Pub Date : 2026-01-19DOI: 10.1016/j.jalgebra.2025.12.025
Rahul Gupta , Amalendu Krishna
We prove a duality theorem for the p-adic étale motivic cohomology of the complement of a divisor on a smooth projective variety over a finite field of characteristic p. We apply this theorem to prove several finiteness results for the Brauer group of normal surfaces and their regular loci over finite fields. In particular, we show that the Artin conjecture about the finiteness of the Brauer group for smooth projective surfaces over a finite field implies the same for all projective surfaces over the field. We also show that the Tate conjecture for divisors on smooth projective surfaces over finite fields implies its analog for normal projective surfaces over such fields.
{"title":"Duality theorem over finite fields and applications to Brauer groups","authors":"Rahul Gupta , Amalendu Krishna","doi":"10.1016/j.jalgebra.2025.12.025","DOIUrl":"10.1016/j.jalgebra.2025.12.025","url":null,"abstract":"<div><div>We prove a duality theorem for the <em>p</em>-adic étale motivic cohomology of the complement of a divisor on a smooth projective variety over a finite field of characteristic <em>p</em>. We apply this theorem to prove several finiteness results for the Brauer group of normal surfaces and their regular loci over finite fields. In particular, we show that the Artin conjecture about the finiteness of the Brauer group for smooth projective surfaces over a finite field implies the same for all projective surfaces over the field. We also show that the Tate conjecture for divisors on smooth projective surfaces over finite fields implies its analog for normal projective surfaces over such fields.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 450-509"},"PeriodicalIF":0.8,"publicationDate":"2026-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025647","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 : 2026-01-16DOI: 10.1016/j.jalgebra.2026.01.004
Dave Benson , Srikanth B. Iyengar , Henning Krause , Julia Pevtsova
For a point in the spectrum of the cohomology ring of a finite group G over a field k, we calculate the spectrum for the subcategory of dualisable objects inside the tensor triangulated category of -local and -torsion objects in the (big) stable module category of the group algebra kG.
{"title":"The spectrum of local dualisable modular representations","authors":"Dave Benson , Srikanth B. Iyengar , Henning Krause , Julia Pevtsova","doi":"10.1016/j.jalgebra.2026.01.004","DOIUrl":"10.1016/j.jalgebra.2026.01.004","url":null,"abstract":"<div><div>For a point <span><math><mi>p</mi></math></span> in the spectrum of the cohomology ring of a finite group <em>G</em> over a field <em>k</em>, we calculate the spectrum for the subcategory of dualisable objects inside the tensor triangulated category of <span><math><mi>p</mi></math></span>-local and <span><math><mi>p</mi></math></span>-torsion objects in the (big) stable module category of the group algebra <em>kG</em>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 78-97"},"PeriodicalIF":0.8,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146001823","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 : 2026-01-16DOI: 10.1016/j.jalgebra.2026.01.021
Nathan Blacher
We study noncommutative rings whose proper subrings all satisfy the same chain condition. We show that if every proper subring of a ring R is right Noetherian, then R is either right Noetherian or the trivial extension of by the Prüfer p-group for a prime p. We also prove that if every proper subring of R is right Artinian, then R is either right Artinian or . For commutative rings, both results were proved by Gilmer and Heinzer in 1992. Our result for right Artinian subrings only generalises the absolute case of their commutative result. We generalise the full result (when only certain subrings are right Artinian) in the context of PI rings.
{"title":"Rings whose subrings are all Noetherian or Artinian","authors":"Nathan Blacher","doi":"10.1016/j.jalgebra.2026.01.021","DOIUrl":"10.1016/j.jalgebra.2026.01.021","url":null,"abstract":"<div><div>We study noncommutative rings whose proper subrings all satisfy the same chain condition. We show that if every proper subring of a ring <em>R</em> is right Noetherian, then <em>R</em> is either right Noetherian or the trivial extension of <span><math><mi>Z</mi></math></span> by the Prüfer <em>p</em>-group for a prime <em>p</em>. We also prove that if every proper subring of <em>R</em> is right Artinian, then <em>R</em> is either right Artinian or <span><math><mi>Z</mi></math></span>. For commutative rings, both results were proved by Gilmer and Heinzer in 1992. Our result for right Artinian subrings only generalises the absolute case of their commutative result. We generalise the full result (when only certain subrings are right Artinian) in the context of PI rings.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 362-371"},"PeriodicalIF":0.8,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025655","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 : 2026-01-16DOI: 10.1016/j.jalgebra.2026.01.016
Xiaojun Yan , Xiuwu Zhu
Let be an elliptic curve, K an imaginary quadratic field, and let be a prime that splits in K and at which E has good ordinary reduction. Assume that the residual Galois representation associated with is irreducible. In this paper, we establish new cases of the two-variable Iwasawa main conjecture for E over K. As applications, we obtain more general results on the p-converse theorem and the p-part of the Birch and Swinnerton-Dyer formula in rank at most one.
