In Huybrecht's 'Fourier-Mukai Transforms in Algebraic Geometry', the see-saw principle is stated in Proposition 9.4 as follows.
Let $X$ be an irreducible complete variety over a field $k$, $T$ an integral scheme, and $L \in \operatorname{Pic}(X \times T)$. Suppose $L_t:=L|_{X \times \{t\}}$ is trivial for all closed points $t \in T$. Then there exists a line bundle $M$ on $T$ with $L \cong p^*M$. (where $p: X \times T \to T$ is the canonical projection)
In the proof sketch, it is explained that being trivial is a closed condition on $T$. From this, it is concluded that checking closed points is enough.
I interpreted this closed condition to mean $\{t \in T\;|\;L_t\text{ is trivial}\} \subset T$ is a closed subset. The conclusion then seems to rely on the assumption that closed points are dense, which I don't think is in general true for integral schemes. Perhaps there is some implicit assumption that $T$ is of finite type over $k$? In this case, closed points are indeed dense.
Am I missing the point here or are there missing hypotheses in the theorem statement?
Thank you!
