Journal of Algebra最新文献

{"title":"Lascoux-type resolutions, derived categories, and flips","authors":"Qingyuan Jiang","doi":"10.1016/j.jalgebra.2025.07.053","DOIUrl":"10.1016/j.jalgebra.2025.07.053","url":null,"abstract":"<div><div>This paper introduces Lascoux-type complexes that extend the Lascoux complexes for resolving generic determinantal ideals. These Lascoux-type complexes naturally arise when analyzing the correspondences between two different types of resolutions of singularities of determinantal varieties. We also discuss the applications of these resolutions in various geometric contexts, including blowups, standard flips, virtual flips, and projectivizations.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 417-453"},"PeriodicalIF":0.8,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145010259","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 semisimple proto-Abelian categories associated to inverse monoids","authors":"Alexander Sistko","doi":"10.1016/j.jalgebra.2025.07.052","DOIUrl":"10.1016/j.jalgebra.2025.07.052","url":null,"abstract":"<div><div>Let <em>G</em> be a finite abelian group written multiplicatively, with <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>=</mo><mi>G</mi><mo>⊔</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span> the pointed abelian group formed by adjoining an absorbing element 0. There is an associated finitary, proto-abelian category <span><math><msub><mrow><mi>Vect</mi></mrow><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow></msub></math></span>, whose objects can be thought of as finite-dimensional vector spaces over <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>. The class of <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>-linear monoids are then defined in terms of this category. In this paper, we study the finitary, proto-abelian category <span><math><mi>Rep</mi><mo>(</mo><mi>M</mi><mo>,</mo><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span> of finite-dimensional <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>-linear representations of a <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>-linear monoid <em>M</em>. Although this category is only a slight modification of the usual category of <em>M</em>-modules, it exhibits significantly different behavior for interesting classes of monoids. Assuming that the regular principal factors of <em>M</em> are objects of <span><math><mi>Rep</mi><mo>(</mo><mi>M</mi><mo>,</mo><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span>, we develop a version of the Clifford-Munn-Ponizovskiĭ Theorem and classify the <em>M</em> for which each non-zero object of <span><math><mi>Rep</mi><mo>(</mo><mi>M</mi><mo>,</mo><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span> is a direct sum of simple objects. When <em>M</em> is the endomorphism monoid of an object in <span><math><msub><mrow><mi>Vect</mi></mrow><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow></msub></math></span>, we discuss alternate frameworks for studying its <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>-linear representations and contrast the various approaches.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 354-397"},"PeriodicalIF":0.8,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144926057","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":"Classification of non-degenerate symmetric bilinear and quadratic forms in the Verlinde category Ver4+","authors":"Iz Chen ,&nbsp;Arun S. Kannan ,&nbsp;Krishna Pothapragada","doi":"10.1016/j.jalgebra.2025.08.010","DOIUrl":"10.1016/j.jalgebra.2025.08.010","url":null,"abstract":"<div><div>Although Deligne's theorem classifies all symmetric tensor categories (STCs) with moderate growth over algebraically closed fields of characteristic zero, the classification does not extend to positive characteristic. At the forefront of the study of STCs is the search for an analog to Deligne's theorem in positive characteristic, and it has become increasingly apparent that the Verlinde categories are to play a significant role. Moreover, these categories are largely unstudied, but have already shown very interesting phenomena as both a generalization of and a departure from superalgebra and supergeometry. In this paper, we study <span><math><msubsup><mrow><mi>Ver</mi></mrow><mrow><mn>4</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span>, the simplest non-trivial Verlinde category in characteristic 2. In particular, we classify all isomorphism classes of non-degenerate symmetric bilinear forms and non-degenerate quadratic forms and study the associated Witt semi-ring that arises from the addition and multiplication operations on bilinear forms.