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Let $C$ be a coalgebra. We prove that a right $C$-comodule is an $n$-cosyzygy if and only if it is a Gorenstein $n$-cosyzygy. We study the Gorenstein global dimension of coalgebras and the class of Gorenstein hereditary coalgebras. As a generalization of Gorenstein hereditary coalgebras, we introduce the concept of Gorenstein semihereditary coalgebras and show that a coalgebra $C$ is Gorenstein semihereditary if and only if every finitely cogenerated factor of any injective comodule is Gorenstein injective.
}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v55n2.22.08}, url = {http://global-sci.org/intro/article_detail/jms/20497.html} }Let $C$ be a coalgebra. We prove that a right $C$-comodule is an $n$-cosyzygy if and only if it is a Gorenstein $n$-cosyzygy. We study the Gorenstein global dimension of coalgebras and the class of Gorenstein hereditary coalgebras. As a generalization of Gorenstein hereditary coalgebras, we introduce the concept of Gorenstein semihereditary coalgebras and show that a coalgebra $C$ is Gorenstein semihereditary if and only if every finitely cogenerated factor of any injective comodule is Gorenstein injective.