Pub Date : 2024-10-11DOI: 10.1016/j.jalgebra.2024.09.023
Yiyang Li , Bin Shu , Yufeng Yao
Let G be a connected reductive algebraic group over an algebraically closed field k of prime characteristic p, and . We study the representations of the reductive Lie algebra with p-character χ of standard Levi-form in this note. We obtain similar results about the translation functor and the wall-crossing functor of simple modules parallelling to the representations of algebraic groups (cf. [10, II.Lem.7.20]). Moreover, we get the Loewy lengths of baby Verma modules provided the Vogan Conjecture holds.
设 G 是在素特性 p 的代数闭域 k 上的连通还原代数群，且 g=Lie(G) 。在本注释中，我们将研究具有标准列维形式 p 字符 χ 的还原性列代数 g 的表示。我们得到了与代数群表示平行的简单模块的平移函子和壁交函子的类似结果（参见 [10, II.Lem.7.20]）。此外，只要沃根猜想成立，我们就能得到韦尔马婴孩模块的洛维长度。
{"title":"A note on the Loewy lengths of baby Verma modules for modular Lie algebras","authors":"Yiyang Li , Bin Shu , Yufeng Yao","doi":"10.1016/j.jalgebra.2024.09.023","DOIUrl":"10.1016/j.jalgebra.2024.09.023","url":null,"abstract":"<div><div>Let <em>G</em> be a connected reductive algebraic group over an algebraically closed field <strong>k</strong> of prime characteristic <em>p</em>, and <span><math><mi>g</mi><mo>=</mo><mtext>Lie</mtext><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. We study the representations of the reductive Lie algebra <span><math><mi>g</mi></math></span> with <em>p</em>-character <em>χ</em> of standard Levi-form in this note. We obtain similar results about the translation functor and the wall-crossing functor of simple modules parallelling to the representations of algebraic groups (cf. <span><span>[10, II.Lem.7.20]</span></span>). Moreover, we get the Loewy lengths of baby Verma modules provided the Vogan Conjecture holds.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142526774","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 : 2024-10-11DOI: 10.1016/j.jalgebra.2024.09.019
Be'eri Greenfeld
We prove that there exists an affine Noetherian algebra over a field of characteristic zero which is not finitely presented. Specifically, we prove that the Medvedev–Passman–Resco–Small algebra, recently shown to form a counterexample to the stability problem for Noetherian algebras, is not finitely presented. This answers a question due to Bergman and Irving in characteristic zero, a case explicitly left open by Resco and Small in 1993, who gave a counterexample to this question in positive characteristic.
{"title":"The finite presentation problem for Noetherian algebras","authors":"Be'eri Greenfeld","doi":"10.1016/j.jalgebra.2024.09.019","DOIUrl":"10.1016/j.jalgebra.2024.09.019","url":null,"abstract":"<div><div>We prove that there exists an affine Noetherian algebra over a field of characteristic zero which is not finitely presented. Specifically, we prove that the Medvedev–Passman–Resco–Small algebra, recently shown to form a counterexample to the stability problem for Noetherian algebras, is not finitely presented. This answers a question due to Bergman and Irving in characteristic zero, a case explicitly left open by Resco and Small in 1993, who gave a counterexample to this question in positive characteristic.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554246","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 : 2024-10-11DOI: 10.1016/j.jalgebra.2024.10.003
Arun S. Kannan
The Steinberg tensor product theorem is a fundamental result in the modular representation theory of reductive algebraic groups. It describes any finite-dimensional simple module of highest weight λ over such a group as the tensor product of Frobenius twists of simple modules with highest weights the weights appearing in a p-adic decomposition of λ, thereby reducing the character problem to a finite collection of weights. In recent years this theorem has been extended to various quasi-reductive supergroup schemes. In this paper, we prove the analogous result for the general linear group scheme for any object X in the Verlinde category .
