Gorenstein Hereditary Coalgebras

Abstract

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.