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Cotorsion pairs, torsion pairs, and Sigma-pure-injective cotilting modules

机译:扭力对,扭力对和Sigma纯注入式倾斜模块

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摘要

In this paper we study cotorsion and torsion pairs induced by cotilting modules. We prove the existence of a strong relationship between the Sigma-pure-injectivity of the cotilting module and the property of the induced cotorsion pair to be of finite type. In particular for cotilting modules of injective dimension at most 1, or for noetherian rings, the two notions are equivalent. On the other hand we prove that a torsion pair is cogenerated by a Sigma-pure-injective cotilting module if and only if its heart is a locally noetherian Grothendieck category. Moreover we prove that any ring admitting a Sigma-pure-injective cotilting module of injective dimension at most I is necessarily coherent. Finally, for noetherian rings, we characterize cotilting torsion pairs induced by Sigma-pure-injective cotilting modules.
机译:在本文中,我们研究了由倾斜模块引起的扭转和扭转对。我们证明了在模块的西格玛纯注入性和诱导的扭曲对的性质是有限类型之间存在很强的关系。特别是对于最多为1的内射维的模块或对于Noetherian环,这两个概念是等效的。另一方面,我们证明了,当且仅当其心脏是局部noetherian Grothendieck类别时,才会通过Sigma-pure-jective倾斜模块来共生一个扭转对。此外,我们证明,任何接纳最多为I的Sigma-pure-injective耦合模块的环必定是相干的。最后,对于noetherian环,我们表征了Sigma纯注入式Cotilting模块引起的Cotiling扭转对。

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