Bounds on Gorenstein Dimensions and Exceptional Complete Intersection Maps

Abstract

We prove that if $f:R \rightarrow S$ is a local homomorphism of noetherian local rings of finite flat dimension and $M$ is a non-zero finitely generated $S$-module whose Gorenstein flat dimension over $R$ is bounded by the difference of the embedding dimensions of $R$ and $S$, then $M$ is a totally reflexive $S$-module and $f$ is an exceptional complete intersection map. This is an extension of a result of Brochard, Iyengar, and Khare to Gorenstein flat dimension. We also prove two analogues involving Gorenstein injective dimension.