§ Covariant Hom is left exact
Let's say we have the exact sequence:
Where the first arrow is for inclusion and the third is for projection.
We now want to consider what happens when we have for some target space .
The induced arrows are induced from composition; I will write
to mean "first do , then do ". Hence, my arrows linking to will be
: To first go from to , and then apply to go from to .
This gives us the sequence (which we are to check if it is exact, and which of the left
and right arrow exist):
§ A particular example
As usual, we go to the classic exact sequence:
We now have three interesting choices for our in relation to the above sequence:
(a) , (b) , (c) . Since and are isomorphic as modules, let's study
the case with and