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Periodic Modules and Acyclic Complexes 2 · Sergio Estrada3 ´ Silvana Bazzoni1 · Manuel Cortes-Izurdiaga

Received: 18 January 2019 / Accepted: 23 July 2019 / © Springer Nature B.V. 2019

Abstract We study the behaviour of modules M that fit into a short exact sequence 0 → M → C → M → 0, where C belongs to a class of modules C , the so-called C -periodic modules. We find a rather general framework to improve and generalize some well-known results of Benson and Goodearl and Simson. In the second part we will combine techniques of hereditary cotorsion pairs and presentation of direct limits, to conclude, among other applications, that if M is any module and C is cotorsion, then M will be also cotorsion. This will lead to some meaningful consequences in the category Ch(R) of unbounded chain complexes and in Gorenstein homological algebra. For example we show that every acyclic complex of cotorsion modules has cotorsion cycles, and more generally, every map F → C where C is a complex of cotorsion modules and F is an acyclic complex of flat cycles, is null-homotopic. In other words, every complex of cotorsion modules is dg-cotorsion. Keywords Periodic C -module · Pure C -periodic module · Locally split short exact sequence · Hereditary cotorsion pair · Acyclic complex Mathematics Subject Classification (2010) 16D90 · 16E05 · 16D40 · 16D50 · 18G25 · 18G35

Presented by: Jan Stovicek The first named author is partially supported by grants BIRD163492 and DOR1690814 of Padova University The second named author is partially supported by grants MTM2014-54439, MTM2016-77445-P of Ministerio de Econom´ıa, Industria y Competitividad and FEDER funds The third named author is partially supported by grant 19880/GERM/15 from the Fundaci´on S´eneca-Agencia de Ciencia y Tecnolog´ıa de la Regi´on de Murcia and by grant MTM2016-77445-P of Ministerio de Ciencia, Innovaci´on y Universidades and FEDER funds  Manuel Cort´es-Izurdiaga

[email protected]

Extended author information available on the last page of the article.

S. Bazzoni et al.

1 Introduction Throughout this paper R is an associative ring with identity and all modules will be right R-modules. The goal of this work is the study of periodic and pure periodic modules with respect to an arbitrary class of modules C . More precisely, one of the main objectives we pursue is to know when C -periodic modules (resp. pure C -periodic modules) are trivial, where an Rmodule M is called C -periodic (resp. pure C -periodic) if it fits into an exact sequence (resp. into a pure exact sequence) of the form 0 → M → C → M → 0, with C ∈ C , and it is called trivial if it belongs to C . The origin of this problem comes from the celebrated result by Benson and Goodearl [4, Theorem 2.5] in which they show that each flat Proj-periodic module is trivial (here Proj denotes the class of all projective modules). It is then easy to observe that Benson and Goodearl statement can be reformulated to saying that each pure Proj-periodic module is trivial. This is because M is always flat in each pure short exact