Pub Date : 2024-10-22DOI: 10.1016/j.jpaa.2024.107829
Andriy Regeta
In this note we extend the result from [14] and prove that if S is an affine non-toric -surface of hyperbolic type that admits a -action and X is an affine irreducible variety such that is isomorphic to as an abstract group, then X is a -surface of hyperbolic type. Further, we show that a smooth Danielewski surface , where p has no multiple roots, is determined by its automorphism group seen as an ind-group in the category of affine irreducible varieties.
在本注释中，我们扩展了 [14] 的结果，证明如果 S 是双曲型的仿射非簇状 Gm 曲面，且允许 Ga 作用，而 X 是仿射不可还原变种，使得 Aut(X) 与作为抽象群的 Aut(S) 同构，则 X 是双曲型的 Gm 曲面。此外，我们还证明了光滑的丹尼列夫斯基曲面 Dp={xy=p(z)}⊂A3（其中 p 没有多根）是由它的自形群决定的，这个自形群被视为仿射不可还原变种范畴中的一个内群。
{"title":"Characterization of affine Gm-surfaces of hyperbolic type","authors":"Andriy Regeta","doi":"10.1016/j.jpaa.2024.107829","DOIUrl":"10.1016/j.jpaa.2024.107829","url":null,"abstract":"<div><div>In this note we extend the result from <span><span>[14]</span></span> and prove that if <em>S</em> is an affine non-toric <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>-surface of hyperbolic type that admits a <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span>-action and <em>X</em> is an affine irreducible variety such that <span><math><mi>Aut</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is isomorphic to <span><math><mi>Aut</mi><mo>(</mo><mi>S</mi><mo>)</mo></math></span> as an abstract group, then <em>X</em> is a <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>-surface of hyperbolic type. Further, we show that a smooth Danielewski surface <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>=</mo><mo>{</mo><mi>x</mi><mi>y</mi><mo>=</mo><mi>p</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>}</mo><mo>⊂</mo><msup><mrow><mi>A</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, where <em>p</em> has no multiple roots, is determined by its automorphism group seen as an ind-group in the category of affine irreducible varieties.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107829"},"PeriodicalIF":0.7,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552392","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 : 2024-10-22DOI: 10.1016/j.jpaa.2024.107832
Fei Kong
Let Q be a non-degenerate even lattice, let be the lattice vertex algebra associated to Q, and let be a quantum lattice vertex algebra ([10]). In this paper, we prove that every -module is completely reducible, and the set of simple -modules are in one-to-one correspondence with the set of cosets of Q in its dual lattice.
{"title":"Representations of quantum lattice vertex algebras","authors":"Fei Kong","doi":"10.1016/j.jpaa.2024.107832","DOIUrl":"10.1016/j.jpaa.2024.107832","url":null,"abstract":"<div><div>Let <em>Q</em> be a non-degenerate even lattice, let <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>Q</mi></mrow></msub></math></span> be the lattice vertex algebra associated to <em>Q</em>, and let <span><math><msubsup><mrow><mi>V</mi></mrow><mrow><mi>Q</mi></mrow><mrow><mi>η</mi></mrow></msubsup></math></span> be a quantum lattice vertex algebra (<span><span>[10]</span></span>). In this paper, we prove that every <span><math><msubsup><mrow><mi>V</mi></mrow><mrow><mi>Q</mi></mrow><mrow><mi>η</mi></mrow></msubsup></math></span>-module is completely reducible, and the set of simple <span><math><msubsup><mrow><mi>V</mi></mrow><mrow><mi>Q</mi></mrow><mrow><mi>η</mi></mrow></msubsup></math></span>-modules are in one-to-one correspondence with the set of cosets of <em>Q</em> in its dual lattice.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107832"},"PeriodicalIF":0.7,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552129","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 : 2024-10-22DOI: 10.1016/j.jpaa.2024.107830
Ioannis Emmanouil, Olympia Talelli
In this paper, we examine the Gorenstein dimension of modules over the group algebra kG of a group G with coefficients in a commutative ring k. As a Gorenstein analogue of the classical case, we bound this dimension in terms of the Gorenstein dimension of the underlying k-module and the Gorenstein dimension of G over k. Our method is based on the notion of a characteristic module for G, introduced by the second author, and uses the stability properties of the Gorenstein categories. We also examine the class of hierarchically decomposable groups defined by Kropholler and use the module of bounded -valued functions on such a group G to characterize the Gorenstein flat -modules, in terms of flat modules, and the Gorenstein injective -modules, in terms of injective modules (by complete analogy with the characterization of Gorenstein projective -modules, in terms of projective modules, obtained by Dembegioti and the second author). It follows that, for a group G in Kropholler's class, (a) any Gorenstein projective -module is Gorenstein flat and (b) a -module is Gorenstein flat if its Pontryagin dual module is Gorenstein injective.
