Confused about notation and terminology of coordinate rings

Question

The nodal cubic $C\subset\mathbb{A}^2$ given by $y^2=x^2(x+1)$ has the usual parametrisation $f:\mathbb{A}^1 \to C \subset \mathbb{A}^2$ given by $x=t^2-1, y=t(t^2-1)$. Show that $f$ is finite, that is, $k[\mathbb{A}^1]$ is a finite $k[C]$-module.

I am confused on how $k[\mathbb{A}^1]$ can be a $k[C]$-module. From what I understand $k[\mathbb{A}^1]$ is the coordinate ring $k[x_1]/I(\mathbb{A}^1)$ and $k[C]$ is $k[x_1,x_2]/I(C)$. But in the first case we have functions defined on $\mathbb{A}^1$ and in the second case we have functions defined on $\mathbb{A^2}$. So how can you get a function defined on $\mathbb{A}^1$ by multiplying a function defined on $\mathbb{A}^2$ with a function defined on $\mathbb{A}^1$?

Answer 1

The map $f:\Bbb A^1\to C$ induces a map of coordinate algebras $f^*:k[C]\to k[\Bbb A^1]$ by pullback: given a function $\varphi$ on $C$, we get that $\varphi\circ f$ is a function on $\Bbb A^1$. This ring homomorphism gives $k[\Bbb A^1]$ the structure of a $k[C]$ module, and the question asks whether under this particular module structure, $k[\Bbb A^1]$ is a finite $k[C]$ module.

The goal of this problem is to explicitly write down what that module structure is and then to exhibit a finite basis of $k[\Bbb A^1]$ as a $k[C]$ module. To do that, can you write down where the generators $x,y\in k[C]$ go under $f^*$? Answer under the spoiler text:

$x\mapsto t^2-1$ and $y\mapsto t(t^2-1)$

Now all you have to do is find a finite basis, which I leave in your capable hands.

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Let's take a little time to pick apart your confusion:

But in the first case we have functions defined on $\mathbb{A}^1$ and in the second case we have functions defined on $\mathbb{A^2}$. So how can you get a function defined on $\mathbb{A}^1$ by multiplying a function defined on $\mathbb{A}^2$ with a function defined on $\mathbb{A}^1$?

It is not true that in the second case we have functions defined on $\Bbb A^2$. A function on $C$ can be lifted to a function on $\Bbb A^2$, but not uniquely: adding any multiple of $y^2-x^2(x+1)$ to the lift will change the function as a function on $\Bbb A^2$ but not as a function on $C$.

It is also true that if we have a map $g:\Bbb A^1\to\Bbb A^2$, we can repeat the same sort of idea above: we get a homomorphism of coordinate algebras $g^*:k[\Bbb A^2]\to k[\Bbb A^1]$ by pullback, and this tells us how a function on $\Bbb A^2$ acts on functions on $\Bbb A^1$.