Journal of the London Mathematical Society-Second Series最新文献

{"title":"Infinitesimally rigid Lie foliations with dense leaves","authors":"Stephane Geudens,&nbsp;Florian Zeiser","doi":"10.1112/jlms.70617","DOIUrl":"https://doi.org/10.1112/jlms.70617","url":null,"abstract":"<p>We call a foliation <span></span><math>\u0000 <semantics>\u0000 <mi>F</mi>\u0000 <annotation>$mathcal {F}$</annotation>\u0000 </semantics></math> on a compact manifold infinitesimally rigid if its deformation cohomology <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>H</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>F</mi>\u0000 <mo>,</mo>\u0000 <mi>N</mi>\u0000 <mi>F</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$H^{1}(mathcal {F},Nmathcal {F})$</annotation>\u0000 </semantics></math> vanishes. This paper studies infinitesimal rigidity for a distinguished class of Riemannian foliations, namely Lie foliations with dense leaves. We construct infinitesimally rigid Lie foliations with dense leaves, modeled on any compact semisimple Lie algebra with simple ideals different from <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>so</mi>\u0000 <mo>(</mo>\u0000 <mn>3</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathfrak {so}(3)$</annotation>\u0000 </semantics></math>. To our knowledge, these are the first examples of infinitesimally rigid Riemannian foliations that are not Hausdorff.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148343346","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 marked elliptic root systems with nonreduced quotient","authors":"A. Fialowski,&nbsp;K. Iohara,&nbsp;Y. Saito","doi":"10.1112/jlms.70598","DOIUrl":"https://doi.org/10.1112/jlms.70598","url":null,"abstract":"<p>Saito (Publ. RIMS 21 (1), 1985, 75–179) has introduced a class of root systems called elliptic root systems that lie in the real vector space <span></span><math>\u0000 <semantics>\u0000 <mi>F</mi>\u0000 <annotation>$F$</annotation>\u0000 </semantics></math> with a metric <span></span><math>\u0000 <semantics>\u0000 <mi>I</mi>\u0000 <annotation>$I$</annotation>\u0000 </semantics></math> whose signature is of type <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>l</mi>\u0000 <mo>,</mo>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mn>0</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(l,2,0)$</annotation>\u0000 </semantics></math>. He also classified the pair <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>R</mi>\u0000 <mo>,</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(R,G)$</annotation>\u0000 </semantics></math> of elliptic root systems <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math> with one-dimensional subspaces <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> of the radical of <span></span><math>\u0000 <semantics>\u0000 <mi>I</mi>\u0000 <annotation>$I$</annotation>\u0000 </semantics></math>, under the assumption that the quotient root system <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>R</mi>\u0000 <mo>/</mo>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation>$R/G$</annotation>\u0000 </semantics></math> is reduced. Here, we classify the pairs <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>R</mi>\u0000 <mo>,</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(R,G)$</annotation>\u0000 </semantics></math> where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>R</mi>\u0000 <mo>/</mo>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation>$R/G$</annotation>\u0000 </semantics></math> is nonreduced.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148343043","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":"Canonical models of adjoint foliated structures on surfaces","authors":"Jun Lu,&nbsp;Xiaohang Wu,&nbsp;Shi Xu","doi":"10.1112/jlms.70622","DOIUrl":"https://doi.org/10.1112/jlms.70622","url":null,"abstract":"<p>In this paper, we study adjoint foliated structures of the form <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 <mi>F</mi>\u0000 </msub>\u0000 <mo>+</mo>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 <annotation>$K_{mathcal {F}}+D$</annotation>\u0000 </semantics></math> on algebraic surfaces and their minimal and canonical models. We investigate the effective behavior of the linear systems <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 <mi>m</mi>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>K</mi>\u0000 <mi>F</mi>\u0000 </msub>\u0000 <mo>+</mo>\u0000 <mi>D</mi>\u0000 <mo>)</mo>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <annotation>$|m(K_{mathcal {F}}+D)|$</annotation>\u0000 </semantics></math> for sufficiently divisible <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 <mo>&gt;</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$m&gt;0$</annotation>\u0000 </semantics></math>. As an application, we obtain an effective answer to a boundedness problem for foliated surfaces of general type posed by Hacon and Langer.