely ramified extensions.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 550-578"},"PeriodicalIF":0.8,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865359","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":"Twisted equivalences in spectral algebraic geometry","authors":"Chang-Yeon Chough","doi":"10.1016/j.jalgebra.2025.07.049","DOIUrl":"10.1016/j.jalgebra.2025.07.049","url":null,"abstract":"<div><div>We study twisted derived equivalences for schemes in the setting of spectral algebraic geometry. To this end, we introduce the notion of a twisted equivalence and show that a twisted equivalence for perfect spectral algebraic stacks admitting a quasi-finite presentation supplies an equivalence between the stacks. Our notion thus compensates for the failure of twisted derived equivalences for non-affine schemes to provide an isomorphism of the schemes. In the case of (not necessarily connective) commutative ring spectra, we also prove a spectral analogue of Rickard's theorem, which shows that a derived equivalence of associative rings induces an isomorphism between their centers.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 474-495"},"PeriodicalIF":0.8,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144861235","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 (k,l)-th singular locus moduli algebras of singularities and their derivation Lie algebras","authors":"Guorui Ma ,&nbsp;Stephen S.-T. Yau ,&nbsp;Huaiqing Zuo","doi":"10.1016/j.jalgebra.2025.08.007","DOIUrl":"10.1016/j.jalgebra.2025.08.007","url":null,"abstract":"<div><div>In this paper, we introduce a series of new invariants to singularities. A new conjecture about the non-existence of negative weight derivations of these new <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>l</mi><mo>)</mo></math></span>-th singular locus moduli algebras for weighted homogeneous isolated hypersurface singularities is proposed. We verify this conjecture in some cases.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 673-688"},"PeriodicalIF":0.8,"publicationDate":"2025-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865361","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":"Pushforward of Siegel flag varieties in the Chow ring","authors":"Simon Cooper","doi":"10.1016/j.jalgebra.2025.07.050","DOIUrl":"10.1016/j.jalgebra.2025.07.050","url":null,"abstract":"<div><div>Given a reductive group <em>G</em> over an algebraically closed field and subsets <span><math><mi>I</mi><mo>,</mo><mi>J</mi><mo>⊂</mo><mi>Δ</mi></math></span> of the simple roots Δ determined by a choice of maximal torus and Borel subgroup, there is a closed embedding of flag varieties <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>J</mi></mrow></msub><mo>/</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>J</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>↪</mo><mi>G</mi><mo>/</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>I</mi></mrow></msub></math></span>. In this paper we compute the class of the sub flag variety <span><math><mo>[</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>J</mi></mrow></msub><mo>/</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>J</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>]</mo><mo>∈</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>•</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>/</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>)</mo></math></span> in the Chow ring for the ‘Siegel’ case where <em>G</em> is a general symplectic group of semisimple rank <em>g</em> and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>I</mi></mrow></msub></math></span> is the parabolic stabilising a maximal isotropic subspace. This corresponds, under the isomorphism with the tautological ring of the moduli space of principally polarised abelian varieties <span><math><msup><mrow><mi>T</mi></mrow><mrow><mo>•</mo></mrow></msup><mo>(</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>g</mi></mrow><mrow><mi>tor</mi></mrow></msubsup><mo>)</mo><mo>≅</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>•</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>/</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>)</mo></math></span>, to the generator of the classes in the tautological ring which are supported on the toroidal boundary. This provides basic evidence for a conjecture describing the tautological ring of a Hodge-type Shimura variety.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 523-535"},"PeriodicalIF":0.8,"publicationDate":"2025-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865358","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 structure of the character degree graphs having diameter three","authors":"Silvio Dolfi ,&nbsp;Roghayeh Hafezieh ,&nbsp;Pablo Spiga","doi":"10.1016/j.jalgebra.2025.08.002","DOIUrl":"10.1016/j.jalgebra.2025.08.002","url":null,"abstract":"<div><div>The structure of the character degree graphs <span><math><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, i.e. the prime graphs on the set <span><math><mrow><mi>cd</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of the irreducible character degrees of a finite group <em>G</em>, such that <em>G</em> is solvable and <span><math><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> has diameter three, remains an intriguing area of study. However, a comprehensive understanding of these structures remains elusive. In this paper, we prove some properties and provide an infinite series of examples of this class of graphs, building on the ideas of M. Lewis <span><span>[8]</span></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 337-360"},"PeriodicalIF":0.8,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144831474","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 dual F-signature of Veronese rings","authors":"Vinicius Bouça ,&nbsp;Eliana Tolosa-Villarreal ,&nbsp;Kevin Vasconcellos","doi":"10.1016/j.jalgebra.2025.07.033","DOIUrl":"10.1016/j.jalgebra.2025.07.033","url":null,"abstract":"<div><div>In this paper we calculate explicitly the dual F-signature of the Verorese subrings of a polynomial ring in several variables, using methods of commutative algebra. This gives a positive answer to a recent conjecture proposed by Ilya Smirnov and Kevin Tucker.