The Construction of the Tensor Product of Modules (Pt. III)
Point of Post: This is a continuation of this post.
Once again, this concretely means that if is an -bimodule and a -bimodule then an tensor product of by over is an module and an -biadditive -map which has the property such that given any other -bimodule and an -biadditive -map there exists a unique -map such that .
For example, the previous construction of a tensor product as an abelian group corresponded to -tensor products over .
Once again, we have by mere definition the following important theorem:
Theorem: Let be a -bimodule and a -bimodule. Then, if and are two -tensor products over , of by then there exists a unique -isomorphism such that .
Unfortunately, similar to before there is absolutely no indication that -tensor products over a ring should exist in general. Strangely enough, we have the following theorem that brings together all of the theory:
Theorem: Let be an -bimodule and an -bimodule. Then, the multiplications and extend to give a -bimodule structure on , and moreover is then a -tensor product of by over .
Proof: We begin by showing that there really is a (necessarily unique) -multiplication on for which . Indeed, consider the map given by . We note then that for any , and we have that
so that is -biadditive. Thus, by the universal property of tensor products we get a group homomorphism such that . We then define the multiplication map by and denote, as per usual, as just . We note though that since
and maps on are determined by their values on elements of the form we may conclude that . Similarly, one can check that on elements of the form and and so they are equal on all of . We then proceed to verify the desired axioms
Thus, we see that we have defined a left -multiplication on such that as desired. The exact same process shows that we have a right -multiplication on such that . To prove that everything defines a -bimodule structure on it suffices to show that for all , , and . It of course suffices to do this on simple tensors:
thus this really is a -bimodule structure on .
Let’s now verify that really is an -tensor product of by over . To do this let’s first check that is an -biadditive -map. The fact that it’s -biadditive follows from previous note, and so it suffices to show that it is -biadditive. To do this we merely note that
and so the fact that it’s an -map follows.
Let’s now show that it’s initial in the desired category. To this end suppose that we have some -bimodule and an -biadditive -map . Since is -biadditive we are guaranteed, by construction, a group homomorphism such that , it thus remains to show that is an -map. To do this let be arbitrary and we see then that
Thus, is an -map such that . Since the uniqueness is guaranteed by the construction of the fact that is an -tensor product over follows.
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