Enochs’ conjecture asserts that each covering class of modules (over any ring) has to be closed under direct limits. Although various special cases of the conjecture have been verified, the conjecture remains open in its full generality. In this paper, we prove the conjecture for the classes Filt(𝒮){operatorname{Filt}(mathcal{S})}, where 𝒮{mathcal{S}} consists of ℵn{aleph_{n}}-presented modules for some fixed n<ω{n<omega}. In particular, this applies to the left-hand class of any cotorsion pair generated by a class of ℵn{aleph_{n}}-presented modules. Moreover, we also show that it is consistent with ZFC that Enochs’ conjecture holds for all classes of the form Filt(𝒮){operatorname{Filt}(m
{"title":"Enochs’ conjecture for cotorsion pairs and more","authors":"Silvana Bazzoni, Jan Šaroch","doi":"10.1515/forum-2023-0220","DOIUrl":"https://doi.org/10.1515/forum-2023-0220","url":null,"abstract":"Enochs’ conjecture asserts that each covering class of modules (over any ring) has to be closed under direct limits. Although various special cases of the conjecture have been verified, the conjecture remains open in its full generality. In this paper, we prove the conjecture for the classes <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Filt</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi mathvariant=\"script\">𝒮</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0220_eq_0359.png\" /> <jats:tex-math>{operatorname{Filt}(mathcal{S})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">𝒮</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0220_eq_0332.png\" /> <jats:tex-math>{mathcal{S}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> consists of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"normal\">ℵ</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0220_eq_0211.png\" /> <jats:tex-math>{aleph_{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-presented modules for some fixed <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>n</m:mi> <m:mo><</m:mo> <m:mi>ω</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0220_eq_0475.png\" /> <jats:tex-math>{n<omega}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In particular, this applies to the left-hand class of any cotorsion pair generated by a class of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"normal\">ℵ</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0220_eq_0211.png\" /> <jats:tex-math>{aleph_{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-presented modules. Moreover, we also show that it is consistent with ZFC that Enochs’ conjecture holds for all classes of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Filt</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi mathvariant=\"script\">𝒮</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0220_eq_0359.png\" /> <jats:tex-math>{operatorname{Filt}(m","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139078558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce and study a notion of module nuclear dimension for a C*mathrm{C}^{*}-algebra A which is a C*mathrm{C}^{*}-module over another C*mathrm{C}^{*}-algebra 𝔄{mathfrak{A}} with compatible actions. We show that the module nuclear dimension of A is zero if A is 𝔄{mathfrak{A}}-NF. The converse is shown to hold when 𝔄{mathfrak{A}} is a C(X){C(X)}-algebra with simple fibers, with X compact and totally d
我们引入并研究了 C * mathrm{C}^{*} -代数 A 的模核维度概念。 -代数 A 的模块核维度概念。 上的另一个 C * mathrm{C}^{*} -模块 -代数𝔄 {mathfrak{A}} 的兼容作用。我们证明，如果 A 是 {mathfrak{A}} ，那么 A 的模块核维度为零。 -NF。当𝔄 {mathfrak{A}} 是具有简单纤维的 C ( X ) {C(X)}-代数，且 X 紧凑且完全断开时，反之成立。我们还引入了一个模块分解秩的概念，并证明当 𝔄 {mathfrak{A}} 是单价且简单时，如果 A 的模块分解秩是有限的，那么 A 是 𝔄 {mathfrak{A}} -QD。 -QD。我们研究𝔄 {mathcal{T}_{mathfrak{A}}(A)} 的𝔄 {mathfrak{A}} 的集合 𝒯 ( A ) {mathcal{T}_{mathfrak{A}}(A)} 。 -A 上的有值模量迹，并将 A 的 Cuntz 半群与集合 𝒯 ( A ) {mathcal{T}_{mathfrak{A}}(A)} 上的下半连续仿射函数联系起来。同时，我们还证明了一个模块崔-埃夫罗斯提升定理。我们给出了一类例子的模块核维度估计值。
{"title":"Finite approximation properties of C*-modules III","authors":"Massoud Amini","doi":"10.1515/forum-2023-0283","DOIUrl":"https://doi.org/10.1515/forum-2023-0283","url":null,"abstract":"We introduce and study a notion of module nuclear dimension for a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi mathvariant=\"normal\">C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0204.png\" /> <jats:tex-math>mathrm{C}^{*}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-algebra <jats:italic>A</jats:italic> which is a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi mathvariant=\"normal\">C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0204.png\" /> <jats:tex-math>mathrm{C}^{*}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-module over another <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi mathvariant=\"normal\">C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0204.png\" /> <jats:tex-math>mathrm{C}^{*}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-algebra <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝔄</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0639.png\" /> <jats:tex-math>{mathfrak{A}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with compatible actions. We show that the module nuclear dimension of <jats:italic>A</jats:italic> is zero if <jats:italic>A</jats:italic> is <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝔄</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0639.png\" /> <jats:tex-math>{mathfrak{A}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-NF. The converse is shown to hold when <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝔄</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0639.png\" /> <jats:tex-math>{mathfrak{A}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>C</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>X</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0344.png\" /> <jats:tex-math>{C(X)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-algebra with simple fibers, with <jats:italic>X</jats:italic> compact and totally d","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"8 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139078619","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let 𝔽qd{mathbb{F}_{q}^{d}} be the d-dimensional vector space over the finite field 𝔽q{mathbb{F}_{q}} with q elements. For each non-zero r in 𝔽q{mathbb{F}_{q}} and E⊂𝔽qd{Esubsetmathbb{F}_{q}^{d}}, we define W(r){W(r)} as the number of quadruples (x,y,z,w)∈E4{(x,y,z,w)in E^{4}} such that Q
设 𝔽 q d {mathbb{F}_{q}^{d}} 是有限域 𝔽 q {mathbb{F}_{q}} 上有 q 个元素的 d 维向量空间。对于𝔽 q {mathbb{F}_{q}} 中的每一个非零 r 和 E ⊂ 𝔽 q d {Esubsetmathbb{F}_{q}^{d}} ，我们定义 W ( r ) {mathbb{F}_{q}^{d} 为有限域上、有 q 个元素的 d 维向量空间。 我们定义 W ( r ) {W(r)} 为∈ E 4 {(x,y,z,w)in E^{4}} 中 Q ( x - y ) Q ( z - w ) = r {frac{Q(x-y)}{Q(z-w)}=r} 的四元数 ( x , y , z , w ) ，其中 Q 是 𝔽 q {mathbb{F}_{q}} 上 d 个变量的非退化二次型。 .当 Q ( α ) = ∑ i = 1 d α i 2 {Q(alpha)=sum_{i=1}^{d}alpha_{i}^{2}} 时，α = ( α 1 , ... , α d ) ∈ α 。, α d ) ∈ 𝔽 q d {alpha=(alpha_{1},ldots,alpha_{d})inmathbb{F}_{q}^{d}} Pham (2022) 最近使用群作用机制证明，如果 E ⊂ 𝔽 q 2 {Esubsetmathbb{F}_{q}^{2}} with q ≡ 3 ( mod 4 ) {qequiv 3~{}(operatorname{mod},4)} and | E |≥ C q {|E|geq Cq} ，那么有 W ( r ) 。 对于任意非零平方数 r∈ 𝔽 q {rinmathbb{F}_{q}} ，其中 C 是一个足够大的平方数。 其中 C 是一个足够大的常数，c 是介于 0 和 1 之间的某个数，而 | E | {|E|} 表示集合 E 的万有引力。在本文中，我们将 Pham 在二维中的结果改进并扩展到具有一般非退化二次距离的任意维度。作为我们结果的一个推论，我们还概括了前两位作者和 Parshall (2019) 关于距离集的商集的 Falconer 型问题的尖锐结果。此外，我们还提供了基础集合大小条件的改进常数。新的关键要素是将 W ( r ) {W(r)} 的估计值与 2 d {2d} 维向量空间中的二次同素异形体联系起来。这种方法富有成果，因为它允许我们利用高斯和，而高斯和比标准距离类型问题中出现的克洛斯特曼和更容易处理。
{"title":"The quotient set of the quadratic distance set over finite fields","authors":"Alex Iosevich, Doowon Koh, Firdavs Rakhmonov","doi":"10.1515/forum-2023-0313","DOIUrl":"https://doi.org/10.1515/forum-2023-0313","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> <m:mi>d</m:mi> </m:msubsup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0313_eq_0296.png\" /> <jats:tex-math>{mathbb{F}_{q}^{d}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the <jats:italic>d</jats:italic>-dimensional vector space over the finite field <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0313_eq_0298.png\" /> <jats:tex-math>{mathbb{F}_{q}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:italic>q</jats:italic> elements. For each non-zero <jats:italic>r</jats:italic> in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0313_eq_0298.png\" /> <jats:tex-math>{mathbb{F}_{q}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>E</m:mi> <m:mo>⊂</m:mo> <m:msubsup> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> <m:mi>d</m:mi> </m:msubsup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0313_eq_0186.png\" /> <jats:tex-math>{Esubsetmathbb{F}_{q}^{d}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, we define <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>W</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>r</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0313_eq_0235.png\" /> <jats:tex-math>{W(r)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> as the number of quadruples <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo>,</m:mo> <m:mi>z</m:mi> <m:mo>,</m:mo> <m:mi>w</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:msup> <m:mi>E</m:mi> <m:mn>4</m:mn> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0313_eq_0153.png\" /> <jats:tex-math>{(x,y,z,w)in E^{4}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> such that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mfrac> <m:mrow> <m:mi>Q</m:mi> <m:mo></m:mo> <m:","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"4 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139078676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the non-separating curve complex of every surface of finite type and genus at least three admits an exhaustion by finite rigid sets.
