We explore sequences of iterated blowups of two-dimensional regular local rings. Classical results of Zariski and Abhyankar show that the directed union of blowups of this type is a valuation ring. We show that the value group of such a valuation ring is determined by the irrational number γ that is the value of an infinite continued fraction associated to the sequence of blowups. Mathematics Subject Classification (2010). Primary: 13A18 ; Secondary: 13H05
{"title":"Iterated blowups of two-dimensional regular local rings","authors":"R. Wiegand, S. Wiegand","doi":"10.4171/rsmup/70","DOIUrl":"https://doi.org/10.4171/rsmup/70","url":null,"abstract":"We explore sequences of iterated blowups of two-dimensional regular local rings. Classical results of Zariski and Abhyankar show that the directed union of blowups of this type is a valuation ring. We show that the value group of such a valuation ring is determined by the irrational number γ that is the value of an infinite continued fraction associated to the sequence of blowups. Mathematics Subject Classification (2010). Primary: 13A18 ; Secondary: 13H05","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"72 1","pages":"281-288"},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74073333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In [6], generalizations of the standard notion of purity on p-local abelian groups were defined using functorial methods to create injective resolutions. For example, if λ is a limit ordinal, then for a group G the completion functor LλG determines the notion of Lλ-purity. Another way of constructing a type of purity, called p w -purity, is defined using the functor Q α<λ(G/p G). Properties of this second type of purity are studied; for example, it is shown to be hereditary if and only if λ has countable cofinality. In addition, Lλ and p <λ w -purity are compared in a variety of contexts, for example, in the category of Warfield groups. Mathematics Subject Classification (2010). 20K10, 20K40
在[6]中，使用泛函方法定义了p局部阿贝尔群上的标准纯度概念的推广，以创建内射分辨。例如，如果λ是一个极限序数，那么对于群G，补全函子LλG决定了l λ纯度的概念。用函子Q α<λ(G/p G)定义另一种构造一类纯度的方法，称为p w -纯度。研究了这二类纯度的性质;例如，当且仅当λ具有可数共性时，证明它是遗传的。此外，Lλ和p <λ w -纯度在各种情况下进行了比较，例如，在Warfield群的范畴中。数学学科分类(2010)。20 k10 20 k40
{"title":"A version of purity on local abelian groups","authors":"P. Keef","doi":"10.4171/rsmup/63","DOIUrl":"https://doi.org/10.4171/rsmup/63","url":null,"abstract":"In [6], generalizations of the standard notion of purity on p-local abelian groups were defined using functorial methods to create injective resolutions. For example, if λ is a limit ordinal, then for a group G the completion functor LλG determines the notion of Lλ-purity. Another way of constructing a type of purity, called p w -purity, is defined using the functor Q α<λ(G/p G). Properties of this second type of purity are studied; for example, it is shown to be hereditary if and only if λ has countable cofinality. In addition, Lλ and p <λ w -purity are compared in a variety of contexts, for example, in the category of Warfield groups. Mathematics Subject Classification (2010). 20K10, 20K40","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"20 1","pages":"159-176"},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83698500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let R be any ring and let 0 → A → B → C → 0 be an exact sequence of R-modules which does not split with A and C uniserial. Then either B is indecomposable or B has a decomposition of the form B = B1 ⊕ B2 where B1 and B2 are indecomposable and at least one of them is uniserial. Mathematics Subject Classification (2010).Primary: 16D10; Secondary: 16G20.
{"title":"Extensions of uniserial modules","authors":"G. D'este, Fatma Kaynarca, D. K. Tütüncü","doi":"10.4171/rsmup/57","DOIUrl":"https://doi.org/10.4171/rsmup/57","url":null,"abstract":"Let R be any ring and let 0 → A → B → C → 0 be an exact sequence of R-modules which does not split with A and C uniserial. Then either B is indecomposable or B has a decomposition of the form B = B1 ⊕ B2 where B1 and B2 are indecomposable and at least one of them is uniserial. Mathematics Subject Classification (2010).Primary: 16D10; Secondary: 16G20.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"8 1","pages":"73-86"},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75856174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Instead of the axiom of choice, we assume that every set of reals has the Baire property. It is shown that under this condition the concept of slenderness known from the theory of abelian groups becomes meaningful for vector spaces. Mathematics Subject Classification (2010). 03E60, 03E75, 20K25, 54E52
{"title":"Linear forms in a playful universe","authors":"Burkhard Wald","doi":"10.4171/rsmup/69","DOIUrl":"https://doi.org/10.4171/rsmup/69","url":null,"abstract":"Instead of the axiom of choice, we assume that every set of reals has the Baire property. It is shown that under this condition the concept of slenderness known from the theory of abelian groups becomes meaningful for vector spaces. Mathematics Subject Classification (2010). 03E60, 03E75, 20K25, 54E52","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"33 1","pages":"271-279"},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87498061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We describe torsion free abelian groups of finite rank applying matrices with polyadic entries. This description can be considered as a modification of the classic description by A.I. Mal’cev.