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"114 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148386871","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":"Generalized free wreath products and their operator algebras","authors":"Pierre Fima,&nbsp;Arthur Troupel","doi":"10.1112/jlms.70595","DOIUrl":"https://doi.org/10.1112/jlms.70595","url":null,"abstract":"<p>We develop a new approach on free wreath products, generalizing the constructions of Bichon and of Fima-Pittau. We show stability properties for certain approximation properties such as exactness, Haagerup property, hyperlinearity, and K-amenability. We study qualitative properties of the associated von Neumann algebra: factoriality, primeness, and absence of Cartan subalgebra and we give a formula for Connes' <span></span><math>\u0000 <semantics>\u0000 <mi>T</mi>\u0000 <annotation>$T$</annotation>\u0000 </semantics></math>-invariant. Finally, we give some explicit computations of K-theory groups for C*-algebras of generalized free wreath products.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"113 6","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70595","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148324110","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":"Rickard's derived Morita theory: Review and outlook","authors":"Gustavo Jasso,&nbsp;Henning Krause,&nbsp;Sibylle Schroll","doi":"10.1112/jlms.70577","DOIUrl":"https://doi.org/10.1112/jlms.70577","url":null,"abstract":"<p>We survey the main results in Jeremy Rickard's seminal papers ‘Morita theory for derived categories’ and ‘Derived equivalences and derived functors’. These papers catalysed the later development of the Morita theory of (enhanced) compactly generated triangulated categories by Keller in the algebraic setting and by Schwede and Shipley in the topological setting. We also discuss the role of Rickard's notion of splendid equivalence in the context of Broué's abelian defect group conjecture, and indicate an alternative proof of parts of Rickard's derived Morita theorem that leverages the notion of completion of a triangulated category.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"113 6","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70577","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148282503","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":"Isotopy and equivalence of knots in 3-manifolds","authors":"Paolo Aceto,&nbsp;Corey Bregman,&nbsp;Christopher W. Davis,&nbsp;JungHwan Park,&nbsp;Arunima Ray","doi":"10.1112/jlms.70600","DOIUrl":"https://doi.org/10.1112/jlms.70600","url":null,"abstract":"<p>Two knots <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mi>J</mi>\u0000 <annotation>$J$</annotation>\u0000 </semantics></math> in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <annotation>$S^3$</annotation>\u0000 </semantics></math> are isotopic if and only if they are related by an orientation-preserving diffeomorphism of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <annotation>$S^3$</annotation>\u0000 </semantics></math>. This claim follows from the fact that any orientation-preserving self-diffeomorphism of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <annotation>$S^3$</annotation>\u0000 </semantics></math> is isotopic to the identity. We show that this same idea applies to any prime oriented closed 3-manifold. More precisely, we show that a prime closed oriented 3-manifold contains a pair of equivalent but nonisotopic knots if and only if the (orientation-preserving) mapping class group is nontrivial. When <span></span><math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math> is additionally irreducible we show that an orientation-preserving diffeomorphism of <span></span><math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math> is isotopic to the identity if and only if it preserves all homotopy classes of knots. For knots in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <mo>×</mo>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$S^1times S^2$</annotation>\u0000 </semantics></math> (the only reducible prime oriented 3-manifold) we exhibit infinitely many knots whose isotopy classes are not preserved by the Gluck twist.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"113 6","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70600","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148282322","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":"Independence and strong independence complexes of finite groups","authors":"Andrea Lucchini,&nbsp;Mima Stanojkovski","doi":"10.1112/jlms.70605","DOIUrl":"https://doi.org/10.1112/jlms.70605","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> be a finite group. In [10], two different concepts of independence (namely, <i>independence</i> and <i>strong independence</i>) are introduced for the subsets of <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math>, yielding to the definition of two simplicial complexes whose vertices are the elements of <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math>. The <i>strong independence complex</i> <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mover>\u0000 <mi>Σ</mi>\u0000 <mo>∼</mo>\u0000 </mover>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$tilde{Sigma }(G)$</annotation>\u0000 </semantics></math> turns out to be a subcomplex of the <i>independence complex</i> <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Σ</mi>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$Sigma (G)$</annotation>\u0000 </semantics></math>. We discuss several invariant properties related to these complexes and ask a number of questions inspired by our results and the examples we construct. We study then the particular case of complexes on finite abelian groups, giving a characterization of the finite groups realizing them. In conclusion, answering a question of Peter Cameron, we classify all finite groups in which the two concepts of independence coincide.