Pub Date : 2026-03-01Epub Date: 2026-02-05DOI: 10.1016/j.jpaa.2026.108194
Elisa Postinghel , Artie Prendergast-Smith
We introduce bilinear secant varieties and joins of subvarieties of products of projective spaces, as a generalisation of the classical secant varieties and joins of projective varieties. We show that the bilinear secant varieties of certain rational normal curves of play a central role in the study of the birational geometry of , its blow-up in s points in general position. We show that is log Fano, and we compute its effective and movable cones, for and and for and , and we compute the effective and movable cones of .
{"title":"Bilinear secants and birational geometry of blowups of Pn×Pn+1","authors":"Elisa Postinghel , Artie Prendergast-Smith","doi":"10.1016/j.jpaa.2026.108194","DOIUrl":"10.1016/j.jpaa.2026.108194","url":null,"abstract":"<div><div>We introduce bilinear secant varieties and joins of subvarieties of products of projective spaces, as a generalisation of the classical secant varieties and joins of projective varieties. We show that the bilinear secant varieties of certain rational normal curves of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>×</mo><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> play a central role in the study of the birational geometry of <span><math><msubsup><mrow><mi>X</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></math></span>, its blow-up in <em>s</em> points in general position. We show that <span><math><msubsup><mrow><mi>X</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></math></span> is log Fano, and we compute its effective and movable cones, for <span><math><mi>s</mi><mo>≤</mo><mi>n</mi><mo>+</mo><mn>2</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span> and for <span><math><mi>s</mi><mo>≤</mo><mi>n</mi><mo>+</mo><mn>3</mn></math></span> and <span><math><mi>n</mi><mo>≤</mo><mn>2</mn></math></span>, and we compute the effective and movable cones of <span><math><msubsup><mrow><mi>X</mi></mrow><mrow><mn>6</mn></mrow><mrow><mn>3</mn><mo>,</mo><mn>4</mn></mrow></msubsup></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 3","pages":"Article 108194"},"PeriodicalIF":0.8,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147422566","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}
Pub Date : 2026-03-01Epub Date: 2026-02-23DOI: 10.1016/j.jpaa.2026.108207
Ben Elias , Hankyung Ko , Nicolas Libedinsky , Leonardo Patimo
For any Coxeter system, and any double coset for two standard parabolic subgroups, we introduce a Demazure operator. These operators form a basis for morphism spaces in a category we call the nilCoxeter category, and we also present this category by generators and relations. We prove a generalization to this context of Demazure's celebrated theorem on Frobenius extensions. This generalized theorem serves as a criterion for ensuring the proper behavior of singular Soergel bimodules.
{"title":"Demazure operators for double cosets","authors":"Ben Elias , Hankyung Ko , Nicolas Libedinsky , Leonardo Patimo","doi":"10.1016/j.jpaa.2026.108207","DOIUrl":"10.1016/j.jpaa.2026.108207","url":null,"abstract":"<div><div>For any Coxeter system, and any double coset for two standard parabolic subgroups, we introduce a Demazure operator. These operators form a basis for morphism spaces in a category we call the nilCoxeter category, and we also present this category by generators and relations. We prove a generalization to this context of Demazure's celebrated theorem on Frobenius extensions. This generalized theorem serves as a criterion for ensuring the proper behavior of singular Soergel bimodules.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 3","pages":"Article 108207"},"PeriodicalIF":0.8,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147422586","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}
Pub Date : 2026-03-01Epub Date: 2026-02-04DOI: 10.1016/j.jpaa.2026.108191
Bernd Schober
Based on previous work by the author we deduce that the invariant introduced by Bierstone and Milman in order to give a proof for constructive resolution of singularities in characteristic zero can be determined purely by considering certain polyhedra and their projections.