在本文中，我们研究了系数在交换环 k 中的群 G 的群代数 kG 上的模块的戈伦斯坦维度。作为经典情况下的戈伦斯坦类比，我们用底层 k 模块的戈伦斯坦维度和 k 上 G 的戈伦斯坦维度来约束这个维度。我们的方法基于第二位作者提出的 G 的特征模块概念，并使用了戈伦斯坦范畴的稳定性。我们还研究了由 Kropholler 定义的可分层分解群类，并使用此类群 G 上的有界 Z 值函数模块，以平模块表征了 Gorenstein 平面 ZG 模块，以注入模块表征了 Gorenstein 注入 ZG 模块（与 Dembegioti 和第二作者以投影模块表征 Gorenstein 投影 ZG 模块的方法完全类似）。由此可见，对于 Kropholler 类中的一个群 G，(a) 任何 Gorenstein 射性 ZG 模块都是 Gorenstein 平面模块；(b) 如果一个 ZG 模块的 Pontryagin 对偶模块是 Gorenstein 注入模块，那么这个 ZG 模块就是 Gorenstein 平面模块。
{"title":"Characteristic modules and Gorenstein (co-)homological dimension of groups","authors":"Ioannis Emmanouil, Olympia Talelli","doi":"10.1016/j.jpaa.2024.107830","DOIUrl":"10.1016/j.jpaa.2024.107830","url":null,"abstract":"<div><div>In this paper, we examine the Gorenstein dimension of modules over the group algebra <em>kG</em> of a group <em>G</em> with coefficients in a commutative ring <em>k</em>. As a Gorenstein analogue of the classical case, we bound this dimension in terms of the Gorenstein dimension of the underlying <em>k</em>-module and the Gorenstein dimension of <em>G</em> over <em>k</em>. Our method is based on the notion of a characteristic module for <em>G</em>, introduced by the second author, and uses the stability properties of the Gorenstein categories. We also examine the class of hierarchically decomposable groups defined by Kropholler and use the module of bounded <span><math><mi>Z</mi></math></span>-valued functions on such a group <em>G</em> to characterize the Gorenstein flat <span><math><mi>Z</mi><mi>G</mi></math></span>-modules, in terms of flat modules, and the Gorenstein injective <span><math><mi>Z</mi><mi>G</mi></math></span>-modules, in terms of injective modules (by complete analogy with the characterization of Gorenstein projective <span><math><mi>Z</mi><mi>G</mi></math></span>-modules, in terms of projective modules, obtained by Dembegioti and the second author). It follows that, for a group <em>G</em> in Kropholler's class, (a) any Gorenstein projective <span><math><mi>Z</mi><mi>G</mi></math></span>-module is Gorenstein flat and (b) a <span><math><mi>Z</mi><mi>G</mi></math></span>-module is Gorenstein flat if its Pontryagin dual module is Gorenstein injective.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107830"},"PeriodicalIF":0.7,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552131","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 : 2024-10-16DOI: 10.1016/j.jpaa.2024.107825
Tadahito Harima , Satoru Isogawa , Junzo Watanabe
In this paper we give a new family of complete intersections which have the strong Lefschetz property. The family consists of Artinian algebras defined by ideals generated by power sum symmetric polynomials of consecutive degrees and of certain ideals naturally derived from them. This family has a structure of a binary tree and this observation is a key to prove that all members in it have the strong Lefschetz property.
