{"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}
{"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}
{"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
{"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}
{"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}
{"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}
{"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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