{"title":"A polyhedral approach to the invariant of Bierstone and Milman","authors":"Bernd Schober","doi":"10.1016/j.jpaa.2026.108191","DOIUrl":"10.1016/j.jpaa.2026.108191","url":null,"abstract":"<div><div>Based on previous work by the author we deduce that the invariant introduced by Bierstone and Milman in order to give a proof for constructive resolution of singularities in characteristic zero can be determined purely by considering certain polyhedra and their projections.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 3","pages":"Article 108191"},"PeriodicalIF":0.8,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147422567","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}
Pub Date : 2026-03-01Epub Date: 2026-02-23DOI: 10.1016/j.jpaa.2026.108209
Marcos Escartín-Ferrer
We prove the -conjecture for two families of Artin groups: Artin groups such that there exists a prime number p dividing for every edge e with even label >2 and balanced Artin groups. The family of balanced Artin groups extends two previously studied families: the one considered by Kochloukova in [19] and the family of coherent Artin groups. We state a conjecture on the -invariant for Artin groups satisfying the -conjecture. The conjecture is proven to be true for two significant families: 2-dimensional and coherent Artin groups. In the 2-dimensional case we are able to compute for all and to derive finiteness properties of the derived subgroup.
{"title":"On the Σ1 and Σ2-invariants of Artin groups","authors":"Marcos Escartín-Ferrer","doi":"10.1016/j.jpaa.2026.108209","DOIUrl":"10.1016/j.jpaa.2026.108209","url":null,"abstract":"<div><div>We prove the <span><math><msup><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-conjecture for two families of Artin groups: Artin groups such that there exists a prime number <em>p</em> dividing <span><math><mfrac><mrow><mi>l</mi><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> for every edge <em>e</em> with even label >2 and balanced Artin groups. The family of balanced Artin groups extends two previously studied families: the one considered by Kochloukova in <span><span>[19]</span></span> and the family of coherent Artin groups. We state a conjecture on the <span><math><msup><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-invariant for Artin groups satisfying the <span><math><mi>K</mi><mo>(</mo><mi>π</mi><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-conjecture. The conjecture is proven to be true for two significant families: 2-dimensional and coherent Artin groups. In the 2-dimensional case we are able to compute <span><math><msup><mrow><mi>Σ</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> for all <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> and to derive finiteness properties of the derived subgroup.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 3","pages":"Article 108209"},"PeriodicalIF":0.8,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147422570","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}
Pub Date : 2026-02-01Epub Date: 2026-01-27DOI: 10.1016/j.jpaa.2026.108183
Kensuke Arakawa
We prove that the homotopy theory of monoidal relative categories is equivalent to that of monoidal ∞-categories, and likewise in the symmetric monoidal setting. As an application, we give a concise and complete proof of the fact that every presentably monoidal or presentably symmetric monoidal ∞-category is presented by a monoidal or symmetric monoidal model category, which, in the monoidal case, was sketched by Lurie, and in the symmetric monoidal case, was proved by Nikolaus–Sagave.
{"title":"Monoidal relative categories model monoidal ∞-categories","authors":"Kensuke Arakawa","doi":"10.1016/j.jpaa.2026.108183","DOIUrl":"10.1016/j.jpaa.2026.108183","url":null,"abstract":"<div><div>We prove that the homotopy theory of monoidal relative categories is equivalent to that of monoidal ∞-categories, and likewise in the symmetric monoidal setting. As an application, we give a concise and complete proof of the fact that every presentably monoidal or presentably symmetric monoidal ∞-category is presented by a monoidal or symmetric monoidal model category, which, in the monoidal case, was sketched by Lurie, and in the symmetric monoidal case, was proved by Nikolaus–Sagave.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108183"},"PeriodicalIF":0.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079527","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}
Pub Date : 2026-02-01Epub Date: 2026-01-26DOI: 10.1016/j.jpaa.2026.108181
H. Ananthnarayan , Omkar Javadekar , Rajiv Kumar
Let R be a fibre product of standard graded algebras over a field. We study the structure of syzygies of finitely generated graded R-modules and the Koszul property of R. As an application of this, we show that the existence of an R-module of finite regularity and infinite projective dimension forces R to be Koszul. We also look at the extremal rays of the Betti cone of finitely generated graded R-modules, and show that when , they are spanned by the Betti tables of pure R-modules if and only if R is Cohen–Macaulay with minimal multiplicity.