{"title":"A binary tree of complete intersections with the strong Lefschetz property","authors":"Tadahito Harima , Satoru Isogawa , Junzo Watanabe","doi":"10.1016/j.jpaa.2024.107825","DOIUrl":"10.1016/j.jpaa.2024.107825","url":null,"abstract":"<div><div>In this paper we give a new family of complete intersections which have the strong Lefschetz property. The family consists of Artinian algebras defined by ideals generated by power sum symmetric polynomials of consecutive degrees and of certain ideals naturally derived from them. This family has a structure of a binary tree and this observation is a key to prove that all members in it have the strong Lefschetz property.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107825"},"PeriodicalIF":0.7,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533598","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 : 2024-10-16DOI: 10.1016/j.jpaa.2024.107824
Haicheng Zhang
Let m be a positive integer and be the m-periodic derived category of a finitary hereditary abelian category . Applying the derived Hall numbers of the bounded derived category , we define an m-periodic extended derived Hall algebra for , and use it to give a global, unified and explicit characterization for the algebra structure of Bridgeland's Hall algebra of periodic complexes. Moreover, we also provide an explicit characterization for the odd periodic derived Hall algebra of defined by Xu-Chen [24].
设 m 为正整数，Dm(A) 为有限遗传无性范畴 A 的 m 周期派生范畴。应用有界派生范畴 Db(A) 的派生霍尔数，我们定义了 Dm(A) 的 m 周期扩展派生霍尔代数，并用它给出了布里奇兰周期复数霍尔代数的全局、统一和明确的代数结构特征。此外，我们还为许琛[24]定义的 A 的奇周期派生霍尔代数提供了一个明确的表征。
{"title":"Periodic derived Hall algebras of hereditary abelian categories","authors":"Haicheng Zhang","doi":"10.1016/j.jpaa.2024.107824","DOIUrl":"10.1016/j.jpaa.2024.107824","url":null,"abstract":"<div><div>Let <em>m</em> be a positive integer and <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> be the <em>m</em>-periodic derived category of a finitary hereditary abelian category <span><math><mi>A</mi></math></span>. Applying the derived Hall numbers of the bounded derived category <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>b</mi></mrow></msup><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, we define an <em>m</em>-periodic extended derived Hall algebra for <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, and use it to give a global, unified and explicit characterization for the algebra structure of Bridgeland's Hall algebra of periodic complexes. Moreover, we also provide an explicit characterization for the odd periodic derived Hall algebra of <span><math><mi>A</mi></math></span> defined by Xu-Chen <span><span>[24]</span></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107824"},"PeriodicalIF":0.7,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444585","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 : 2024-10-16DOI: 10.1016/j.jpaa.2024.107823
Alicja Jaworska-Pastuszak, Grzegorz Pastuszak, Grzegorz Bobiński
Assume that K is an algebraically closed field and denote by the Krull-Gabriel dimension of R, where R is a locally bounded K-category (or a bound quiver K-algebra). Assume that C is a tilted K-algebra and are the associated repetitive category, cluster repetitive category and cluster-tilted algebra, respectively. Our first result states that . Since the Krull-Gabriel dimensions of tame locally support-finite repetitive categories are known, we further conclude that . Finally, in the Appendix Grzegorz Bobiński presents a different way of determining the Krull-Gabriel dimension of the cluster-tilted algebras, by applying results of Geigle.