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 738-754"},"PeriodicalIF":0.8,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865364","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":"Ribbon blocks for centraliser algebras of symmetric groups","authors":"Matthew Fayers ,&nbsp;Lorenzo Putignano","doi":"10.1016/j.jalgebra.2025.07.042","DOIUrl":"10.1016/j.jalgebra.2025.07.042","url":null,"abstract":"<div><div>Suppose <span><math><mi>l</mi><mo>,</mo><mi>m</mi></math></span> are natural numbers with <figure><img></figure>, and <span><math><mi>F</mi></math></span> a field of characteristic <em>p</em>, and let <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi>m</mi></mrow><mrow><mi>F</mi></mrow></msubsup></math></span> denote the centraliser of the group algebra <span><math><mi>F</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span> inside <span><math><mi>F</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>. Ellers and Murray give a conjectured classification of the blocks of <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>l</mi><mo>,</mo><mi>m</mi></mrow><mrow><mi>F</mi></mrow></msubsup></math></span>, in terms of the <em>p</em>-blocks of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>. We prove this conjecture for a family of blocks that we call <em>ribbon blocks</em> and <em>belt blocks</em>. These are the blocks containing Specht modules labelled by skew-partitions having no repeated entries in their <em>p</em>-content.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 271-312"},"PeriodicalIF":0.8,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144831477","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 subgroup lattice of polycyclic groups","authors":"Marco Trombetti","doi":"10.1016/j.jalgebra.2025.08.001","DOIUrl":"10.1016/j.jalgebra.2025.08.001","url":null,"abstract":"<div><div>Let <em>G</em> be a polycyclic-by-finite group, and let <em>X</em> be a subgroup of <em>G</em>. It has been proved by Kegel [Math. Ann. 163 (1966), 248–258] that if the image of <em>X</em> is subnormal in every finite quotient of <em>G</em>, then <em>X</em> is actually subnormal in <em>G</em>; while Robinson [Invent. Math. 10 (1970), 38–43] and Wehrfritz [Proc. London Math. Soc. 20:3 (1970), 101–122] proved that a polycyclic-by-finite group is nilpotent provided that all its finite quotients are nilpotent.</div><div>Our first main result (Theorem 2.4) shows that every modular subgroup can be similarly recognized by only looking at the finite quotients of polycyclic-by-finite groups. This extends a theorem of Lennox and Wilson [Arch. Math. (Basel) 28 (1977), 113–116] and improves the main result of Musella [Arch. Math. (Basel) 76 (2001), 161–165] — see also Corollary 2.6.</div><div>Our second main result (Theorem 2.16) provides a detailed description of uniquely complemented subgroups in infinite polycyclic-by-finite groups. This is the first non-trivial characterization of this type of subgroups in the infinite case (see [18], p.142), and it has some surprising consequences. It shows in fact that in infinite polycyclic-by-finite groups, the neutral subgroups coincide with the join-distributive subgroups (Corollary 2.31), and that the meet-quasi-distributive subgroups coincide with the uniquely complements subgroups (Theorem 2.22); thus, we face one of those rare occasions in which some types of subgroups coincide in the infinite case but they do not coincide in the finite case. Further relevant consequences of this result deal with detailed descriptions of meet-distributive and join-distributive subgroups (Theorems 2.28 and 2.26), and with the possibility of recognizing all the previously mentioned types of subgroup starting from their images in the finite quotients (Corollary 2.17, Corollary 2.21, Theorem 2.23 and Theorem 2.29).</div><div>Finally, in the spirit of Baumslag, Cannonito and Miller III [Math. Z. 153 (1977), 117–134], we also provide theoretical algorithms to determine if a given subgroup of a polycyclic-by-finite group is a modular, join-distributive, meet-(quasi-) distributive, uniquely complemented, and neutral (Corollary 2.9 and Theorem 2.32).</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 422-446"},"PeriodicalIF":0.8,"publicationDate":"2025-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144861232","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":"Periodicity shadows I. A new approach to combinatorics of periodic algebras","authors":"Adam Skowyrski","doi":"10.1016/j.jalgebra.2025.07.025","DOIUrl":"10.1016/j.jalgebra.2025.07.025","url":null,"abstract":"<div><div>This article is devoted to introduce a new notion of periodicity shadow, which appeared naturally in the study of combinatorics of tame symmetric algebras of period four, or more generally, algebras of generalized quaternion type <span><span>[18]</span></span>. For any such an algebra Λ, we consider its shadow <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>Λ</mi></mrow></msub></math></span>, which is the (signed) adjacency matrix of the Gabriel quiver of Λ. Studying properties of shadows <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>Λ</mi></mrow></msub></math></span> leads us to the definition of the periodicity shadow, which is basically, a skew-symmetric integer matrix satisfying certain set of conditions motivated by the properties of shadows <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>Λ</mi></mrow></msub></math></span>. This turned out to be a very useful tool for describing the combinatorics of Gabriel quivers of algebras of generalized quaternion type, not only of algebras with small Gabriel quivers (i.e. up to 6 vertices), which it was originally desined for. In this paper, we introduce and briefly discuss this notion and present one of its theoretical applications, which shows how significant it is. Namely, the main result of this paper describes the global shape of the Gabriel quivers of algebras of generalized quaternion type, as quivers obtained from some basic shadows by attaching 2-cycles, and moreover, position of the 2-cycles is restricted by precise rules (see the Main Theorem). Computational aspects are reported in the second part <span><span>[3]</span></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 1-37"},"PeriodicalIF":0.8,"publicationDate":"2025-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144886413","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}