{"title":"Betti cones over fibre products","authors":"H. Ananthnarayan , Omkar Javadekar , Rajiv Kumar","doi":"10.1016/j.jpaa.2026.108181","DOIUrl":"10.1016/j.jpaa.2026.108181","url":null,"abstract":"<div><div>Let <em>R</em> be a fibre product of standard graded algebras over a field. We study the structure of syzygies of finitely generated graded <em>R</em>-modules and the Koszul property of <em>R</em>. As an application of this, we show that the existence of an <em>R</em>-module of finite regularity and infinite projective dimension forces <em>R</em> to be Koszul. We also look at the extremal rays of the Betti cone of finitely generated graded <em>R</em>-modules, and show that when <span><math><mtext>depth</mtext><mo>(</mo><mi>R</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>, they are spanned by the Betti tables of pure <em>R</em>-modules if and only if <em>R</em> is Cohen–Macaulay with minimal multiplicity.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108181"},"PeriodicalIF":0.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079528","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}
Pub Date : 2026-02-01Epub Date: 2026-01-27DOI: 10.1016/j.jpaa.2026.108186
Elad Paran , Tran Nam Son
We study the images of polynomial maps over algebraically closed division rings. Our first result generalizes the classical Ax-Grothendieck theorem: We show that if are elements of the free associative algebra generated by variables over an algebraically closed division ring D of finite dimension over its center F, and if the induced map is injective, then f must be surjective. With no condition on the dimension over the center, our second result is that if p is either an element in with zero constant term such that , or a nonconstant polynomial in . Furthermore, we also establish some Waring type results. For instance, for any integer , we prove that every matrix in can be expressed as a difference of pairs of multiplicative commutators of elements from , provided again that D is finite-dimensional over F.
{"title":"Images of polynomial maps and the Ax-Grothendieck theorem over algebraically closed division rings","authors":"Elad Paran , Tran Nam Son","doi":"10.1016/j.jpaa.2026.108186","DOIUrl":"10.1016/j.jpaa.2026.108186","url":null,"abstract":"<div><div>We study the images of polynomial maps over algebraically closed division rings. Our first result generalizes the classical Ax-Grothendieck theorem: We show that if <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> are elements of the free associative algebra <span><math><mi>D</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>m</mi></mrow></msub><mo>〉</mo></math></span> generated by <span><math><mi>m</mi><mo>≥</mo><mn>1</mn></math></span> variables over an algebraically closed division ring <em>D</em> of finite dimension over its center <em>F</em>, and if the induced map <span><math><mi>f</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo><mo>:</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>→</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> is injective, then <em>f</em> must be surjective. With no condition on the dimension over the center, our second result is that <span><math><mi>p</mi><mo>(</mo><mi>D</mi><mo>)</mo><mo>=</mo><mi>D</mi></math></span> if <em>p</em> is either an element in <span><math><mi>F</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>m</mi></mrow></msub><mo>〉</mo></math></span> with zero constant term such that <span><math><mi>p</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>≠</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span>, or a nonconstant polynomial in <span><math><mi>F</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span>. Furthermore, we also establish some Waring type results. For instance, for any integer <span><math><mi>n</mi><mo>></mo><mn>1</mn></math></span>, we prove that every matrix in <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>D</mi><mo>)</mo></math></span> can be expressed as a difference of pairs of multiplicative commutators of elements from <span><math><mi>p</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>D</mi><mo>)</mo><mo>)</mo></math></span>, provided again that <em>D</em> is finite-dimensional over <em>F</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108186"},"PeriodicalIF":0.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079526","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}
Pub Date : 2026-02-01Epub Date: 2026-01-12DOI: 10.1016/j.jpaa.2026.108172
Kaiyue He
We introduce a new numerical invariant associated to a finite-length R-module M and an ideal I in an Artinian local ring R. This invariant measures the ratio between and . We establish fundamental relationships between this invariant and the Betti numbers of the module under the assumption of the Tor modules vanishing. In particular, we use this invariant to establish a freeness criterion for modules under certain Tor vanishing conditions. The criterion applies specifically to the class of I-free modules — those modules M for which is isomorphic to a direct sum of copies of . Lastly, we apply these results to the canonical module, proving that, under certain conditions on the ring structure, when the zeroth Betti number is greater than or equal to the first Betti number of the canonical module, then the ring is Gorenstein. This partially answers a question posed by Jorgensen and Leuschke concerning the relationship between Betti numbers of the canonical module and Gorenstein properties.
