I have now been staring at this particular sentence for way too long, and thus will start using any and all communication lines I can find to get assistance. Either I’m being way too stupid, or the author neglects to mention some salient detail.
Setup: is a group homomorphism, , . can be given the structure of a -module by pulling back through , i.e. we define for and .
So far it’s all crystal clear for me. However, it then turns out that we’re highly interested in using a morphism and I cannot for the life of me find out how such beasts are guaranteed to exist. If it where , I wouldn’t have any problems with it; but then the stuff I need/want to do with it don’t work out.