Is $X\times_ST$ a Noetherian scheme?

Question

Let $X$ be a Noetherian scheme. Given two morphisms of schemes $\pi:T\to S$ and $f:X\to S$, if $\pi$ is flat, is $X\times_ST$ a Noetherian scheme?

Answer 1

No it's not. But I assume the you asked this question for the problem in this picture i.e. Exercise 5.1.16 in Qingliu's Algebraic Geometry and Arithmetic Curve. For this exercise, the key step is to show that $f_{T,*}$ preserves quasi-coherent sheaves when $X$ is Noetherian. Here is the idea:

1. Since being quasi-coherent is a local property, we may assume that $T$ is affine.
2. By shrinking $T$, we may further assume $\pi(T)\subset U$ for some open affine $U\subset S$.
3. We have $X_T=(X_U)_T$. And $X_U=f^{-1}(U)$ is an open subscheme of the Noetherian scheme $X$, so $X_U$ is also Noetherian. We may suppose that $S=U$ is affine.
4. Since $X$ is Noetherian, $X$ can be covered by a finite open affine covering s.t. each intersection is covered by a finite open affine covering.
5. So $X_T$ can also be covered by a finite open affine covering s.t. each intersection is covered by a finite open affine covering (since $S$ and $T$ are affine now).
6. And this is the ultimate condition we need to show that $f_{T,*}$ preserves quasi-coherent sheaves in Proposition 5.1.14.