{"title":"A Bounded Below, Noncontractible, Acyclic Complex Of Projective Modules","authors":"L. Positselski","doi":"10.1007/s10474-024-01414-1","DOIUrl":null,"url":null,"abstract":"<div><p>We construct examples of bounded below, noncontractible, acyclic\ncomplexes of finitely generated projective modules over some rings <i>S</i>,\nas well as bounded above, noncontractible, acyclic complexes of\ninjective modules.\n The rings <i>S</i> are certain rings of infinite matrices with entries in\nthe rings of commutative polynomials or formal power series in\ninfinitely many variables.\n In the world of comodules or contramodules over coalgebras over\nfields, similar examples exist over the cocommutative symmetric\ncoalgebra of an infinite-dimensional vector space.\n A simpler, universal example of a bounded below, noncontractible,\nacyclic complex of free modules with one generator, communicated to\nthe author by Canonaco, is included at the end of the paper.</p></div>","PeriodicalId":50894,"journal":{"name":"Acta Mathematica Hungarica","volume":"172 2","pages":"324 - 345"},"PeriodicalIF":0.6000,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10474-024-01414-1.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Hungarica","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10474-024-01414-1","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We construct examples of bounded below, noncontractible, acyclic
complexes of finitely generated projective modules over some rings S,
as well as bounded above, noncontractible, acyclic complexes of
injective modules.
The rings S are certain rings of infinite matrices with entries in
the rings of commutative polynomials or formal power series in
infinitely many variables.
In the world of comodules or contramodules over coalgebras over
fields, similar examples exist over the cocommutative symmetric
coalgebra of an infinite-dimensional vector space.
A simpler, universal example of a bounded below, noncontractible,
acyclic complex of free modules with one generator, communicated to
the author by Canonaco, is included at the end of the paper.
我们构造了在某些环 S 上有限生成的射影模块的有界下、非收缩、非循环复数的例子，以及射影模块的有界上、非收缩、非循环复数的例子。环 S 是某些无限矩阵环，其条目是无限多变量的交换多项式或形式幂级数环。在域上煤球上的组合模或反组合模的世界里，在无穷维向量空间的可交换对称煤球上也有类似的例子。本文末尾还有一个更简单、更普遍的例子，即由卡诺纳科（Canonaco）向作者提供的一个生成器的有界下、非可收缩、非循环自由模块复合物。
期刊介绍：
Acta Mathematica Hungarica is devoted to publishing research articles of top quality in all areas of pure and applied mathematics as well as in theoretical computer science. The journal is published yearly in three volumes (two issues per volume, in total 6 issues) in both print and electronic formats. Acta Mathematica Hungarica (formerly Acta Mathematica Academiae Scientiarum Hungaricae) was founded in 1950 by the Hungarian Academy of Sciences.
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