假设 K 是一个代数闭域，用 KG(R) 表示 R 的克鲁尔-加布里埃尔维数，其中 R 是一个局部有界 K 范畴（或有界四元组 K-代数）。假设 C 是倾斜 K 代数，Cˆ,Cˇ,C˜ 分别是相关的重复范畴、簇重复范畴和簇倾斜代数。我们的第一个结果表明，KG(C˜)=KG(Cˇ)≤KG(Cˆ)。由于驯服局部支持无限重复范畴的克鲁尔-加布里埃尔维数是已知的，我们进一步得出结论：KG(C˜)=KG(Cˇ)=KG(Cˆ)∈{0,2,∞}。最后，在附录中，格热戈兹-波宾斯基（Grzegorz Bobiński）运用盖格尔（Geigle）的结果，提出了另一种确定簇倾斜代数的克鲁尔-加布里埃尔维度的方法。
{"title":"On Krull-Gabriel dimension of cluster repetitive categories and cluster-tilted algebras","authors":"Alicja Jaworska-Pastuszak, Grzegorz Pastuszak, Grzegorz Bobiński","doi":"10.1016/j.jpaa.2024.107823","DOIUrl":"10.1016/j.jpaa.2024.107823","url":null,"abstract":"<div><div>Assume that <em>K</em> is an algebraically closed field and denote by <span><math><mi>KG</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> the Krull-Gabriel dimension of <em>R</em>, where <em>R</em> is a locally bounded <em>K</em>-category (or a bound quiver <em>K</em>-algebra). Assume that <em>C</em> is a tilted <em>K</em>-algebra and <span><math><mover><mrow><mi>C</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>,</mo><mover><mrow><mi>C</mi></mrow><mrow><mo>ˇ</mo></mrow></mover><mo>,</mo><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> are the associated repetitive category, cluster repetitive category and cluster-tilted algebra, respectively. Our first result states that <span><math><mi>KG</mi><mo>(</mo><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>)</mo><mo>=</mo><mi>KG</mi><mo>(</mo><mover><mrow><mi>C</mi></mrow><mrow><mo>ˇ</mo></mrow></mover><mo>)</mo><mo>≤</mo><mi>KG</mi><mo>(</mo><mover><mrow><mi>C</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span>. Since the Krull-Gabriel dimensions of tame locally support-finite repetitive categories are known, we further conclude that <span><math><mi>KG</mi><mo>(</mo><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>)</mo><mo>=</mo><mi>KG</mi><mo>(</mo><mover><mrow><mi>C</mi></mrow><mrow><mo>ˇ</mo></mrow></mover><mo>)</mo><mo>=</mo><mi>KG</mi><mo>(</mo><mover><mrow><mi>C</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo><mo>∈</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>∞</mo><mo>}</mo></math></span>. Finally, in the Appendix Grzegorz Bobiński presents a different way of determining the Krull-Gabriel dimension of the cluster-tilted algebras, by applying results of Geigle.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107823"},"PeriodicalIF":0.7,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444584","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}
Pub Date : 2024-10-16DOI: 10.1016/j.jpaa.2024.107826
Czesław Bagiński, Kamil Zabielski
As a result of impressive research [5], D. García-Lucas, Á. del Río and L. Margolis defined an infinite series of non-isomorphic 2-groups G and H, whose group algebras and over the field are isomorphic, solving negatively the long-standing Modular Isomorphism Problem (MIP). In this note we give a different perspective on their examples and show that they are special cases of a more general construction. We also show that this type of construction for does not provide a similar counterexample to the MIP.
