tensoring coordinate ring.

Question

Let $B = \mathbb R[x,y]$ where $x^2 + y^2 = 1$ (called the coordinate ring of the unit circle).\

I want to prove that $\mathbb C \otimes_{\mathbb R} B$ is a UFD. I know that we are tensoring up with scalars but how this will help me in proving that it is a UFD.

I know that: a unique factorization domain is an atomic integral domain in which factorization to irreducibles is unique (up to associates). And a commutative ring is atomic if each $r \in R$ is a finite product of irreducibles in R.

Any help is appreciated.

Answer 1

Ok, here's the plan:

1. Show that $\mathbb{C} \otimes_{\mathbb{R}} B \cong \mathbb{C}[x,y]/(x^2+y^2-1)$. One direction of this isomorphism is determined by $z\otimes p \mapsto zp$ when $z \in \mathbb{C}$ and $p \in B$. Show that this extends to a ring homomorphism $\mathbb{C} \otimes_{\mathbb{R}} B \to \mathbb{C}[x,y]/(x^2+y^2-1)$ and find an inverse!
2. Show that $\mathbb{C}[x,y]/(x^2+y^2-1) \cong \mathbb{C}[t,t^{-1}]$. To do this, we need to construct homomorphisms $\mathbb{C}[x,y]/(x^2+y^2-1) \to \mathbb{C}[t,t^{-1}]$ and $\mathbb{C}[t,t^{-1}] \to \mathbb{C}[x,y]/(x^2+y^2-1)$ which compose to give the identities. The only (good) way to build maps between these $\mathbb{C}$-algebras is to use their universal properties. A $\mathbb{C}$-algebra homomorphism from $\mathbb{C}[x,y]/(x^2+y^2-1)$ is uniquely determined by the images of $x$ and $y$ (let's call these $a$ and $b$, respectively), with the restriction that $a^2 + b^2 = 1$. A $\mathbb{C}$-algebra homomorphism from $\mathbb{C}[t,t^{-1}]$ is uniquely determined by the image of $t$, with the restriction that the image of $t$ must be a unit.
   Here are some hints: in order for the map $\mathbb{C}[t,t^{-1}] \to \mathbb{C}[x,y]/(x^2+y^2-1)$ to be surjective, the image of $t$ must generate the codomain as a $\mathbb{C}$-algebra. In order for the map $\mathbb{C}[x,y]/(x^2+y^2-1) \to \mathbb{C}[t,t^{-1}]$ to be surjective, the image must contain $t$ and $t^{-1}$. In particular, you need $t$ to correspond to a unit in $\mathbb{C}[x,y]/(x^2+y^2-1)$ -- think about the factorization of $x^2 + y^2$ over $\mathbb{C}$ to find this unit!
3. Show that $\mathbb{C}[t,t^{-1}]$ is a UFD, because it's a localization of a UFD.

Hope this helps! Happy to fill in more details if you get stuck somewhere.