斯坦伯格张量积定理是还原代数群的模块表示理论中的一个基本结果。它描述了在这样一个群上的任何有限维最高权重简单模块λ，作为最高权重简单模块的弗罗贝纽斯捻的张量积，其权重出现在λ的p-adic分解中，从而将特征问题简化为权重的有限集合。近年来，这一定理被扩展到各种准还原超群方案。在本文中，我们证明了一般线性群方案 GL(X) 对于 Verlinde 范畴 Verp 中任何对象 X 的类似结果。
{"title":"The Steinberg tensor product theorem for general linear group schemes in the Verlinde category","authors":"Arun S. Kannan","doi":"10.1016/j.jalgebra.2024.10.003","DOIUrl":"10.1016/j.jalgebra.2024.10.003","url":null,"abstract":"<div><div>The Steinberg tensor product theorem is a fundamental result in the modular representation theory of reductive algebraic groups. It describes any finite-dimensional simple module of highest weight <em>λ</em> over such a group as the tensor product of Frobenius twists of simple modules with highest weights the weights appearing in a <em>p</em>-adic decomposition of <em>λ</em>, thereby reducing the character problem to a finite collection of weights. In recent years this theorem has been extended to various quasi-reductive supergroup schemes. In this paper, we prove the analogous result for the general linear group scheme <span><math><mi>G</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> for any object <em>X</em> in the Verlinde category <span><math><msub><mrow><mi>Ver</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444751","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 : 2024-10-11DOI: 10.1016/j.jalgebra.2024.09.022
Jitendra Bajpai , Daniele Dona , Martin Nitsche
We explore the thinness of hypergeometric groups of type and by applying a new approach of computer-assisted ping-pong. We prove the thinness of 17 hypergeometric groups with maximally unipotent monodromy in , completing the classification of all 40 such groups into arithmetic and thin cases.
In addition, we establish the thinness of an additional 46 hypergeometric groups in , and of three hypergeometric groups in , completing the classification of all hypergeometric groups. To the best of our knowledge, this article produces the first 63 examples in the cyclotomic family of Zariski dense non-arithmetic hypergeometric monodromy groups of real rank three.
{"title":"Thin monodromy in Sp(4) and Sp(6)","authors":"Jitendra Bajpai , Daniele Dona , Martin Nitsche","doi":"10.1016/j.jalgebra.2024.09.022","DOIUrl":"10.1016/j.jalgebra.2024.09.022","url":null,"abstract":"<div><div>We explore the thinness of hypergeometric groups of type <span><math><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>4</mn><mo>)</mo></math></span> and <span><math><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>6</mn><mo>)</mo></math></span> by applying a new approach of computer-assisted ping-pong. We prove the thinness of 17 hypergeometric groups with maximally unipotent monodromy in <span><math><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>6</mn><mo>)</mo></math></span>, completing the classification of all 40 such groups into arithmetic and thin cases.</div><div>In addition, we establish the thinness of an additional 46 hypergeometric groups in <span><math><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>6</mn><mo>)</mo></math></span>, and of three hypergeometric groups in <span><math><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>4</mn><mo>)</mo></math></span>, completing the classification of all <span><math><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>4</mn><mo>)</mo></math></span> hypergeometric groups. To the best of our knowledge, this article produces the first 63 examples in the cyclotomic family of Zariski dense non-arithmetic hypergeometric monodromy groups of real rank three.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142442412","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 : 2024-10-11DOI: 10.1016/j.jalgebra.2024.10.005
Eliezer Batista , William Hautekiet , Paolo Saracco , Joost Vercruysse
Making the first steps towards a classification of simple partial comodules, we give a general construction for partial comodules of a Hopf algebra H using central idempotents in right coideal subalgebras and show that any 1-dimensional partial comodule is of that form. We conjecture that in fact all finite-dimensional simple partial H-comodules arise this way. For for some finite group G, we give conditions for the constructed partial comodule to be simple, and we determine when two of them are isomorphic. If , then our construction recovers the work of M. Dokuchaev and N. Zhukavets [12]. We also study the partial modules and comodules of the non-commutative non-cocommutative Kac-Paljutkin algebra .
作为简单部分协元分类的第一步，我们利用右共边子代数中的中心幂，给出了霍普夫代数 H 部分协元的一般构造，并证明任何一维部分协元都是这种形式。我们猜想，事实上所有有限维简单部分 H-协元都是这样产生的。对于某个有限群 G 的 H=kG，我们给出了所构造的部分组合数为简单组合数的条件，并确定了其中两个组合数同构的情况。如果 H=kG⁎，那么我们的构造就恢复了 M. Dokuchaev 和 N. Zhukavets [12] 的工作。我们还研究了非交换非交换 Kac-Paljutkin 代数 A 的部分模块和组合模块。
{"title":"Towards a classification of simple partial comodules of Hopf algebras","authors":"Eliezer Batista , William Hautekiet , Paolo Saracco , Joost Vercruysse","doi":"10.1016/j.jalgebra.2024.10.005","DOIUrl":"10.1016/j.jalgebra.2024.10.005","url":null,"abstract":"<div><div>Making the first steps towards a classification of simple partial comodules, we give a general construction for partial comodules of a Hopf algebra <em>H</em> using central idempotents in right coideal subalgebras and show that any 1-dimensional partial comodule is of that form. We conjecture that in fact all finite-dimensional simple partial <em>H</em>-comodules arise this way. For <span><math><mi>H</mi><mo>=</mo><mi>k</mi><mi>G</mi></math></span> for some finite group <em>G</em>, we give conditions for the constructed partial comodule to be simple, and we determine when two of them are isomorphic. If <span><math><mi>H</mi><mo>=</mo><mi>k</mi><msup><mrow><mi>G</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, then our construction recovers the work of M. Dokuchaev and N. Zhukavets <span><span>[12]</span></span>. We also study the partial modules and comodules of the non-commutative non-cocommutative Kac-Paljutkin algebra <span><math><mi>A</mi></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529839","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}
We study the set of common –rational solutions of “smooth” systems of multivariate symmetric polynomials with coefficients in a finite field . We show that, under certain conditions, the set of common solutions of such polynomial systems over the algebraic closure of has a “good” geometric behavior. This allows us to obtain precise estimates on the corresponding number of common –rational solutions. In the case of hypersurfaces we are able to strengthen the results. We illustrate the interest of these estimates through their application to certain classical combinatorial problems over finite fields.