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 220-262"},"PeriodicalIF":0.8,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144926060","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":"Generalization of Gurjar's hyperplane section theorem to arbitrary analytic varieties and AmAC classes","authors":"A.J. Parameswaran ,&nbsp;Mohit Upmanyu","doi":"10.1016/j.jalgebra.2025.07.047","DOIUrl":"10.1016/j.jalgebra.2025.07.047","url":null,"abstract":"<div><div>This paper aims to generalize the hyperplane section Theorem of R.V. Gurjar to arbitrary (local) analytic varieties, even if the intersection with hyperplanes is not necessarily isolated.</div><div>In the case of formal varieties, we generalize the statement to work for different classes of hypersurfaces other than hyperplanes. We call the classes of functions (which are subsets of the formal power series ring) defining these classes of hypersurface algebraic <span><math><mi>m</mi></math></span>-adically closed (A<span><math><mi>m</mi></math></span>AC) classes.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 101-126"},"PeriodicalIF":0.8,"publicationDate":"2025-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144892976","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":"About the algebraic closure of formal power series in several variables","authors":"Michel Hickel,&nbsp;Mickaël Matusinski","doi":"10.1016/j.jalgebra.2025.08.004","DOIUrl":"10.1016/j.jalgebra.2025.08.004","url":null,"abstract":"<div><div>Let <em>K</em> be a field of characteristic zero. We deal with the algebraic closure of the field of fractions of the ring of formal power series <span><math><mi>K</mi><mo>[</mo><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>r</mi></mrow></msub><mo>]</mo><mo>]</mo></math></span>, <span><math><mi>r</mi><mo>≥</mo><mn>2</mn></math></span>. More precisely, we view the latter as a subfield of an iterated Puiseux series field <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span>. On the one hand, given <span><math><msub><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> which is algebraic, we provide an algorithm that reconstructs the space of all polynomials which annihilates <span><math><msub><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> up to a certain order (arbitrarily high). On the other hand, given a polynomial <span><math><mi>P</mi><mo>∈</mo><mi>K</mi><mo>[</mo><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>r</mi></mrow></msub><mo>]</mo><mo>]</mo><mo>[</mo><mi>y</mi><mo>]</mo></math></span> with simple roots, we derive a closed form formula for the coefficients of a root <span><math><msub><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> in terms of the coefficients of <em>P</em> and a fixed initial part of <span><math><msub><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 263-353"},"PeriodicalIF":0.8,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144926055","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":"Cosets of normal subgroups and union of two conjugacy classes","authors":"Antonio Beltrán","doi":"10.1016/j.jalgebra.2025.08.009","DOIUrl":"10.1016/j.jalgebra.2025.08.009","url":null,"abstract":"<div><div>Let <em>G</em> be a finite group, <em>N</em> a normal subgroup of <em>G</em> and <span><math><mi>x</mi><mo>∈</mo><mi>G</mi><mo>−</mo><mi>N</mi></math></span>. We discuss when the coset <em>Nx</em> is contained in the union of two conjugacy classes, <em>K</em> and <em>D</em>, of <em>G</em>. We show that <em>N</em> need not be solvable, and can even be non-abelian simple, but in these cases, <em>K</em> and <em>D</em> must have the same cardinality, and the non-solvable structure of <em>N</em> is restricted. The non-abelian principal factors of <em>G</em> contained in <em>N</em> are then isomorphic to <span><math><mi>S</mi><mo>×</mo><mo>⋯</mo><mo>×</mo><mi>S</mi></math></span>, where <em>S</em> is a simple group of Lie type of odd characteristic.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 689-702"},"PeriodicalIF":0.8,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865362","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":"Finite free irreducible conformal modules over a class of Z+-graded Lie conformal superalgebras","authors":"Jianzhi Han ,&nbsp;Yumeng Zhan","doi":"10.1016/j.jalgebra.2025.08.006","DOIUrl":"10.1016/j.jalgebra.2025.08.006","url":null,"abstract":"<div><div>In the present paper, we consider a class of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>-graded Lie conformal superalgebras <span><math><mi>R</mi><mo>=</mo><msub><mrow><mo>⊕</mo></mrow><mrow><mi>i</mi><mo>∈</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></msub><mo>(</mo><mi>C</mi><mo>[</mo><mo>∂</mo><mo>]</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>⊕</mo><mi>C</mi><mo>[</mo><mo>∂</mo><mo>]</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span>, each of which has a basis <span><math><mo>{</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mi>i</mi><mo>∈</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>}</mo><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span> as a free <span><math><mi>C</mi><mo>[</mo><mo>∂</mo><mo>]</mo></math></span>-module