I am trying to prove that morphisms of finite type are stable under base change, but I am having some trouble moving from the case where everything is affine to the general case. Suppose $f:X \rightarrow Y$ is a morphism of finite type and $Y'$ is a $Y$-scheme. I want to show that the morphism $g: X \times_Y Y' \rightarrow Y'$ is of finite type. In the case that $X$, $Y$, and $Y'$ are affine, I understand why this is true. For the general case, by a lemma in Liu's book, it is enough to show that there is an affine open cover $\{V_i\}_i$ of $Y'$ such that for each $i$, $g^{-1}(V_i)$ is a finite union of affine open subsets $U_{ij}$ such that for each $i$ and $j$, $O_X(U_{ij})$ is a finitely generated algebra over $O_Y(V_i)$. Here is my attempt at proving this.
Choose an affine open cover $\{V_i\}_i$ of $Y'$. Is it true that $g^{-1}(V_i)=X \times_Y V_i$? I think this should follow from how we constructed the fibered product by gluing. Since $f:X \rightarrow Y$ is of finite type, we may choose an affine open cover $\{Y_j\}$ of $Y$ such that $f^{-1}(Y_j)$ is covered by a finite number of affine opens $W_{jk}$. Now, $W_{jk} \times_Y V_i$ are open subschemes that cover $X \times_Y V_i$, but since $Y$ is not necessarily affine, these schemes are not necessarily affine, right? Furthermore, if we are using the $W_{jk}$ to cover all of $X$, there could be infinitely many of them. To make the $W_{jk} \times_Y V_i$ affine, we could further cover them with $W_{jk} \times_{Y_k} V_i$, but we are not guaranteed finitely many $Y_k$ either, so while these schemes will be affine, there will not necessarily be finitely many. I have been having some trouble with these sorts of arguments where one can immediately reduce to the affine case, and some help here would be greatly appreciated.
Best Answer
You can do a slight change to make your argument work and give you an affine cover of $X\times_Y Y'$ as follows. Call $h$ the base change morphism $Y'\to Y$. Consider a cover of $Y$ by open affines $V_i=Spec(A_i)$ and cover any $h^{-1}(V_i)$ by open affines $V_{ij}=Spec(A_{ij})$ of $Y'$. The preimage of $V_{ij}$ by $g$ is as you said $X\times_Y V_{ij}$ which by the properties of the fibered product coincides with $X_i\times_{V_i}V_{ij}$, where $X_i=f^{-1}(V_i)$ (this is crucial to end up with affine schemes as you will see in a moment). You can cover $X_i$ with open affines $X_{ik}=Spec(B_{ik})$ with all $B_{ik}$ finite type $A_i$-algebras. So $g^{-1}(V_{ij})$ is covered by the affines $X_{ik}\times_{V_i}V_{ij}=Spec(B_{ik}\otimes_{A_i}A_{ij})$, which are $A_{ij}$-algebras of finite type. This proves that the base change of $f$ is locally of finite type (actually we didn't use that $f$ is quasi compact, so we proved that locally of finite type morphisms are stable under base change).
If now $f$ is quasi compact, you just need a finite number of $X_{ik}$ to cover $X_{i}$ and so a finite number of $X_{ik}\times_{V_i}V_{ij}$ is enough to cover $g^{-1}(V_{ij})$, proving the quasi-compactness of $g$.