Could someone provide me a link to the proof of the adjointness of Hom and Tensor. I did an extensive google search but could not find anything self contained that presented the proof in full generality (or at least the generality I know). Let $R\to S$ be a ring homomorphism, let $M,N$ be $S$-modules and $Q$ an $R$-module. Then, we have $\textrm{Hom}_R(M\otimes_S N,Q) \cong \textrm{Hom}_S(M,\textrm{Hom}_R(N,Q)$
Adjointness of Hom and Tensor
13
$\begingroup$
abstract-algebra
commutative-algebra
category-theory
tensor-products
-
1Rotman, An introduction to Homological Algebra, 2nd edition, proves it, and calls it the "adjoint isomorphism theorem". – 2011-04-09
1 Answers
10
Let $f \in Hom_R(M\otimes_S N,Q)$. We define $g \in Hom_S(M,Hom_R(N,Q))$ by:
$g(m)(n)=f(m \otimes n)$
Similarly, if $g$ is defined, we can easily define $f$.
I'll leave it to you to prove that this map between $f$ and $g$ actually goes to the appropriate sets, but this is the basic argument. As mentioned by one of the comments, it does depend on how you define the tensor product.