separation property for schemes Let $A$ be a ring, then we have a scheme $T=\mathrm{Spec}A[y_i; i\in I]$ where $I$ is an index set. Let $B$ be a closed subset of $T$ and $t\in T-B$ be a point of $T$, then I'd like to ask how to find $g\in A[y_i; i\in I]$ such that $B\subset V(g)$ and $t\notin V(g)$.
This is used in a proof due to B. Poonen that universally closed morphism of schemes is quasi-compact.
Any help would be appreciated.

The $D(g)$ (for $g\in A[y_i:i\in I]$) form a basis for the topology on $T$, and $T - B$ is open. So there is a $g$ such that $t\in D(g)\subset T - B$. This means $t\notin V(g)$ and $B\subset V(g)$.
