Cotorsion pairs and Enochs Conjecture for object ideals

Abstract

Let $\mathcal{I}$ and $\mathcal{J}$ be object ideals in an exact category $(\mathcal{A};\mathcal{E})$. It is proved that $(\mathcal{I},\mathcal{J})$ is a perfect ideal cotorsion pair if and only if $(\mathrm{Ob}(\mathcal{I}),\mathrm{Ob}(\mathcal{J}))$ is a perfect cotorsion pair, where $\mathrm{Ob}(\mathcal{I})$ and $\mathrm{Ob}(\mathcal{J})$ is the objects of $\mathcal{I}$ and $\mathcal{J}$, respectively. If in addition $(\mathcal{A};\mathcal{E})$ has enough projective objects and injective objects, and $\mathcal{J}$ is enveloping, then $(\mathcal{I},\mathcal{J})$ is a complete ideal cotorsion pair if and only if $(\mathrm{Ob}(\mathcal{I}),\mathrm{Ob}(\mathcal{J}))$ is a complete cotorsion pair. This gives a partial answer to the question posed by Fu, Guil Asensio, Herzog and Torrecillas. Moreover, for any object ideal $\mathcal{I}$ in the category of left $R$-modules, it is proved that $\mathcal{I}$ satisfies Enochs Conjecture if and only if $\mathrm{Ob}(\mathcal{I})$ satisfies Enochs Conjecture. Applications are given to projective morphisms and ideal cotorsion pairs $(\mathcal{I},\mathcal{J})$ of object ideals under certain conditions.