  For any integer $n\ge 0$ and any ring $R$, \ $(\mathcal {PGF}_n, \ \mathcal
P_n^\perp \cap \mathcal {PGF}^{\perp})$ proves to be a complete hereditary
cotorsion pair in $R$-Mod, where $\mathcal {PGF}$ is the class of PGF modules,
introduced by J. \v{S}aroch and J. \v{S}\'{t}ov\'{i}\v{c}ek, and \ $\mathcal
{PGF}_n$ is the class of $R$-modules of PGF dimension $\le n$. For any Artin
algebra $R$, \ $(\mathcal {GP}_n, \ \mathcal P_n^\perp \cap \mathcal
{GP}^{\perp})$ proves to be a complete and hereditary cotorsion pair in
$R$-Mod, where $\mathcal {GP}_n$ is the class of modules of Gorenstein
projective dimension $\le n$. These cotorsion pairs induce two chains of
hereditary Hovey triples \ $(\mathcal {PGF}_n, \ \mathcal P_n^\perp, \ \mathcal
{PGF}^{\perp})$ and \ $(\mathcal {GP}_n, \ \mathcal P_n^\perp, \ \mathcal
{GP}^{\perp})$, and the corresponding homotopy categories in the same chain are
the same. It is observed that some complete cotorsion pairs in $R$-Mod can
induce complete cotorsion pairs in some special extension closed subcategories
of $R$-Mod. Then corresponding results in exact categories $\mathcal {PGF}_n$,
\ $\mathcal {GP}_n$, \ $\mathcal {GF}_n$, \ $\mathcal {PGF}^{<\infty}$, \
$\mathcal {GP}^{<\infty}$ and $\mathcal {GF}^{<\infty}$, are also obtained. As
a byproduct, $\mathcal{PGF} = \mathcal {GP}$ for a ring $R$ if and only if
$\mathcal{PGF}^\perp\cap\mathcal{GP}_n=\mathcal P_n$ for some $n$.
