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Abstract
In this paper, we introduce and study a notion of Gorenstein AC-injective dimension for complexes of left modules over associative rings. We show first that the class of complexes with finite Gorenstein AC-injective dimension is exactly the class of complexes admitting a complete $$\mathcal {AC}$$ -coresolution. Then the interaction between the corresponding relative and Tate cohomologies of complexes is given. Finally, the relationships between Gorenstein AC-injective dimensions and injective dimensions for complexes are given.
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