Special precovering classes in comma categories

Abstract

Let $T$ be a right exact functor from an abelian category $\mathcal{B}$ into another abelian category $\mathcal{A}$. Then there exists a functor $\mathbf{p}$ from the product category $\mathcal{A}\times \mathcal{B}$ to the comma category $(T\downarrow \mathcal{A})$. In this paper, we study the property of the extension closure of some classes of objects in $(T\downarrow \mathcal{A})$, the exactness of the functor $\mathbf{p}$ and the detail description of orthogonal classes of a given class $\mathbf{p}(\mathcal{X},\mathcal{Y})$ in $(T\downarrow \mathcal{A})$. Moreover, we characterize when special precovering classes in abelian categories $\mathcal{A}$ and $\mathcal{B}$ can induce special precovering classes in $(T\downarrow \mathcal{A})$. As an application, we prove that under suitable cases, the class of Gorenstein projective left $\Lambda$-modules over a triangular matrix ring $\Lambda =\left(\begin{array}{cc}R& M\\ O& S\end{array}\right)$ is special precovering if and only if both the classes of Gorenstein projective left $R$-modules and left $S$-modules are special precovering. Consequently, we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them.

To appear in SCIENCE CHINA Mathematics