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.