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SOME COVERS AND ENVELOPES IN THE CHAIN COMPLEX CATEGORY OF R-MODULES

Part of: Model theory Homological algebra

Published online by Cambridge University Press:  31 August 2011

ZHANPING WANG  and
ZHONGKUI LIU
ZHANPING WANG*
Affiliation:
Department of Mathematics, Northwest Normal University, Lanzhou 730070, PR China (email: wangzp@nwnu.edu.cn)
ZHONGKUI LIU
Affiliation:
Department of Mathematics, Northwest Normal University, Lanzhou 730070, PR China (email: liuzk@nwnu.edu.cn)
*
✉For correspondence; e-mail: wangzp@nwnu.edu.cn
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Abstract

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We study the existence of some covers and envelopes in the chain complex category of R-modules. Let (𝒜,ℬ) be a cotorsion pair in R-Mod and let ℰ𝒜 stand for the class of all exact complexes with each term in 𝒜. We prove that (ℰ𝒜,ℰ𝒜⊥) is a perfect cotorsion pair whenever 𝒜 is closed under pure submodules, cokernels of pure monomorphisms and direct limits and so every complex has an ℰ𝒜-cover. As an application we show that every complex of R-modules over a right coherent ring R has an exact Gorenstein flat cover. In addition, the existence of -covers and -envelopes of special complexes is considered where and denote the classes of all complexes with each term in 𝒜 and ℬ, respectively.

Keywords

cotorsion paircoverenvelopeGorenstein flat complex

MSC classification

Secondary: 18G35: Chain complexes 03C35: Categoricity and completeness of theories
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 90 , Issue 3 , June 2011 , pp. 385 - 401
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

Supported by the National Natural Science Foundation of China (10961021).

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