我们证明，每一个有限类型、至少三属的曲面的非分离曲线复曲面都可以通过有限刚集穷尽。
{"title":"Finite rigid sets of the non-separating curve complex","authors":"Rodrigo De Pool","doi":"10.1515/forum-2023-0024","DOIUrl":"https://doi.org/10.1515/forum-2023-0024","url":null,"abstract":"We prove that the non-separating curve complex of every surface of finite type and genus at least three admits an exhaustion by finite rigid sets.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"2 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139080147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that if X is a complete metric space with uniform relative normal structure and G is a subgroup of the isometry group of X with bounded orbits, then there is a point in X fixed by every isometry in G. As a corollary, we obtain a theorem of U. Lang (2013) concerning injective metric spaces. A few applications of this theorem are given to the problems of inner derivations. In particular, we show that if L1(μ){L_{1}(mu)} is an essential Banach L1(G){L_{1}(G)}-bimodule, then any continuous derivation δ:L1(G)→L∞(μ){delta:L_{1}(G)rightarrow L_{infty}(mu)} is inner. This extends a theorem of B. E. Johnson (1991) asserting that the convolution algebra L1(G){L_{1}(G)} is weakly am
我们证明，如果 X 是具有均匀相对法向结构的完全度量空间，而 G 是 X 的有界轨道等值群的一个子群，那么 X 中存在一个被 G 中的每个等值固定的点。这个定理在内层衍生问题上有一些应用。特别是，我们证明了如果 L 1 ( μ ) {L_{1}(mu)} 是一个本质的巴纳赫 L 1 ( G ) {L_{1}(G)} - 二模子，那么任何连续求导 δ : L 1 ( G ) → L ∞ ( μ ) {delta:L_{1}(G)rightarrow L_{infty}(mu)} 都是内求导。这扩展了 B. E. Johnson (1991) 的一个定理，即如果 G 是局部紧凑群，卷积代数 L 1 ( G ) {L_{1}(G)} 是弱可变的。
{"title":"A fixed point theorem for isometries on a metric space","authors":"Andrzej Wiśnicki","doi":"10.1515/forum-2023-0193","DOIUrl":"https://doi.org/10.1515/forum-2023-0193","url":null,"abstract":"We show that if <jats:italic>X</jats:italic> is a complete metric space with uniform relative normal structure and <jats:italic>G</jats:italic> is a subgroup of the isometry group of <jats:italic>X</jats:italic> with bounded orbits, then there is a point in <jats:italic>X</jats:italic> fixed by every isometry in <jats:italic>G</jats:italic>. As a corollary, we obtain a theorem of U. Lang (2013) concerning injective metric spaces. A few applications of this theorem are given to the problems of inner derivations. In particular, we show that if <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>μ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0193_eq_0087.png\" /> <jats:tex-math>{L_{1}(mu)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is an essential Banach <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0193_eq_0086.png\" /> <jats:tex-math>{L_{1}(G)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-bimodule, then any continuous derivation <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>δ</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>→</m:mo> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mi mathvariant=\"normal\">∞</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>μ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0193_eq_0136.png\" /> <jats:tex-math>{delta:L_{1}(G)rightarrow L_{infty}(mu)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is inner. This extends a theorem of B. E. Johnson (1991) asserting that the convolution algebra <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0193_eq_0086.png\" /> <jats:tex-math>{L_{1}(G)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is weakly am","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"8 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139078683","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lars Winther Christensen, Sergio Estrada, Li Liang, Peder Thompson, Junpeng Wang
Distinctive characteristics of Iwanaga–Gorenstein rings are typically understood through their intrinsic symmetry. We show that several of those that pertain to the Gorenstein global dimensions carry over to the one-sided situation, even without the noetherian hypothesis. Our results yield new relations among homological invariants related to the Gorenstein property, not only Gorenstein global dimensions but also the suprema of projective/injective dimensions of injective/projective modules and finitistic dimensions.