and give the classification of finite (free) irreducible conformal modules over these <span><math><mi>R</mi></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 406-421"},"PeriodicalIF":0.8,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144831476","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":"Cartier–Gabriel–Kostant theorem for relative Rota–Baxter operators","authors":"Haixing Zhu ,&nbsp;Yujie Di","doi":"10.1016/j.jalgebra.2025.08.008","DOIUrl":"10.1016/j.jalgebra.2025.08.008","url":null,"abstract":"<div><div>Let <em>H</em> and <em>K</em> be two cocommutative Hopf algebras over an algebraically closed field <span><math><mi>F</mi></math></span> of characteristic 0. In this paper, we first prove any relative Rota–Baxter operator <span><math><mi>T</mi><mo>:</mo><mi>K</mi><mo>⟶</mo><mi>H</mi></math></span> to be isomorphic to a relative Rota–Baxter operator <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>:</mo><mi>U</mi><mo>(</mo><mi>P</mi><mo>(</mo><mi>K</mi><mo>)</mo><mo>)</mo><mo>♯</mo><mi>F</mi><mo>[</mo><mi>G</mi><mo>(</mo><mi>K</mi><mo>)</mo><mo>]</mo><mo>⟶</mo><mi>U</mi><mo>(</mo><mi>P</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>)</mo><mo>♯</mo><mi>F</mi><mo>[</mo><mi>G</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>]</mo></math></span>, which is determined by a relative Rota–Baxter operator <span><math><mi>R</mi><mo>:</mo><mi>P</mi><mo>(</mo><mi>K</mi><mo>)</mo><mo>⟶</mo><mi>P</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> on the Lie algebra <span><math><mi>P</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span>, and a relative Rota–Baxter operator <span><math><mi>B</mi><mo>:</mo><mi>G</mi><mo>(</mo><mi>K</mi><mo>)</mo><mo>⟶</mo><mi>G</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> on the group <span><math><mi>G</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span>. Next we prove that Rota–Baxter operators on <em>H</em> are in one to one correspondence with Rota–Baxter pairs on the Lie algebra <span><math><mi>P</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> and the group <span><math><mi>G</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span>. These results should be viewed as the Rota–Baxter operator version of classical Cartier–Gabriel–Kostant structure theorem. Finally, we prove that this well-known structure theorem also holds for post-Hopf algebras and Hopf braces.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 775-800"},"PeriodicalIF":0.8,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865366","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":"Instanton sheaves on ruled Fano 3-folds of Picard rank 2 and index 1","authors":"Ozhan Genc ,&nbsp;Marcos Jardim","doi":"10.1016/j.jalgebra.2025.07.051","DOIUrl":"10.1016/j.jalgebra.2025.07.051","url":null,"abstract":"<div><div>We study rank 2 <em>h</em>-instanton sheaves on projective threefolds. We demonstrate that any orientable rank 2, non-locally free <em>h</em>-instanton sheaf with defect 0 on a threefold can be obtained as an elementary transformation of a locally free <em>h</em>-instanton sheaf. Our focus then shifts to ruled Fano threefolds of Picard rank 2 and index 1, of which there are five deformation classes. We establish the existence of orientable rank 2 <em>h</em>-instanton bundles on such varieties. Additionally, we prove the existence of Ulrich bundles on such varieties, which correspond to <em>h</em>-instanton sheaves of minimum charge.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 579-607"},"PeriodicalIF":0.8,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865980","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":"Logarithmic Hochschild co/homology via formality of derived intersections","authors":"Márton Hablicsek ,&nbsp;Leo Herr ,&nbsp;Francesca Leonardi","doi":"10.1016/j.jalgebra.2025.07.048","DOIUrl":"10.1016/j.jalgebra.2025.07.048","url":null,"abstract":"<div><div>We define log Hochschild co/homology for log schemes that behaves well for simple normal crossing pairs <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> or toroidal singularities.</div><div>We prove a Hochschild-Kostant-Rosenberg isomorphism for log smooth schemes, as well as an equivariant version for log orbifolds. We define cyclic homology and compute it in simple cases. We show that log Hochschild co/homology is invariant under log alterations.</div><div>Our main technical result in log geometry shows the tropicalization (Artin fan) of a product of log schemes <span><math><mi>X</mi><mo>×</mo><mi>Y</mi></math></span> is usually the product of the tropicalizations of <em>X</em> and <em>Y</em>. This and the machinery of <em>formality</em> of derived intersections facilitate a geometric approach to log Hochschild.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 127-175"},"PeriodicalIF":0.8,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144902156","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}