拉回正合范畴中两个态射合成的核与余像

Kernel and Coimage of Two Composite Morphisms in Pullback Exact Category

在Abelian范畴中,如果f:A→B和g:B→C是两个态射,则存在(1)Im f∩Ker g=f(Ker gf);(2)Im f+Ker g=g-1(Im gf).虽然在拉回正合范畴(C,E)中一般没有像的概念,但也有与(1)(2)性质相类似的结论,这就是Ker f→Ker gf→Ker g×BCoim f和Ker g→Ker gЦDCoim f→Coim gf均为E-短正合列,其中D=Ker g×BCoim gf.

Let f：A→B and g：B→C be two morphisms in an Abeilian category.Then there exist：（1）Im f∩Ker g=f（Ker gf）;（2）Im f＋Ker g=g-1（Im gf）.Uncommonly,there aren＇t the version of image in the pull-back exact category（C,E）,but the corresponding properties are obtained.That is Ker f→Ker gf→Ker g×BCoim f and Ker g→Ker gЦDCoim f→Coim gf are E-short exact sequence,where D=Ker g×BCoim f.