Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X39620131215Gorenstein projective objects in Abelian categories10791097462ENH.ChengDepartment of Mathematics, Nanjing University,
Nanjing 210093, ChinaX.ZhuDepartment of Mathematics, Nanjing University,
Nanjing 210093, ChinaJournal Article20120701Let $mathcal {A}$ be an abelian category with enough projective <br />objects and $mathcal {X}$ be a full subcategory of <br />$mathcal {A}$. We define Gorenstein projective objects with respect <br />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 <br />Gorensein projective objects are related in a nice way. In <br />particular, if $mathcal {P}(mathcal {A})subseteqmathcal {X}$, we <br />show that $Xin Ch(mathcal {A})$ is Gorenstein projective with respect to $mathcal{Y}_{mathcal{X}}$ if and only if $X^{i}$ is Gorenstein <br />projective with respect to $mathcal {X}$ for each $i$, when $mathcal {X}$ is a self-orthogonal <br />class or $X$ is $Hom(-,mathcal {X})$-exact. Subsequently, we <br />consider the relationships of Gorenstein projective dimensions between them. As an <br />application, if $mathcal {A}$ is of finite left Gorenstein projective <br />global dimension with respect to $mathcal{X}$ and contains an injective <br />cogenerator, then we find a new <br />model structure on $Ch(mathcal {A})$ by Hovey's results in cite{Ho} .http://bims.iranjournals.ir/article_462_bdb19acaedd836465e241a86c9c3a04e.pdf