Frobenius functors and Gorenstein flat dimensions

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.

Keywords:

• Frobenius bimodule
• Frobenius functor
• Gorenstein flat dimension
• Gorenstein flat module

MATHEMATICS SUBJECT CLASSIFICATION:

• 13B02
• 16G50
• 18G25

Additional information

Funding

This research was partially supported by the NSF of China (Grants No. 11501257, 11671069, 11771202, 11771212), Qing Lan Project of Jiangsu Province and Jiangsu Government Scholarship for Overseas Studies (JS-2019-328). The authors are grateful to the referee for the valuable comments.