{"title":"One-sided Gorenstein rings","authors":"Lars Winther Christensen, Sergio Estrada, Li Liang, Peder Thompson, Junpeng Wang","doi":"10.1515/forum-2023-0303","DOIUrl":"https://doi.org/10.1515/forum-2023-0303","url":null,"abstract":"Distinctive characteristics of Iwanaga–Gorenstein rings are typically understood through their intrinsic symmetry. We show that several of those that pertain to the Gorenstein global dimensions carry over to the one-sided situation, even without the noetherian hypothesis. Our results yield new relations among homological invariants related to the Gorenstein property, not only Gorenstein global dimensions but also the suprema of projective/injective dimensions of injective/projective modules and finitistic dimensions.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"34 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139078512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The automorphism group of a map on a surface acts naturally on its flags (triples of incident vertices, edges, and faces). We will study the action of the automorphism group of a map on its edges. A map is semi-equivelar if all of its vertices have the same type of face-cycles. A semi-equivelar toroidal map refers to a semi-equivelar map embedded on a torus. If a map has k edge orbits under its automorphism group, it is referred to as a k-edge orbital or k-orbital. Specifically, it is referred to as an edge-transitive map if k=1{k=1}. If any two edges have the same edge-symbol, a map is said to be edge-homogeneous. Every edge-homogeneous toroidal map has an edge-transitive cover, as proved in [A. Orbanić, D. Pellicer, T. Pisanski and T. W. Tucker, Edge-transitive maps of low genus, Ars Math. Contemp. 4 2011, 2, 385–402]. In this article, we show the existence and give a classification of k-edge covers of semi-equivelar toroidal maps.
曲面上的映射的自形群自然地作用于它的旗（入射顶点、边和面的三元组）。我们将研究映射的自形群对其边的作用。如果一个映射的所有顶点都有相同类型的面循环，那么这个映射就是半等式映射。半等边环面映射指的是嵌入在环面上的半等边映射。如果一个映射在其自动形群下有 k 个边轨道，则称为 k 边轨道或 k 轨道。具体来说，如果 k = 1 {k=1} ，则称为边跨映射。如果任意两条边的边符号相同，则称为边同构映射。Orbanić, D. Pellicer, T. Pisanski and T. W. Tucker, Edge-transitive maps of low genus, Ars Math.Contemp.4 2011, 2, 385-402].在本文中，我们证明了半等价环面映射的 k 边盖的存在并给出了其分类。
{"title":"Archimedean toroidal maps and their edge covers","authors":"Arnab Kundu, Dipendu Maity","doi":"10.1515/forum-2023-0168","DOIUrl":"https://doi.org/10.1515/forum-2023-0168","url":null,"abstract":"The automorphism group of a map on a surface acts naturally on its flags (triples of incident vertices, edges, and faces). We will study the action of the automorphism group of a map on its edges. A map is semi-equivelar if all of its vertices have the same type of face-cycles. A semi-equivelar toroidal map refers to a semi-equivelar map embedded on a torus. If a map has <jats:italic>k</jats:italic> edge orbits under its automorphism group, it is referred to as a <jats:italic>k</jats:italic>-edge orbital or <jats:italic>k</jats:italic>-orbital. Specifically, it is referred to as an edge-transitive map if <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0168_eq_0915.png\" /> <jats:tex-math>{k=1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. If any two edges have the same edge-symbol, a map is said to be edge-homogeneous. Every edge-homogeneous toroidal map has an edge-transitive cover, as proved in [A. Orbanić, D. Pellicer, T. Pisanski and T. W. Tucker, Edge-transitive maps of low genus, Ars Math. Contemp. 4 2011, 2, 385–402]. In this article, we show the existence and give a classification of <jats:italic>k</jats:italic>-edge covers of semi-equivelar toroidal maps.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"27 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139078746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We improve the work [R. L. Frank and J. Sabin, Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces, Adv. Math. 317 2017, 157–192] concerning the spectral cluster bounds for orthonormal systems at <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0254_eq_0261.png" /> <jats:tex-math>{p=infty}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, on the flat torus and spaces of nonpositive sectional curvature, by shrinking the spectral band from <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:msup> <m:mi>λ</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>,</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>λ</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0254_eq_0140.png" /> <jats:tex-math>{[lambda^{2},(lambda+1)^{2})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> to <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:msup> <m:mi>λ</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>,</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>λ</m:mi> <m:mo>+</m:mo> <m:mrow> <m:mi>ϵ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>λ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0254_eq_0142.png" /> <jats:tex-math>{[lambda^{2},(lambda+epsilon(lambda))^{2})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ϵ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>λ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0254_eq_0166.png" /> <jats:tex-math>{epsilon(lambda)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a function of λ that goes to 0 as λ goes to <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">∞</m:mi> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0254_eq_0184.png" /> <jats:tex-math>{i
我们改进了 [R. L. Frank 和 J. Sabin, Schatten 空间中正态系统和振荡积分算子的谱群边界, Adv.L. Frank and J. Sabin, Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces, Adv. Math.317 2017, 157-192] concerning the spectral cluster bounds for orthonormal systems at p = ∞ {p=infty} , on the flat torus and spaces of non-finfty}. 通过将谱带从[ λ 2 , ( λ + 1 ) 2 ) 缩小，在平环面和非正断面曲率空间上，正交系统的谱簇边界 {[lambda^{2},(lambda+1)^{2})}缩小到[λ 2 , ( λ + ϵ ( λ ) 2 ) {[lambda^{2},(lambda+epsilon(lambda))^{2})} 其中 ϵ ( λ ) {epsilon(lambda)} 是 λ 的函数，当 λ 变为 ∞ {infty} 时，该函数变为 0。为了实现这一目标，我们引用了 [J. Bourgain, P. Shao] 中开发的方法。Bourgain, P. Shao, C. D. Sogge and X. Yao, On L p L^{p} -resolvent estimates and the density density} 中开发的方法。 -紧凑黎曼流形的残余估计和特征值密度，Comm.Math.333 2015, 3, 1483-1527].
{"title":"Improved spectral cluster bounds for orthonormal systems","authors":"Tianyi Ren, An Zhang","doi":"10.1515/forum-2023-0254","DOIUrl":"https://doi.org/10.1515/forum-2023-0254","url":null,"abstract":"We improve the work [R. L. Frank and J. Sabin, Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces, Adv. Math. 317 2017, 157–192] concerning the spectral cluster bounds for orthonormal systems at <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mi mathvariant=\"normal\">∞</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0254_eq_0261.png\" /> <jats:tex-math>{p=infty}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, on the flat torus and spaces of nonpositive sectional curvature, by shrinking the spectral band from <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:msup> <m:mi>λ</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>,</m:mo> <m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>λ</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0254_eq_0140.png\" /> <jats:tex-math>{[lambda^{2},(lambda+1)^{2})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> to <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:msup> <m:mi>λ</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>,</m:mo> <m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>λ</m:mi> <m:mo>+</m:mo> <m:mrow> <m:mi>ϵ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>λ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0254_eq_0142.png\" /> <jats:tex-math>{[lambda^{2},(lambda+epsilon(lambda))^{2})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ϵ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>λ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0254_eq_0166.png\" /> <jats:tex-math>{epsilon(lambda)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a function of λ that goes to 0 as λ goes to <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">∞</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0254_eq_0184.png\" /> <jats:tex-math>{i","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"258 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138692558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let M ( d , χ ) {M(d,chi)} , with ( d , χ ) = 1 {(d,chi)=1} , be the moduli space of semistable sheaves on ℙ 2 {mathbb{P}^{2}} supported on curves of degree d and with Euler characteristic χ. The cohomology ring H * ( M ( d , χ ) , ℤ ) {H^{*}(M(d,chi),mathbb{Z})} of M ( d , χ ) {M(d,chi)} is isomorphic to its Chow ring A * ( M ( d , χ ) ) {A^{*}(M(d,chi))} by Markman’s result. Pi and Shen have described a minimal generating set of A * ( M ( d , χ ) ) {A^{*}(M(d,chi))} consisting of 3 d - 7 {3d-7} generators, which they also showed to have no relation in A ≤ d - 2 ( M ( d , χ ) ) {A^{leq d-2}(M(d,chi))} . We compute the two Betti numbers b 2 ( d - 1 ) {b_{2(d-1)}} and b 2 d {b_{2d}} of M ( d , χ ) {M(d,chi)} , and as a corollary we show that the generators given by Pi and Shen have no relations in A ≤ d - 1 ( M ( d , χ ) ) {A^{leq d-1}(M(d,chi))} , but do have three linearly independent relations in A d ( M ( d , χ ) ) {A^{d}(M(d,chi))} .