{"title":"Smooth symmetric systems over a finite field and applications","authors":"Nardo Giménez , Guillermo Matera , Mariana Pérez , Melina Privitelli","doi":"10.1016/j.jalgebra.2024.09.011","DOIUrl":"10.1016/j.jalgebra.2024.09.011","url":null,"abstract":"<div><div>We study the set of common <span><math><msub><mrow><mi>F</mi></mrow><mrow><mspace></mspace><mi>q</mi></mrow></msub></math></span>–rational solutions of “smooth” systems of multivariate symmetric polynomials with coefficients in a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mspace></mspace><mi>q</mi></mrow></msub></math></span>. We show that, under certain conditions, the set of common solutions of such polynomial systems over the algebraic closure of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mspace></mspace><mi>q</mi></mrow></msub></math></span> has a “good” geometric behavior. This allows us to obtain precise estimates on the corresponding number of common <span><math><msub><mrow><mi>F</mi></mrow><mrow><mspace></mspace><mi>q</mi></mrow></msub></math></span>–rational solutions. In the case of hypersurfaces we are able to strengthen the results. We illustrate the interest of these estimates through their application to certain classical combinatorial problems over finite fields.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529841","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 : 2024-10-11DOI: 10.1016/j.jalgebra.2024.09.021
Yuta Kozakai , Arashi Sakai
Let G be a finite group, N a normal subgroup of G, and k a field of characteristic . In this paper, we formulate the brick version of Clifford's theorem under suitable assumptions and prove it by using the theory of wide subcategories. As an application of our theorem, we consider the restrictions of semibricks and two-term simple-minded collections under the assumption that the index of the normal subgroup N in G is a p-power.
设 G 是有限群，N 是 G 的正则子群，k 是特征 p>0 的域。在本文中，我们在适当的假设条件下提出了砖版克利福德定理，并利用广义子类理论证明了这一定理。作为我们定理的一个应用，我们考虑了在 G 中正态子群 N 的索引是 p 幂的假设下半砖和两期简明集合的限制。
{"title":"Clifford's theorem for bricks","authors":"Yuta Kozakai , Arashi Sakai","doi":"10.1016/j.jalgebra.2024.09.021","DOIUrl":"10.1016/j.jalgebra.2024.09.021","url":null,"abstract":"<div><div>Let <em>G</em> be a finite group, <em>N</em> a normal subgroup of <em>G</em>, and <em>k</em> a field of characteristic <span><math><mi>p</mi><mo>></mo><mn>0</mn></math></span>. In this paper, we formulate the brick version of Clifford's theorem under suitable assumptions and prove it by using the theory of wide subcategories. As an application of our theorem, we consider the restrictions of semibricks and two-term simple-minded collections under the assumption that the index of the normal subgroup <em>N</em> in <em>G</em> is a <em>p</em>-power.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444749","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 : 2024-10-11DOI: 10.1016/j.jalgebra.2024.08.036
Sumana Hatui , Gurleen Kaur , Sahanawaj Sabnam
The twisted group ring isomorphism problem (TGRIP) is a variation of the classical group ring isomorphism problem. It asks whether the ring structure of the twisted group ring determines the group up to isomorphism. In this article, we study the TGRIP for direct product and central product of groups. We provide some criteria to answer the TGRIP for groups by answering the TGRIP for certain associated quotients. As an application of these results, we provide several examples. Finally, we answer the TGRIP for extra-special p-groups, and for all but five groups of order , where is prime.
