Let $R$ be a commutative Noetherian ring and let $I$ be a semidualizing ideal of $R$. In this paper, it is shown that the $G_{I}$-projective, $G_{I}$-injective, and $G_{I}$-flat dimensions agree with $Gpd _{R\bowtie I}(-)$, $Gid _{R\bowtie I}(-)$, and $Gfd _{R\bowtie I}(-)$, respectively. Also, it is proved that for a non-negative integer $n$ if $\sup \{\mathcal{GP}_{I}-pd _{R}(M) \mid M\in \mathcal{M}(R) \}\leq n$ (or $\sup \{\mathcal{GI}_{I}-id _{R}(M) \mid M\in \mathcal{M}(R) \}\leq n)$, then for every projective $(R\bowtie I)$-module $P$ we have $id _{R\bowtie I}(P)\leq n$, and for every injective $(R\bowtie I)$-module $E$ we have $pd_{R\bowtie I}(E)\leq n$.