利用具有多进元的矩阵描述有限秩无扭转阿贝尔群。这种描述可以看作是A.I. Mal 'cev对经典描述的修改。
{"title":"A matrix description for torsion free abelian groups of finite rank","authors":"A. Fomin","doi":"10.4171/rsmup/60","DOIUrl":"https://doi.org/10.4171/rsmup/60","url":null,"abstract":"We describe torsion free abelian groups of finite rank applying matrices with polyadic entries. This description can be considered as a modification of the classic description by A.I. Mal’cev.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"12 1","pages":"115-128"},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75249322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For actions of m commuting endomorphisms of a torsion abelian group we compute the algebraic entropy and the algebraic receptive entropy, showing that the latter one takes finite positive values in many cases when the former one vanishes. Mathematics Subject Classification (2010). 20M10, 20K30, 22A26, 22D05.
{"title":"Algebraic entropies of commuting endomorphisms of torsion abelian groups","authors":"A. Biś, D. Dikranjan, A. Bruno, L. Stoyanov","doi":"10.4171/rsmup/55","DOIUrl":"https://doi.org/10.4171/rsmup/55","url":null,"abstract":"For actions of m commuting endomorphisms of a torsion abelian group we compute the algebraic entropy and the algebraic receptive entropy, showing that the latter one takes finite positive values in many cases when the former one vanishes. Mathematics Subject Classification (2010). 20M10, 20K30, 22A26, 22D05.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"140 1","pages":"45-60"},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81711070","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
P. Danchev, B. Goldsmith, K. Rangaswamy, L. Salce, L. Strüngmann
{"title":"Special issue dedicated to László Fuchs on the Occasion of his 95th Birthday","authors":"P. Danchev, B. Goldsmith, K. Rangaswamy, L. Salce, L. Strüngmann","doi":"10.4171/rsmup/51","DOIUrl":"https://doi.org/10.4171/rsmup/51","url":null,"abstract":"","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"13 1","pages":"1-8"},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87355487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Here “group” means abelian group. Compact connected groups contain ı-subgroups, that is, compact totally disconnected subgroups with torus quotients, which are essential ingredients in the important Resolution Theorem, a description of compact groups. Dually, full free subgroups of discrete torsion-free groups of finite rank are studied in order to obtain a comprehensive picture of the abundance of ı-subgroups of a protorus. Associated concepts are also considered. Mathematics Subject Classification (2010). Primary: 22C05; Secondary: 20K15, 22B05.
{"title":"Free subgroups with torsion quotients and profinite subgroups with torus quotients","authors":"Wayne Lewis, P. Loth, A. Mader","doi":"10.4171/rsmup/64","DOIUrl":"https://doi.org/10.4171/rsmup/64","url":null,"abstract":"Here “group” means abelian group. Compact connected groups contain ı-subgroups, that is, compact totally disconnected subgroups with torus quotients, which are essential ingredients in the important Resolution Theorem, a description of compact groups. Dually, full free subgroups of discrete torsion-free groups of finite rank are studied in order to obtain a comprehensive picture of the abundance of ı-subgroups of a protorus. Associated concepts are also considered. Mathematics Subject Classification (2010). Primary: 22C05; Secondary: 20K15, 22B05.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"54 35 1","pages":"177-195"},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78649996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let R be a local one-dimensional domain, with maximal ideal M, which is not a valuation domain. We investigate the class of the finitely generated mixed R-modules of Warfield type, so called since their construction goes back to R. B. Warfield. We prove that these R-modules have local endomorphism rings, hence they are indecomposable. We examine the torsion part t(M) of a Warfield type module M , investigating the natural property t(M) ⊂ MM . This property is related to b/a being integral over R, where a and b are elements of R that define M . We also investigate M/t(M) and determine its minimum number of generators. Mathematics Subject Classification (2010). 13G05, 13A15, 13A17.
{"title":"Finitely generated mixed modules of Warfield type","authors":"P. Zanardo","doi":"10.4171/rsmup/71","DOIUrl":"https://doi.org/10.4171/rsmup/71","url":null,"abstract":"Let R be a local one-dimensional domain, with maximal ideal M, which is not a valuation domain. We investigate the class of the finitely generated mixed R-modules of Warfield type, so called since their construction goes back to R. B. Warfield. We prove that these R-modules have local endomorphism rings, hence they are indecomposable. We examine the torsion part t(M) of a Warfield type module M , investigating the natural property t(M) ⊂ MM . This property is related to b/a being integral over R, where a and b are elements of R that define M . We also investigate M/t(M) and determine its minimum number of generators. Mathematics Subject Classification (2010). 13G05, 13A15, 13A17.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"23 1","pages":"289-302"},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78700544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
U. Albrecht, Francisco Javier Santillán-Covarrubias
This paper discusses maximal and almost maximal rings in a noncommutative setting. Annihilators ideals in chain rings and their relationship to the concept of self-injectivity are investigated. In particular, a two-sided chain ring is right self-injective if and only if it is right co-Hopfian and a left maximal ring. Finally, localizations of chain rings are discussed. Mathematics Subject Classification (2010). Primary 16L30; Secondary 16D50; 16P70
{"title":"A note on almost maximal chain rings","authors":"U. Albrecht, Francisco Javier Santillán-Covarrubias","doi":"10.4171/rsmup/52","DOIUrl":"https://doi.org/10.4171/rsmup/52","url":null,"abstract":"This paper discusses maximal and almost maximal rings in a noncommutative setting. Annihilators ideals in chain rings and their relationship to the concept of self-injectivity are investigated. In particular, a two-sided chain ring is right self-injective if and only if it is right co-Hopfian and a left maximal ring. Finally, localizations of chain rings are discussed. Mathematics Subject Classification (2010). Primary 16L30; Secondary 16D50; 16P70","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"9 1","pages":"1-11"},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88565115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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