Abstract
We prove that if the Frobenius functor F (from the category of left R-modules to the category of left S-modules) is faithful, then for any R-module X, the Gorenstein flat dimension of X is equal to the Gorenstein flat dimension of F(X), which is motivated by a result of Nakayama and Tsuzuku about relations between Frobenius extensions and flat dimension of modules. It is a well-known metatheorem of Holm that every result in homological algebra has a counterpart in Gorenstein homological algebra. Note that the flat dimension of X can be different from that of F(X). Our main result provides a counterexample to the converse of Holm’s metatheorem. In addition, some applications are given.
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