Assume $A$ and $B$ are finite type $k$-algebras over a field.
Consider the tensor product $C = A\otimes_k B$. There are maps $A\to C$ and $B\to C$.
Given a prime ideal $p \subseteq C$, I can contract it into $A$ and $B$ using these maps, and obtain prime ideals $p_A\subseteq A$, and $p_B\subseteq B$.
Now, I can localize $A_{p_A}$, $B_{p_B}$, and again take tensor product: $A_{p_A}\otimes_k B_{p_B}$.
On the other hand, I can localize $C_p$. Is there any clear relation between these two rings? Is there a natural map between them? Are they isomorphic?
Here is my thought on the matter:
We have a natural map, $A\otimes_k B \to A_{p_A} \otimes_k B_{p_B}$. If we could prove that elements not in $p$ maps under this function to unit elements, then by the universal property of localization we will obtain a map $C_p \to A_{p_A} \otimes_k B_{p_B}$, but it is not clear to me that this is the case.