{"title":"Betti numbers for modules over Artinian local rings","authors":"Kaiyue He","doi":"10.1016/j.jpaa.2026.108172","DOIUrl":"10.1016/j.jpaa.2026.108172","url":null,"abstract":"<div><div>We introduce a new numerical invariant <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> associated to a finite-length <em>R</em>-module <em>M</em> and an ideal <em>I</em> in an Artinian local ring <em>R</em>. This invariant measures the ratio between <span><math><mi>λ</mi><mo>(</mo><mi>I</mi><mi>M</mi><mo>)</mo></math></span> and <span><math><mi>λ</mi><mo>(</mo><mi>M</mi><mo>/</mo><mi>I</mi><mi>M</mi><mo>)</mo></math></span>. We establish fundamental relationships between this invariant and the Betti numbers of the module under the assumption of the Tor modules vanishing. In particular, we use this invariant to establish a freeness criterion for modules under certain Tor vanishing conditions. The criterion applies specifically to the class of <em>I</em>-free modules — those modules <em>M</em> for which <span><math><mi>M</mi><mo>/</mo><mi>I</mi><mi>M</mi></math></span> is isomorphic to a direct sum of copies of <span><math><mi>R</mi><mo>/</mo><mi>I</mi></math></span>. Lastly, we apply these results to the canonical module, proving that, under certain conditions on the ring structure, when the zeroth Betti number is greater than or equal to the first Betti number of the canonical module, then the ring is Gorenstein. This partially answers a question posed by Jorgensen and Leuschke concerning the relationship between Betti numbers of the canonical module and Gorenstein properties.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108172"},"PeriodicalIF":0.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981621","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}
Pub Date : 2026-02-01Epub Date: 2026-02-04DOI: 10.1016/j.jpaa.2026.108192
Valerio Buttinelli
We study the projective normality of the projective bundle of an Ulrich vector bundle embedded through the complete linear system of its tautological line bundle. The focus will be on Ulrich bundles defined over curves, surfaces with and hypersurfaces of dimension 2 and 3.
{"title":"On the projective normality of Ulrich bundles on some low-dimensional varieties","authors":"Valerio Buttinelli","doi":"10.1016/j.jpaa.2026.108192","DOIUrl":"10.1016/j.jpaa.2026.108192","url":null,"abstract":"<div><div>We study the projective normality of the projective bundle of an Ulrich vector bundle embedded through the complete linear system of its tautological line bundle. The focus will be on Ulrich bundles defined over curves, surfaces with <span><math><mi>q</mi><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>=</mo><mn>0</mn></math></span> and hypersurfaces of dimension 2 and 3.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108192"},"PeriodicalIF":0.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174132","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}
Pub Date : 2026-02-01Epub Date: 2026-02-04DOI: 10.1016/j.jpaa.2026.108189
Jiacheng Tang
Solid abelian groups, as introduced by Dustin Clausen and Peter Scholze, form a subcategory of all condensed abelian groups satisfying some “completeness” conditions and having favourable categorical properties. Given a profinite ring R, there is an associated condensed ring which is solid. We show that the natural embedding of profinite R-modules into solid -modules preserves Ext and tensor products, as well as the fact that profinite rings are analytic.
{"title":"Profinite and solid cohomology","authors":"Jiacheng Tang","doi":"10.1016/j.jpaa.2026.108189","DOIUrl":"10.1016/j.jpaa.2026.108189","url":null,"abstract":"<div><div>Solid abelian groups, as introduced by Dustin Clausen and Peter Scholze, form a subcategory of all condensed abelian groups satisfying some “completeness” conditions and having favourable categorical properties. Given a profinite ring <em>R</em>, there is an associated condensed ring <span><math><munder><mrow><mi>R</mi></mrow><mo>_</mo></munder></math></span> which is solid. We show that the natural embedding of profinite <em>R</em>-modules into solid <span><math><munder><mrow><mi>R</mi></mrow><mo>_</mo></munder></math></span>-modules preserves Ext and tensor products, as well as the fact that profinite rings are analytic.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108189"},"PeriodicalIF":0.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174131","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}
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