作为令人印象深刻的研究成果[5]，加西亚-卢卡斯（D. García-Lucas）、德尔里奥（Á. del Río）和马格里斯（L. Margolis）定义了非同构 2 群 G 和 H 的无限序列，它们在 F=F2 上的群代数 FG 和 FH 是同构的，从而消极地解决了长期存在的模块同构问题（MIP）。在本论文中，我们将从另一个角度来分析它们的例子，并证明它们是一种更普遍构造的特例。我们还证明，p>2 的这种构造并没有为 MIP 提供类似的反例。
{"title":"The Modular Isomorphism Problem – the alternative perspective on counterexamples","authors":"Czesław Bagiński, Kamil Zabielski","doi":"10.1016/j.jpaa.2024.107826","DOIUrl":"10.1016/j.jpaa.2024.107826","url":null,"abstract":"<div><div>As a result of impressive research <span><span>[5]</span></span>, D. García-Lucas, Á. del Río and L. Margolis defined an infinite series of non-isomorphic 2-groups <em>G</em> and <em>H</em>, whose group algebras <span><math><mi>F</mi><mi>G</mi></math></span> and <span><math><mi>F</mi><mi>H</mi></math></span> over the field <span><math><mi>F</mi><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are isomorphic, solving negatively the long-standing Modular Isomorphism Problem (MIP). In this note we give a different perspective on their examples and show that they are special cases of a more general construction. We also show that this type of construction for <span><math><mi>p</mi><mo>></mo><mn>2</mn></math></span> does not provide a similar counterexample to the MIP.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107826"},"PeriodicalIF":0.7,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533602","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 : 2024-10-15DOI: 10.1016/j.jpaa.2024.107827
Greg Stevenson
We give an example of a commutative coherent ring of infinite global dimension such that the category of perfect complexes has finite Rouquier dimension.
我们举例说明了一个具有无限全维度的交换相干环，其完备复数范畴具有有限的鲁基尔维度。
{"title":"Rouquier dimension versus global dimension","authors":"Greg Stevenson","doi":"10.1016/j.jpaa.2024.107827","DOIUrl":"10.1016/j.jpaa.2024.107827","url":null,"abstract":"<div><div>We give an example of a commutative coherent ring of infinite global dimension such that the category of perfect complexes has finite Rouquier dimension.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107827"},"PeriodicalIF":0.7,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444583","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}
Pub Date : 2024-10-09DOI: 10.1016/j.jpaa.2024.107818
Sergio Pavon
We introduce the notion of torsion-simple objects in an abelian category: these are the objects which are always either torsion or torsion-free with respect to any torsion pair. We present some general results concerning their properties, and then proceed to investigate the notion in various contexts, such as the category of modules over an artin algebra or a commutative noetherian ring, and the category of quasi-coherent sheaves over the projective line.
{"title":"Torsion-simple objects in abelian categories","authors":"Sergio Pavon","doi":"10.1016/j.jpaa.2024.107818","DOIUrl":"10.1016/j.jpaa.2024.107818","url":null,"abstract":"<div><div>We introduce the notion of torsion-simple objects in an abelian category: these are the objects which are always either torsion or torsion-free with respect to any torsion pair. We present some general results concerning their properties, and then proceed to investigate the notion in various contexts, such as the category of modules over an artin algebra or a commutative noetherian ring, and the category of quasi-coherent sheaves over the projective line.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107818"},"PeriodicalIF":0.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426023","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 : 2024-10-09DOI: 10.1016/j.jpaa.2024.107821
Zhenxing Di , Liping Li , Li Liang
In this paper, the concept of compatible weak factorization systems in general categories is introduced as a counterpart of compatible complete cotorsion pairs in abelian categories. We describe a method to construct model structures on general categories via two compatible weak factorization systems satisfying certain conditions, and hence, generalize a very useful result by Gillespie for abelian model structures. As particular examples, we show that weak factorization systems associated to some classical model structures (for example, the Kan-Quillen model structure on ) satisfy these conditions.
{"title":"Compatible weak factorization systems and model structures","authors":"Zhenxing Di , Liping Li , Li Liang","doi":"10.1016/j.jpaa.2024.107821","DOIUrl":"10.1016/j.jpaa.2024.107821","url":null,"abstract":"<div><div>In this paper, the concept of compatible weak factorization systems in general categories is introduced as a counterpart of compatible complete cotorsion pairs in abelian categories. We describe a method to construct model structures on general categories via two compatible weak factorization systems satisfying certain conditions, and hence, generalize a very useful result by Gillespie for abelian model structures. As particular examples, we show that weak factorization systems associated to some classical model structures (for example, the Kan-Quillen model structure on <span><math><mi>sSet</mi></math></span>) satisfy these conditions.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107821"},"PeriodicalIF":0.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426022","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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