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Gorenstein projective, injective and flat complexes. (Chinese. English summary) Zbl 1240.16010

Summary: It is shown in the paper that for a general associative ring \(R\), any complex \(C\) of \(R\)-modules is Gorenstein projective if and only if each \(R\)-module \(C^m\) is Gorenstein projective for all \(m\in\mathbb Z\); as immediate consequences of the result, Gorenstein projective dimensions of complexes are characterized. Furthermore, if \(C\) is an exact complex of \(R\)-modules such that this sequence remains exact when the functor \(\operatorname{Hom}(-,Q)\) is applied to it for any projective \(R\)-module \(Q\), then \(C\) is Gorenstein projective if and only if each \(\text{Ker}(\delta^m_C)\) is Gorenstein projective for all \(m\in\mathbb Z\). Similarly, Gorenstein injective and Gorenstein flat versions of all these results are given.

MSC:

16E05 Syzygies, resolutions, complexes in associative algebras
16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
18G05 Projectives and injectives (category-theoretic aspects)
16E10 Homological dimension in associative algebras
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
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