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

In this paper we study cotorsion and torsion pairs induced by cotilting modules. We prove the existence of a strong relationship between the Σ-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 Σ-pure injective cotilting module if and only if its heart is a locally noetherian Grothendieck category. Moreover we prove that any ring admitting a Σ-pure injective cotilting module of injective dimension at most 1 is necessarily coherent. Finally, for noetherian rings, we characterize cotilting torsion pairs induced by Σ-pure injective cotilting modules.