Gorenstein projective objects in Abelian categories

Document Type : Research Paper


Department of Mathematics, Nanjing University, Nanjing 210093, China


Let $mathcal {A}$ be an abelian category with enough projective
objects and $mathcal {X}$ be a full subcategory of
$mathcal {A}$. We define Gorenstein projective objects with respect
to $mathcal {X}$ and $mathcal{Y}_{mathcal{X}}$, respectively, where $mathcal{Y}_{mathcal{X}}$=${ Yin Ch(mathcal {A})| Y$ is acyclic and $Z_{n}Yinmathcal{X}}$. We point out that under certain hypotheses, these two
Gorensein projective objects are related in a nice way. In
particular, if $mathcal {P}(mathcal {A})subseteqmathcal {X}$, we
show that $Xin Ch(mathcal {A})$ is Gorenstein projective with respect to $mathcal{Y}_{mathcal{X}}$ if and only if $X^{i}$ is Gorenstein
projective with respect to $mathcal {X}$ for each $i$, when $mathcal {X}$ is a self-orthogonal
class or $X$ is $Hom(-,mathcal {X})$-exact. Subsequently, we
consider the relationships of Gorenstein projective dimensions between them. As an
application, if $mathcal {A}$ is of finite left Gorenstein projective
global dimension with respect to $mathcal{X}$ and contains an injective
cogenerator, then we find a new
model structure on $Ch(mathcal {A})$ by Hovey's results in cite{Ho} .


Main Subjects