Support of a coherent sheaf over a fiber product scheme

Question

I'm trying to prove the following fact which I don't know if it is true since I am not able to find a counterexample:

Let $X,S$ be two $K$-scheme of finite type with $K$ an algebraically closed field. Let $\mathcal{E}$ be a coherent sheaf over $X\times_{Spec(K)} S=X\times S$ flat over $S$. Then, $Supp(\mathcal{E})\subset X\times S$ is proper over $S$

I know this is obviously false when we take the fibre product of $X$ and $S$ over another scheme $Y$, say. I'm wondering if in this particular case it is true that $\pi_S:X\times S\to S$ is of finite type (perhaps by using that being of finite type is preserved under base change) so that we can use the following lemma https://stacks.math.columbia.edu/tag/0CYL from StackExchange.

If the above fact it's not true, could you please give me a counterexample?

Answer 1

Take $X = S = \mathbb{A}^1$, $Z = \{xy = 1\} \subset X \times S$ (where $x$ is the coordinate on $X$ and $y$ on $S$), and let $\mathcal{E}$ be the structure sheaf of $Z$.