Let $R$ be a commutative ring. I want to show that $R[x] \otimes_R R[y] $ is isomorphic to $R[x,y]$ as $R$-algebras.
First, I tried as follows:
$R[x] \otimes _R R[y]$ has a universal property (as a $R$-module), that for any bilinear map $f: R[x] \times R[y] \to M$ with $M$ an $R$-module, $f$ induces a unique $R$-module homomorphism $ \bar f : R[x] \otimes _R R[y] \to M$. I showed that $R[x,y]$ also have this property (as an $R$-module), thereby showing $R[x] \otimes_R R[y] $ is isomorphic to $R[x,y]$ as $R$-modules.
$R[x,y]$ also has a universal property (as a ring), that any ring homomorphism $R \to S$ can be unqiuely extended to a ring homomorphism $R[x,y] \to S$. I showed that $ R[x] \otimes_R R[y] $ also have this property (as a ring), and hence $R[x] \otimes_R R[y] $ is isomorphic to $R[x,y]$ as rings.
But I think these two does not imply that $R[x] \otimes_R R[y] $ is isomorphic to $R[x,y]$ as $R$-algebras. Namely, I don't expect that the following statement is true:
For two $R$-algebras $A$ and $B$, if $A \simeq B$ as $R$-modules and as rings, then $A \simeq B$ also as $R$-algebras.
How do I have to show that $R[x] \otimes_R R[y] \simeq R[x,y]$ as $R$-algebras? Should I just have to construct an explicit isomorphism?
Best Answer
$R[x] \otimes_R R[y]$ is an $R$-algebra
Both $R[x]$ and $R[y]$ are $R$-algebras. The map
$$ R[x] \times R[y] \times R[x] \times R[y] \rightarrow R[x] \otimes R[y] \\ (r, s, r', s') \mapsto (rr') \otimes (ss') $$
is multilinear and, using associativity of the tensor product, induces a linear map
$$ (R[x] \otimes R[y]) \otimes (R[x] \otimes R[y]) \rightarrow R[x] \otimes R[y] $$
By the universal mapping property, this map corresponds to a bilinear map
$$ (R[x] \otimes R[y]) \times (R[x] \otimes R[y]) \rightarrow R[x] \otimes R[y] \\ (r \otimes s, r' \otimes s') \mapsto (r\otimes s)\cdot (r'\otimes s') $$
where $(r\otimes s)\cdot (r'\otimes s') := (rr') \otimes (ss')$. That gives us a multiplication on $R[x] \otimes R[y]$, turning it into an $R$-algebra.
$R[x] \otimes_R R[y]$ and $R[x,y]$ are isomorphic as $R$-algebras
To show this, construct the isomorphisms. The first we get from the universal mapping property of the tensor product:
$$ F : R[x] \otimes R[y] \rightarrow R[x,y], r \otimes s \mapsto rs $$
For the other direction, define
$$ G : R[x,y] \rightarrow R[x]\otimes R[y], x^i y^j \mapsto (x^i \otimes y^j) $$
on monomials and extend linearly.
Both $x^iy^j$ and $x^i \otimes y^j$ generate $R[x,y]$ and $R[x]\otimes R[y]$ as $R$-modules respectively, and on the generators we have
$$ (F\circ G)(x^iy^j) = F(x^i \otimes y^j) = x^i y^j \\ (G \circ F)(x^i \otimes y^j) = G(x^i y^j) = x^i \otimes y^j $$
So $F$ and $G$ are inverse to each other, and F respects the ring structure:
$$ \begin{align} F((r\otimes s)\cdot(r' \otimes s')) &= F((rr' \otimes (ss')) \\ &= rr'ss' \\ &= F(r\otimes s)\,F(r'\otimes s') \end{align} $$