逗号范畴中的特殊预覆盖类
Special precovering classes in comma categories
摘要：
设$T$是从一个阿贝尔范畴$\mathscr{B}$到另一个阿贝尔范畴$\mathscr{A}$的右正合函子.然后存在从产品类别$\mathscr{A}\times\mathscr{B}$到逗号类别$(T\downarrow\mathscr{A})$的函子${\bf p}$。本文研究了$(T\downarrow\mathscr{A})$中某些对象类的扩张闭包的性质、函子${\bf p}$的正确性以及$(T\downarrow\mathscr{A})$中给定类${\bf p}(\mathcal{X},\mathcal{Y})$的正交类的详细描述.此外，我们刻画了交换范畴$\mathscr{A}$和$\mathscr{B}$中的特殊预覆盖类何时能诱导$(T\downarrow\mathscr{A})$中的特殊预覆盖类。作为应用，我们证明了在适当的情形下，三角矩阵环$Λ=\left(\begin{smallmatrix}R & M \\ O & S \end{smallmatrix} \right)$上的Gorenstein投射左$Λ$ -模类是特殊预覆盖当且仅当Gorenstein投射左$R$ -模类和左$S$ -模类都是特殊预覆盖.因此，我们产生了大量的各种各样的环的例子，使类Gorenstein投射模是特殊的precovering在他们身上。
英文摘要：
Let $T$ be a right exact functor from an abelian category $\mathscr{B}$ into another abelian category $\mathscr{A}$. Then there exists a functor ${\bf p}$ from the product category $\mathscr{A}\times\mathscr{B}$ to the comma category $(T\downarrow\mathscr{A})$. In this paper, we study the property of the extension closure of some classes of objects in $(T\downarrow\mathscr{A})$, the exactness of the functor ${\bf p}$ and the detail description of orthogonal classes of a given class ${\bf p}(\mathcal{X},\mathcal{Y})$ in $(T\downarrow\mathscr{A})$. Moreover, we characterize when special precovering classes in abelian categories $\mathscr{A}$ and $\mathscr{B}$ can induce special precovering classes in $(T\downarrow\mathscr{A})$. As an application, we prove that under suitable cases, the class of Gorenstein projective left $Λ$-modules over a triangular matrix ring $Λ=\left(\begin{smallmatrix}R & M \\ O & S \end{smallmatrix} \right)$ 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.
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