{"title":"Some Betti numbers of the moduli of 1-dimensional sheaves on ℙ2","authors":"Yao Yuan","doi":"10.1515/forum-2023-0111","DOIUrl":"https://doi.org/10.1515/forum-2023-0111","url":null,"abstract":"Abstract Let M ( d , χ ) {M(d,chi)} , with ( d , χ ) = 1 {(d,chi)=1} , be the moduli space of semistable sheaves on ℙ 2 {mathbb{P}^{2}} supported on curves of degree d and with Euler characteristic χ. The cohomology ring H * ( M ( d , χ ) , ℤ ) {H^{*}(M(d,chi),mathbb{Z})} of M ( d , χ ) {M(d,chi)} is isomorphic to its Chow ring A * ( M ( d , χ ) ) {A^{*}(M(d,chi))} by Markman’s result. Pi and Shen have described a minimal generating set of A * ( M ( d , χ ) ) {A^{*}(M(d,chi))} consisting of 3 d - 7 {3d-7} generators, which they also showed to have no relation in A ≤ d - 2 ( M ( d , χ ) ) {A^{leq d-2}(M(d,chi))} . We compute the two Betti numbers b 2 ( d - 1 ) {b_{2(d-1)}} and b 2 d {b_{2d}} of M ( d , χ ) {M(d,chi)} , and as a corollary we show that the generators given by Pi and Shen have no relations in A ≤ d - 1 ( M ( d , χ ) ) {A^{leq d-1}(M(d,chi))} , but do have three linearly independent relations in A d ( M ( d , χ ) ) {A^{d}(M(d,chi))} .","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"40 5","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138519659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A positive rig is a commutative and unitary semi-ring A such that 1+x{1+x} is invertible for every x∈A{xin A}. We show that the category of positive rigs shares many properties with that of K-algebras for a (non-algebraically closed) field K. In particular, it is coextensive and, although we do not have an analogue of Hilbert’s basis theorem for positive rigs, we show that every finitely presentable positive rig is a finite direct product of directly indecomposable ones. We also describe free positive rigs as rigs of rational functions with non-negative rational coefficients, and we give a characterization of the positive rigs with a unique maximal ideal.
{"title":"Positive rigs","authors":"Matías Menni","doi":"10.1515/forum-2022-0271","DOIUrl":"https://doi.org/10.1515/forum-2022-0271","url":null,"abstract":"A <jats:italic>positive rig</jats:italic> is a commutative and unitary semi-ring <jats:italic>A</jats:italic> such that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>x</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0271_eq_0090.png\" /> <jats:tex-math>{1+x}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is invertible for every <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi>A</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0271_eq_0449.png\" /> <jats:tex-math>{xin A}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We show that the category of positive rigs shares many properties with that of <jats:italic>K</jats:italic>-algebras for a (non-algebraically closed) field <jats:italic>K</jats:italic>. In particular, it is coextensive and, although we do not have an analogue of Hilbert’s basis theorem for positive rigs, we show that every finitely presentable positive rig is a finite direct product of directly indecomposable ones. We also describe free positive rigs as rigs of rational functions with non-negative rational coefficients, and we give a characterization of the positive rigs with a unique maximal ideal.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"102 ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138519645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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