Describing a particular scheme

Question

I want to describe the affine scheme $Spec(\mathbb{C}(t) \otimes_\mathbb{C} \mathbb{C}(t))$. My task is to especially show that this scheme as infinetly many points, but I'm more interested in how to approach this in general. I'm not familiar in working with schemes, so I would be thankful for someone illustrating the approach to me, or to point me in the right direction. Thanks a lot!

Answer 1

For clarity I'll write $A=\mathbb C(s)\otimes_\mathbb C \mathbb C(t)$ and then describe the scheme $X=\operatorname {Spec}A$.
The trick is to realise that $\mathbb C(s)$ is the fraction field $\mathbb C(s)=S^{-1}\mathbb C[s]$ where $S=\mathbb C[s]\setminus \{0\}$ and similarly $\mathbb C(t)=T^{-1}\mathbb C[t]$ with $T=\mathbb C[t]\setminus \{0\}$. We then have: $$A=\mathbb C(s)\otimes_\mathbb C \mathbb C(t)=S^{-1}\mathbb C[s]\otimes_\mathbb C S^{-1}\mathbb C[t]=(ST)^{-1}(\mathbb C[s]\otimes_\mathbb C \mathbb C[t])=(ST)^{-1}\mathbb C[s,t]$$ Hence $A$ consists of rational functions of the form $$\phi(s,t)=\frac {P(s,t)}{q(s)r(t)} \quad \operatorname {with} \quad P(s,t)\in \mathbb C[s,t], \:q(s)\in \mathbb C[s]\setminus \{0\}, \: r(t)\in \mathbb C[t] \setminus \{0\}$$ The points of the scheme $X=\operatorname {Spec}A$ then consists of the prime ideals $\mathfrak p\subset \mathbb C[s,t]$ disjoint from $ST$, i.e. containing no $q(s)\neq 0$ and no $r(t)\neq 0$.
So, finally it is easy to describe the scheme $X$ but the result is rather strange:

The scheme $X$ is obtained from the affine plane $\mathbb A^2_\mathbb C$ by deleting all its closed points $\langle s-a,t-b\rangle $, all generic points $\langle s-a\rangle $ of the "vertical" lines $s-a=0$ and all generic points $\langle t-b\rangle $ of the "horizontal" lines $t-b=0$.
Thus $X$ consists of the generic point $\langle 0\rangle $ and the generic points $\langle f(s,t)\rangle $of all irreducible curves $f(s,t)=0$ which are neither a "horizontal" nor a "vertical" line.

In particular notice that the Krull dimension of $X$ is $1$.
That $X$ is obtained from $\mathbb A^2_\mathbb C$ after one has removed all its classical points reminds me of the Cheshire-Cat in Alice in Wonderland whose grin remains after he disappears...