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WEAK INJECTIVE AND WEAK FLAT COMPLEXES

Part of: Homological algebra

Published online by Cambridge University Press:  21 July 2015

ZENGHUI GAO  and
ZHAOYONG HUANG
ZENGHUI GAO
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, P.R. China College of Mathematics, Chengdu University of Information Technology, Chengdu 610225, Sichuan Province, P.R. China e-mail: gaozenghui@cuit.edu.cn
ZHAOYONG HUANG
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, P.R. China e-mail: huangzy@nju.edu.cn
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Abstract

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Let R be an arbitrary ring. We introduce and study a generalization of injective and flat complexes of modules, called weak injective and weak flat complexes of modules respectively. We show that a complex C is weak injective (resp. weak flat) if and only if C is exact and all cycles of C are weak injective (resp. weak flat) as R-modules. In addition, we discuss the weak injective and weak flat dimensions of complexes of modules. Finally, we show that the category of weak injective (resp. weak flat) complexes is closed under pure subcomplexes, pure epimorphic images and direct limits. As a result, we then determine the existence of weak injective (resp. weak flat) covers and preenvelopes of complexes.

MSC classification

Primary: 18G35: Chain complexes
Secondary: 18G15: Ext and Tor, generalizations, Künneth formula 18G20: Homological dimension
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 3 , September 2016 , pp. 539 - 557
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

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