## Adjoint Isomorphism for Tensor and Hom (Pt. I)

**Point of Post: **In this post we show that the tensor functor and Hom functor form an adjoint pair.

*Motivation*

In this post we prove one of the fundamental theorems of module theory–that the Hom and tensor functors form an adjoint pair. I do not want to spend a great deal of time explaining what this precisely means, because I will eventually discuss adjoint functors in their full generality, and anything I say here will be nothing more than a specific, less enlightened case of what I say there. That said, saying nothing at all would be a crime. In a very practical sense the adjointness of Hom and tensor says that there is a one-to-one correspondence of maps of the form and maps of the form . The vague reasoning for this should be clear enough, namely a mapping can be thought of as a mapping (via the universal characterization) and the rest is the common techinique of “currying” where a two-variable map can be thought of as a mapping to a set of maps. Namely, if we have a map fixing one of the variables, say , gives us a map . Thus, we can think that a function is nothing more than a paramateried set of functions or a mapping . In fact, as an equality of sets this is eqiuvalent to saying that or, with cardinal numbers, that . The amazing thing is that we aren’t going to just get a bijection of some typese of Hom sets, but an isomorphism. This connection between tensor and Hom will often allow us to make arguments about one in terms of the other, often allowing us to take difficult problems involving, say, tensor and make them into easy looking problems involving Hom.

*Adjointness*

We begin by noting how the advent of bimodules allows us to endow Hom sets with more than an abelian group structure. In particular:

If be an -bimodule and a left -module. Then, is a left -module with .

If is an -bimodule and a right -module then is a right -module with .

If is a right -module and an -bimodule then is a left -module with .

If is a left -module and an -bimodule then is a right -module with .

These are all easily verified. While these may seem awfully contrived (we just multiplied where we could, on the side that made associativity work out) they do have meaning, as the following theorem will show. Moreover, if is commutative and and we give every module in sight the usual -bimodule structure then these all collapse to the same definition we have mentioned before: .

**Theorem(Adjointness for Hom and Tensor): ***Let be an -bimodule, an -bimodule, and an -bimodule. Then, there exists group isomorphisms*

*such that fixing any two and letting the third range over the appropriate set of modules makes the corresponding set of isomorphisms and into natural equivalences between the associated functors.*

**Proof: **We define, given an -map a map by . For completeness, let’s check that really is an -map . To see that is a group homomorphism we merely note that

checking that is a right -map then comes via

Thus, we see that really is a right -map . Let’s now verify that the correspondence really is an -map. To do this we must merely note that

**References:**

[1] Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

[2] Rotman, Joseph J. *Advanced Modern Algebra*. Providence, RI: American Mathematical Society, 2010. Print.

[3] Blyth, T. S. *Module Theory.* Clarendon, 1990. Print.

[4] Lang, Serge. *Algebra*. Reading, MA: Addison-Wesley Pub., 1965. Print.

[5] Grillet, Pierre A. *Abstract Algebra*. New York: Springer, 2007. Print.

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