{"title":"On the twisted group ring isomorphism problem for a class of groups","authors":"Sumana Hatui , Gurleen Kaur , Sahanawaj Sabnam","doi":"10.1016/j.jalgebra.2024.08.036","DOIUrl":"10.1016/j.jalgebra.2024.08.036","url":null,"abstract":"<div><div>The twisted group ring isomorphism problem (TGRIP) is a variation of the classical group ring isomorphism problem. It asks whether the ring structure of the twisted group ring determines the group up to isomorphism. In this article, we study the TGRIP for direct product and central product of groups. We provide some criteria to answer the TGRIP for groups by answering the TGRIP for certain associated quotients. As an application of these results, we provide several examples. Finally, we answer the TGRIP for extra-special <em>p</em>-groups, and for all but five groups of order <span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>5</mn></mrow></msup></math></span>, where <span><math><mi>p</mi><mo>≥</mo><mn>5</mn></math></span> is prime.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529895","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 : 2024-10-11DOI: 10.1016/j.jalgebra.2024.09.014
Jinlei Dong, Fang Li
In this paper, we give presentations of the mapping class groups of marked surfaces stabilizing boundaries for any genus. Note that in the existing works, the mapping class groups of marked surfaces were the isotopy classes of homeomorphisms fixing boundaries pointwise. The condition for stabilizing boundaries of mapping class groups makes the requirement for mapping class groups to fix boundaries pointwise to be unnecessary.
As an application of presentations of the mapping class groups of marked surfaces stabilizing boundaries, we obtain the presentation of the cluster automorphism group of a cluster algebra from a feasible surface .
Lastly, for the case (1) 4-punctured sphere, the cluster automorphism group of a cluster algebra from the surface is characterized. Since cluster automorphism groups of cluster algebras from those surfaces were given in [1] in the cases (2) the once-punctured 4-gon and (3) the twice-punctured digon, we indeed give presentations of cluster automorphism groups of cluster algebras from surfaces which are not feasible.
{"title":"Presentations of mapping class groups and an application to cluster algebras from surfaces","authors":"Jinlei Dong, Fang Li","doi":"10.1016/j.jalgebra.2024.09.014","DOIUrl":"10.1016/j.jalgebra.2024.09.014","url":null,"abstract":"<div><div>In this paper, we give presentations of the mapping class groups of marked surfaces stabilizing boundaries for any genus. Note that in the existing works, the mapping class groups of marked surfaces were the isotopy classes of homeomorphisms fixing boundaries pointwise. The condition for stabilizing boundaries of mapping class groups makes the requirement for mapping class groups to fix boundaries pointwise to be unnecessary.</div><div>As an application of presentations of the mapping class groups of marked surfaces stabilizing boundaries, we obtain the presentation of the cluster automorphism group of a cluster algebra from a feasible surface <span><math><mo>(</mo><mi>S</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span>.</div><div>Lastly, for the case (1) 4-punctured sphere, the cluster automorphism group of a cluster algebra from the surface is characterized. Since cluster automorphism groups of cluster algebras from those surfaces were given in <span><span>[1]</span></span> in the cases (2) the once-punctured 4-gon and (3) the twice-punctured digon, we indeed give presentations of cluster automorphism groups of cluster algebras from surfaces which are not feasible.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142526773","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 : 2024-10-11DOI: 10.1016/j.jalgebra.2024.09.028
Ioannis Emmanouil, Olympia Talelli
We study properties of modules that appear as syzygies of acyclic complexes of projective, injective, flat or flat-cotorsion modules and obtain criteria for these complexes to be contractible or totally acyclic. Our results illustrate the importance of strongly fp-injective modules in the study of these properties. We examine implications of the existence of complete resolutions (in a certain weak sense) and the finiteness of the Gorenstein projective dimension of pure-projective modules. We also use the orthogonality in the homotopy category between complexes of flat modules and pure acyclic complexes of cotorsion modules, in order to study the syzygies of acyclic complexes of flat modules. Finally, we present some applications of our results to group rings, regarding complete resolutions and cohomological periodicity.
{"title":"Triviality criteria for unbounded complexes","authors":"Ioannis Emmanouil, Olympia Talelli","doi":"10.1016/j.jalgebra.2024.09.028","DOIUrl":"10.1016/j.jalgebra.2024.09.028","url":null,"abstract":"<div><div>We study properties of modules that appear as syzygies of acyclic complexes of projective, injective, flat or flat-cotorsion modules and obtain criteria for these complexes to be contractible or totally acyclic. Our results illustrate the importance of strongly fp-injective modules in the study of these properties. We examine implications of the existence of complete resolutions (in a certain weak sense) and the finiteness of the Gorenstein projective dimension of pure-projective modules. We also use the orthogonality in the homotopy category between complexes of flat modules and pure acyclic complexes of cotorsion modules, in order to study the syzygies of acyclic complexes of flat modules. Finally, we present some applications of our results to group rings, regarding complete resolutions and cohomological periodicity.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444750","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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