{"title":"Main conjectures for non-CM elliptic curves at good ordinary primes","authors":"Xiaojun Yan , Xiuwu Zhu","doi":"10.1016/j.jalgebra.2026.01.016","DOIUrl":"10.1016/j.jalgebra.2026.01.016","url":null,"abstract":"<div><div>Let <span><math><mi>E</mi><mo>/</mo><mi>Q</mi></math></span> be an elliptic curve, <em>K</em> an imaginary quadratic field, and let <span><math><mi>p</mi><mo>></mo><mn>2</mn></math></span> be a prime that splits in <em>K</em> and at which <em>E</em> has good ordinary reduction. Assume that the residual Galois representation associated with <span><math><mo>(</mo><mi>E</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> is irreducible. In this paper, we establish new cases of the two-variable Iwasawa main conjecture for <em>E</em> over <em>K</em>. As applications, we obtain more general results on the <em>p</em>-converse theorem and the <em>p</em>-part of the Birch and Swinnerton-Dyer formula in rank at most one.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 372-402"},"PeriodicalIF":0.8,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025656","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 : 2026-01-16DOI: 10.1016/j.jalgebra.2025.12.026
Lisa Mandal, Md. Ali Zinna
In this article we prove the following results:
(1) A smooth surface in a smooth affine -algebra with trivial conormal bundle is a set theoretic complete intersection, if its class in the Grothendieck group is torsion.
(2) A smooth hypersurface in an affine variety of dimension over can be described set theoretically by equations.
{"title":"Set-theoretic complete intersection for smooth surfaces in a smooth affine algebra","authors":"Lisa Mandal, Md. Ali Zinna","doi":"10.1016/j.jalgebra.2025.12.026","DOIUrl":"10.1016/j.jalgebra.2025.12.026","url":null,"abstract":"<div><div>In this article we prove the following results:</div><div>(1) A smooth surface in a smooth affine <span><math><msub><mrow><mover><mrow><mi>F</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>p</mi></mrow></msub></math></span>-algebra with trivial conormal bundle is a set theoretic complete intersection, if its class in the Grothendieck group is torsion.</div><div>(2) A smooth hypersurface in an affine variety of dimension <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span> over <span><math><msub><mrow><mover><mrow><mi>F</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>p</mi></mrow></msub></math></span> can be described set theoretically by <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> equations.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 36-53"},"PeriodicalIF":0.8,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146001820","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 : 2026-01-16DOI: 10.1016/j.jalgebra.2026.01.007
Robert Auffarth
Given an embedding of a projective variety into projective space, we study the structure of the space of all linear projections that, when composed with the embedding, give a Galois morphism from the variety to a projective space of the same dimension.
{"title":"Galois subspaces for projective varieties","authors":"Robert Auffarth","doi":"10.1016/j.jalgebra.2026.01.007","DOIUrl":"10.1016/j.jalgebra.2026.01.007","url":null,"abstract":"<div><div>Given an embedding of a projective variety into projective space, we study the structure of the space of all linear projections that, when composed with the embedding, give a Galois morphism from the variety to a projective space of the same dimension.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 62-77"},"PeriodicalIF":0.8,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146001822","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 : 2026-01-16DOI: 10.1016/j.jalgebra.2026.01.005
Neeraj Deshmukh , Jack Hall
We construct smooth presentations of algebraic stacks that are local epimorphisms in the Morel–Voevodsky -homotopy category. As a consequence, we show that the motive of a smooth stack (in Voevodsky's triangulated category of motives) has many of the same properties as the motive of a smooth scheme.
{"title":"On the motivic homotopy type of algebraic stacks","authors":"Neeraj Deshmukh , Jack Hall","doi":"10.1016/j.jalgebra.2026.01.005","DOIUrl":"10.1016/j.jalgebra.2026.01.005","url":null,"abstract":"<div><div>We construct smooth presentations of algebraic stacks that are local epimorphisms in the Morel–Voevodsky <span><math><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-homotopy category. As a consequence, we show that the motive of a smooth stack (in Voevodsky's triangulated category of motives) has many of the same properties as the motive of a smooth scheme.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 790-804"},"PeriodicalIF":0.8,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146075489","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 : 2026-01-16DOI: 10.1016/j.jalgebra.2026.01.015
Meirav Amram , Cheng Gong , Jia-Li Mo , János Kollár
This paper considers some algebraic surfaces that can deform to planar Zappatic surfaces with a unique singularity of type . We prove that the Galois covers of these surfaces are all simply connected of general type, for . We also give a formula for a local Zappatic singularity of a Zappatic surface of type . As an application, we prove that such surfaces do not exist for . Furthermore, Kollár improves the result to in Appendix A.
