Point of Post: This is a continuation of this post.
Point of Post: In this post we show how one can use partial exactness to actually compute, explicitly, the isomorphism type of certain tensor products.
Up until this point I have committed a sort of embarrassing crime–I have yet to explicitly compute a tensor product! This is not good, because I strongly believe that if one doesn’t stop and compute some things, one can get a little lost in the abstraction. So, this post is devoted primarily to discussing a way in which we can use partial exactness to explicitly compute the isomorphism type of some actual examples. Roughly the idea is that if we can express a module in terms of an exact sequence of the form then we know that is the cokernel of the initial, right exactness tells us then that is exact and so will be the cokernel of the first map. After showing some examples where this is useful we shall describe a general technique for computing tensor products by using “free presentations” which ultimately amount to expressing a module in terms of generators and relations.