## Functorial Properties of the Tensor Product (Pt. II)

**Point of Post: **This is a continuation of this post.

What we’d now like to prove is that is that the tensor product functor, fixed in either variable, is *right exact *in the sense that right exact sequences get carried to right exact sequences. More precisely:

**Theorem: ***Let be left -modules. If *

*is an exact sequence of left -modules and a right -module, then*

*is an exact sequence of abelian groups.*

**Proof: **Let’s first check that , but this follows immediately since

(the last equality holding since additive functors take zero maps to zero maps). Let’s now prove that . This is the hard part. We have already shown that and so from first principles we can factor through . In other words, we are able to get a map such that if is the natural projection, then . If is an isomorphism then we’d have that . We can prove this by construction an explicit inverse . To do this we note that since is surjective we can find a (set!) map such that and . We are then able to define

We claim that this map is -biadditive. Indeed, we need to prove that is equal to . To do this it suffices to prove that

is in . To do this we note that

and so and so . Since is -biadditive we are afforded a map

such that . Thus, we find that

and

Thus, is a two-sided inverse for , and so is an isomorphism and so the fact that and so exactness in the middle follows.

It lastly suffices to prove that is surjective. To do this, let in be arbitrary. Since is surjective there exists some such that and so . Thus, hits all the simple tensors in and since the simple tensors generate and is a -map surjectivity follows.

Practically the same proof shows that:

**Theorem: ***If are right -modules and we have an exact sequence*

*is an exact sequence of -maps, and if is a left -module, then*

*is an exact sequence of abelian groups.*

**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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