{"title":"Deformations of Zappatic surfaces and their Galois covers","authors":"Meirav Amram , Cheng Gong , Jia-Li Mo , János Kollár","doi":"10.1016/j.jalgebra.2026.01.015","DOIUrl":"10.1016/j.jalgebra.2026.01.015","url":null,"abstract":"<div><div>This paper considers some algebraic surfaces that can deform to planar Zappatic surfaces with a unique singularity of type <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We prove that the Galois covers of these surfaces are all simply connected of general type, for <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>. We also give a formula for a local Zappatic singularity of a Zappatic surface of type <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. As an application, we prove that such surfaces do not exist for <span><math><mi>n</mi><mo>></mo><mn>30</mn></math></span>. Furthermore, Kollár improves the result to <span><math><mi>n</mi><mo>></mo><mn>9</mn></math></span> in Appendix <span><span>A</span></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 710-731"},"PeriodicalIF":0.8,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146075492","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 : 2026-01-16DOI: 10.1016/j.jalgebra.2025.12.024
Anoot Kumar Yadav , Kumari Saloni
<div><div>Let <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> be a Cohen-Macaulay local ring of dimension <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span> and <em>I</em> an <span><math><mi>m</mi></math></span>-primary ideal. In this paper, we prove that <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>J</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>≤</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>+</mo><mi>ℓ</mi><mo>(</mo><mi>A</mi><mo>/</mo><mi>I</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span> if <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span> for <span><math><mi>i</mi><mo>≥</mo><mn>4</mn></math></span> where <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>J</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo></math></span> is the reduction number of <em>I</em> with respect to <em>J</em> and <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo></math></span> are the Hilbert coefficients. Our result affirms a conjecture of M.E. Rossi. We also prove that (i) <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>≤</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>)</mo></math></span> for any <em>I</em>-admissible filtration <span><math><mi>I</mi></math></span> and (ii) <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>≤</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><mi>ℓ</mi><mo>(</mo><mi>A</mi><mo>/</mo><mi>I</mi><mo>)</mo><mo>)</mo></math></span> for an integrally closed ideal <em>I</em>. The above bound for <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo></math></span> in the case of <span><math><mi>m</mi></math></span>-primary ideals is better than the earlier known bounds in our knowledge. Further, the respective
{"title":"A note on a conjecture of Rossi for reduction numbers of ideals and their Ratliff-Rush filtration","authors":"Anoot Kumar Yadav , Kumari Saloni","doi":"10.1016/j.jalgebra.2025.12.024","DOIUrl":"10.1016/j.jalgebra.2025.12.024","url":null,"abstract":"<div><div>Let <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> be a Cohen-Macaulay local ring of dimension <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span> and <em>I</em> an <span><math><mi>m</mi></math></span>-primary ideal. In this paper, we prove that <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>J</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>≤</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>+</mo><mi>ℓ</mi><mo>(</mo><mi>A</mi><mo>/</mo><mi>I</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span> if <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span> for <span><math><mi>i</mi><mo>≥</mo><mn>4</mn></math></span> where <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>J</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo></math></span> is the reduction number of <em>I</em> with respect to <em>J</em> and <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo></math></span> are the Hilbert coefficients. Our result affirms a conjecture of M.E. Rossi. We also prove that (i) <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>≤</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>)</mo></math></span> for any <em>I</em>-admissible filtration <span><math><mi>I</mi></math></span> and (ii) <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>≤</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><mi>ℓ</mi><mo>(</mo><mi>A</mi><mo>/</mo><mi>I</mi><mo>)</mo><mo>)</mo></math></span> for an integrally closed ideal <em>I</em>. The above bound for <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo></math></span> in the case of <span><math><mi>m</mi></math></span>-primary ideals is better than the earlier known bounds in our knowledge. Further, the respective","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 213-239"},"PeriodicalIF":0.8,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025651","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 : 2026-01-14DOI: 10.1016/j.jalgebra.2026.01.002
Andrew Ng
We observe a criterion for groups to have vanishing virtual first Betti number and use it to give infinitely many examples of torsion-free, finitely generated, residually finite groups which aren't virtually diffuse. This answers a question raised by Kionke and Raimbault.
{"title":"Virtual first Betti number of GGS groups","authors":"Andrew Ng","doi":"10.1016/j.jalgebra.2026.01.002","DOIUrl":"10.1016/j.jalgebra.2026.01.002","url":null,"abstract":"<div><div>We observe a criterion for groups to have vanishing virtual first Betti number and use it to give infinitely many examples of torsion-free, finitely generated, residually finite groups which aren't virtually diffuse. This answers a question raised by Kionke and Raimbault.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"693 ","pages":"Pages 54-61"},"PeriodicalIF":0.8,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146001821","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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