I assume rings are commutative. Given surjective ring morphisms $f:A\to C$, $g:B\to C$, I wonder if it's possible to determine the prime ideals of the pullback $A\times_C B$ in terms of the prime ideals of $A,B,C$. This question was asked with in addition the assumptions that $A,B,C$ are Noetherian in the post below:
Prime ideals and maximal ideals of the Pullback of rings
I am not entirely convinced by the answers provided, and I wonder if someone knows how to finish the answers.
First answer: Note that $\operatorname{Spec}(A\times_B C)$ is the push-out in the category of schemes.
My question is: why is $\operatorname{Spec}(A\times_B C)$ also a push-out in the category of topological spaces? According to the post below, this is generally not the case, but is it still true in this case? Maybe with the added assumption that $A,B,C$ are Noetherian?
Does $\operatorname{Spec}$ preserve pushouts?
Second answer: Inspired by how we know how the prime ideals of $A\times B$ (product) look like, and noting that $A\times_C B\to A$ and $A\times_C B\to B$ is surjective, we know that these projection maps map prime ideals to $A,\mathfrak p$ (with $\mathfrak p\in\operatorname{Spec}(A)$) and $B,\mathfrak q$ (with $\mathfrak q\in\operatorname{Spec}(B)$ respectively.
My question is: how do we know now how the prime ideals of $A\times_C B$ look like? In one argument I know for the primes of $A\times B$, we use that $(1,0),(0,1)\in A\times B$, which is not necessarily the case for $A\times_C B$. So the argument, if it exists, has to be different. Any idea?
BTW, I am aware that asking two questions might go against the rules of this website (in which case I will split the post in two separate posts), but since these questions are so tightly related, I'm wondering if maybe we can answer both simultaneously.