Covering Morphisms in a Pushout-Pullback Diagram

DOI：10.3770/j.issn:1000-341X.2010.03.018

 作者 单位 周德旭 福建师范大学数学系, 福建 福州 350007

在由四个同态$f:A\rightarrow B$, $g:A\rightarrow C$, $\alpha:C\rightarrow D$ 以及 $\beta:B\rightarrow D$构成的推出-拉回图上, 我们给出了这四个模的覆盖之间的关系.若ker$f\in I({\mathscr{L}})$，则$g:A\rightarrow C$为${\mathscr{L}}$-覆盖性同态当且仅当$\beta:B\rightarrow D$为${\mathscr{L}}$-覆盖性同态. 若每个模都有${\mathscr{L}}$-预覆盖且ker$f\in I({\mathscr{L}})$, 则$A$和$C$有同构的${\mathscr{L}}$-预覆盖当且仅当$B$和$D$也有同构的${\mathscr{L}}$-预覆盖.

In a pushout-pullback diagram, which consists of four morphisms $f:A\rightarrow B$, $g:A\rightarrow C$, $\alpha:C\rightarrow D$ and $\beta:B\rightarrow D$, we give some relations among the covers of these four modules. If ker$f\in I({\mathscr{L}})$, then $g:A\rightarrow C$ is ${\mathscr{L}}$-covering if and only if $\beta:B\rightarrow D$ is ${\mathscr{L}}$-covering. If every module has an ${\mathscr{L}}$-precover and ker$f\in I({\mathscr{L}})$, then $A$ and $C$ have isomorphic ${\mathscr{L}}$-precovers if and only if $B$ and $D$ have isomorphic ${\mathscr{L}}$-precovers.