Thread: short exact sequences

If you have a homomorphism (a,b,c) of two short exact sequences, and if a and c are surjective, why does it follow that b is surjective?

I have played with it a lot, and I can show that if a and c are injective, then b is injective, but for some reason I can't find the trick to this one. Any hint would be appreciated. Thanks.

I got, no worries.

Originally Posted by jimmybuffet

I got, no worries.

Why don't you share your solution? It might help someone else.

-Dan

the main trick is that after you choose an element x in the module that b is supposed to map onto, you consider the image of x-b(y) under the bottom right map of the ses, where y is the obvious element you get in the domain of b from the two surjective maps on the right and top right of the ses. anyway it maps to zero, and hence is in the kernel of that map, and hence in the image of the one before that by exactness, blah blah. ha, it is hard to explain, but it's simple--i just don't know how to make graphics of short exact sequences in my reply so i tried to do it in words.

p.s. the reason i came back to this forum is that i was going to ask another question, but then i solved that one before i sent the question. it's kind of like playing chess, where you see a good move right as you're making a bad one.