Adjoint Isomorphism for Tensor and Hom (Pt. II)
Point of Post: This is a continuation of this post.
From all of this we can conclude that is a well-defined map (yay!). Let’s now verify that it is a group isomorphism. To see that is a homomorphism we merely note that
and since were arbitrary the desired equality holds. Lastly, let’s check that is a bijection. Injectivity is simple enough, if then we see that and agree on all simple tensors in and so (because the simple tensors are a generating set) must be equal on all of . For surjectivity, let be an arbitrary -map. Define the map given by . We claim that is an -biadditive -bimap. Indeed:
Thus, by the universal properties of the tensor product lifts to an -map such that . Note then that for all and so . Thus, surjectivity, and thus bijectivity, and thus isomorphism follow.
Let us fix and show that in is a natural transformation. Indeed, suppose that is an -map then we wish to show the commutativity of the following diagram
but this is easy:
The other naturalities follow similarly.
The case for is basically the same as the case for , and so we admit it.
Ok, so getting a little more down to earth, the above basically says that if we take a commutative ring and and take three -modules (thought of, in the usual way, as -bimodules) and noting that and are both -maps in this context, that’s pretty easy to see
for example, we have the following corollary (which is the case we’ll most often use):
Theorem: Let be a commutative unital ring and be unital -modules. Then, the above maps and give -isomorphisms
 Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.
 Rotman, Joseph J. Advanced Modern Algebra. Providence, RI: American Mathematical Society, 2010. Print.
 Blyth, T. S. Module Theory. Clarendon, 1990. Print.
 Lang, Serge. Algebra. Reading, MA: Addison-Wesley Pub., 1965. Print.
 Grillet, Pierre A. Abstract Algebra. New York: Springer, 2007. Print.
No comments yet.