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"113 6","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70605","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148282230","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":"Analyticity up to the boundary for the Prandtl and Navier–Stokes equations","authors":"Wei-Xi Li,&nbsp;Tong Yang,&nbsp;Ping Zhang","doi":"10.1112/jlms.70604","DOIUrl":"https://doi.org/10.1112/jlms.70604","url":null,"abstract":"<p>We study the space-time analyticity of solutions to both the Prandtl and Navier–Stokes equations in the half-space. The analyticity estimates are local in time variable and global in space variable up to the boundary, with the lower bound of analyticity radii agreeing with that for the classical heat equation. The space-time analytic smoothing effect holds for the Navier–Stokes equations with finite Sobolev regular initial data, and for the Prandtl equations with real-analytic initial data in tangential direction.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"113 6","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148281989","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":"Invariants of the finite orthogonal groups in odd dimension and even characteristic","authors":"H. E. A. Campbell,&nbsp;R. J. Shank,&nbsp;D. L. Wehlau","doi":"10.1112/jlms.70597","DOIUrl":"https://doi.org/10.1112/jlms.70597","url":null,"abstract":"<p>We describe the ring of invariants for the finite orthogonal groups in odd dimension and even characteristic acting on the defining representation. We construct a minimal algebra generating set and describe the relations amongst the generators. This ring of invariants is shown to be a complete intersection and thus is Cohen–Macaulay. This extends the previous computation of Kropholler, Mohseni Rajaei, and Segal valid over the field of order 2.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"113 6","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148281991","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":"Primitivity testing in free group algebras via duality","authors":"Matan Seidel,&nbsp;Danielle Ernst-West,&nbsp;Doron Puder","doi":"10.1112/jlms.70599","DOIUrl":"https://doi.org/10.1112/jlms.70599","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;annotation&gt;$K$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be a field and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;annotation&gt;$F$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; a free group. By a classical result of Cohn and Lewin, the free group algebra &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;mfenced&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;/mfenced&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Kleft[Fright]$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a &lt;i&gt;free ideal ring&lt;/i&gt; (FIR): a ring over which the submodules of free modules are themselves free, and of a well-defined rank. Given a finitely generated right ideal &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;I&lt;/mi&gt;\u0000 &lt;mo&gt;⩽&lt;/mo&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;mfenced&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;/mfenced&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Ileqslant Kleft[Fright]$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and an element &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;f&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;mi&gt;I&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$fin I$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, we give an explicit algorithm determining whether &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;f&lt;/mi&gt;\u0000 &lt;annotation&gt;$f$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is part of some basis of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;I&lt;/mi&gt;\u0000 &lt;annotation&gt;$I$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. More generally, given free &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;mo&gt;[&lt;/mo&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;mo&gt;]&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$K[F]$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-modules &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mo&gt;⩽&lt;/mo&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Mleqslant N$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, we provide algorithms determining whether &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;annotation&gt;$M$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a free summand of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;annotation&gt;$N$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"113 6","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70599","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148281990","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}