Let $S=Spec(A)$ and $S'=Spec(B)$ be two affine schemes for some rings $A$ and $B$ such that there is a morphism of schemes $f:S'\rightarrow S$. For any $S$-scheme $X$, one can consider the fiber product $X\times_S S'$ of $X$ and $S'$ over $S$.
If we assume that $X$ is given by a set of equations $(E)$ in $A$, what are the equations which define the $S'$-scheme $X\times_S S'$? is it the equations in $B$ which are obtained by applying to $(E)$ the morphism of rings induced by $f$ ? I can this be written properly?
Another construction which is even more simple : assuming that $Y$ is an $S'$-scheme, $Y$ can be considered as an $S$-scheme via $Y\longrightarrow S'\longrightarrow S$ (composing by $f$). I have two questions about this construction : first in the same way i did for fiber products, is it possible to find the equations which define $Y$ as an $S$ variety from whose which define it as an $S'$ variety ?
Finally something that seems reasonnable to me : $Z$ is an $S$-scheme, and you consider the fiber product $T=Z\times_S S'$ as an $S'$-scheme. Is the scheme $T$ consider as an $S$-scheme with the previous construction isomorphic to $Z$ as an $S$-scheme? I think its the case just because of the definition of the fiber product, but i would like to be sure.
Best Answer
First paragraph: yes. Second paragraph: getting equations is going to be gross in general, although it depends on the nature of $S'$ and $S$. Third paragraph: no.
Maybe this will be a motivating example. Let $S$ be Spec of an algebraically closed field, and let $S'$ be Spec of a much larger algebraically closed field (e.g. adjoin a transcendental to your field, then algebraically close that). If you take an $S'$ scheme and "think of it as an $S$-scheme" you will get a monstrosity that's not of finite type (e.g. in the last paragraph you are asserting that $\mathbb{C}[t]$ is isomorphic to $\overline{\mathbb{Q}}[t]$ as a $\overline{\mathbb{Q}}$-algebra).
The lesson to take away is that base change is the reasonable, good functor, and the "forgetful functor" is the ugly one, the opposite of the way you'd think it would be on first glance.