{"title":"过滤游戏和潜在的投影模块","authors":"Sean D. Cox","doi":"10.4064/fm237-10-2022","DOIUrl":null,"url":null,"abstract":"The notion of a \\textbf{$\\boldsymbol{\\mathcal{C}}$-filtered} object, where $\\mathcal{C}$ is some (typically small) collection of objects in a Grothendieck category, has become ubiquitous since the solution of the Flat Cover Conjecture around the year 2000. We introduce the \\textbf{$\\boldsymbol{\\mathcal{C}}$-Filtration Game of length $\\boldsymbol{\\omega_1}$} on a module, paying particular attention to the case where $\\mathcal{C}$ is the collection of all countably presented, projective modules. We prove that Martin's Maximum implies the determinacy of many $\\mathcal{C}$-Filtration Games of length $\\omega_1$, which in turn imply the determinacy of certain Ehrenfeucht-Fraisse games of length $\\omega_1$; this allows a significant strengthening of a theorem of Mekler-Shelah-Vaananen \\cite{MR1191613}. Also, Martin's Maximum implies that if $R$ is a countable hereditary ring, the class of \\textbf{$\\boldsymbol{\\sigma}$-closed potentially projective modules}---i.e., those modules that are projective in some $\\sigma$-closed forcing extension of the universe---is closed under $<\\aleph_2$-directed limits. We also give an example of a (ZFC-definable) class of abelian groups that, under the ordinary subgroup relation, constitutes an Abstract Elementary Class (AEC) with Lowenheim-Skolem number $\\aleph_1$ in some models in set theory, but fails to be an AEC in other models of set theory.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Filtration games and potentially projective modules\",\"authors\":\"Sean D. Cox\",\"doi\":\"10.4064/fm237-10-2022\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The notion of a \\\\textbf{$\\\\boldsymbol{\\\\mathcal{C}}$-filtered} object, where $\\\\mathcal{C}$ is some (typically small) collection of objects in a Grothendieck category, has become ubiquitous since the solution of the Flat Cover Conjecture around the year 2000. We introduce the \\\\textbf{$\\\\boldsymbol{\\\\mathcal{C}}$-Filtration Game of length $\\\\boldsymbol{\\\\omega_1}$} on a module, paying particular attention to the case where $\\\\mathcal{C}$ is the collection of all countably presented, projective modules. We prove that Martin's Maximum implies the determinacy of many $\\\\mathcal{C}$-Filtration Games of length $\\\\omega_1$, which in turn imply the determinacy of certain Ehrenfeucht-Fraisse games of length $\\\\omega_1$; this allows a significant strengthening of a theorem of Mekler-Shelah-Vaananen \\\\cite{MR1191613}. Also, Martin's Maximum implies that if $R$ is a countable hereditary ring, the class of \\\\textbf{$\\\\boldsymbol{\\\\sigma}$-closed potentially projective modules}---i.e., those modules that are projective in some $\\\\sigma$-closed forcing extension of the universe---is closed under $<\\\\aleph_2$-directed limits. We also give an example of a (ZFC-definable) class of abelian groups that, under the ordinary subgroup relation, constitutes an Abstract Elementary Class (AEC) with Lowenheim-Skolem number $\\\\aleph_1$ in some models in set theory, but fails to be an AEC in other models of set theory.\",\"PeriodicalId\":55138,\"journal\":{\"name\":\"Fundamenta Mathematicae\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2020-10-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Fundamenta Mathematicae\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4064/fm237-10-2022\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fundamenta Mathematicae","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4064/fm237-10-2022","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Filtration games and potentially projective modules
The notion of a \textbf{$\boldsymbol{\mathcal{C}}$-filtered} object, where $\mathcal{C}$ is some (typically small) collection of objects in a Grothendieck category, has become ubiquitous since the solution of the Flat Cover Conjecture around the year 2000. We introduce the \textbf{$\boldsymbol{\mathcal{C}}$-Filtration Game of length $\boldsymbol{\omega_1}$} on a module, paying particular attention to the case where $\mathcal{C}$ is the collection of all countably presented, projective modules. We prove that Martin's Maximum implies the determinacy of many $\mathcal{C}$-Filtration Games of length $\omega_1$, which in turn imply the determinacy of certain Ehrenfeucht-Fraisse games of length $\omega_1$; this allows a significant strengthening of a theorem of Mekler-Shelah-Vaananen \cite{MR1191613}. Also, Martin's Maximum implies that if $R$ is a countable hereditary ring, the class of \textbf{$\boldsymbol{\sigma}$-closed potentially projective modules}---i.e., those modules that are projective in some $\sigma$-closed forcing extension of the universe---is closed under $<\aleph_2$-directed limits. We also give an example of a (ZFC-definable) class of abelian groups that, under the ordinary subgroup relation, constitutes an Abstract Elementary Class (AEC) with Lowenheim-Skolem number $\aleph_1$ in some models in set theory, but fails to be an AEC in other models of set theory.
期刊介绍:
FUNDAMENTA MATHEMATICAE concentrates on papers devoted to
Set Theory,
Mathematical Logic and Foundations of Mathematics,
Topology and its Interactions with Algebra,
Dynamical Systems.