Les S be a scheme. Does there exist a faithfully flat morphism T to S with T a locally Noetherian scheme?

Note that if $T\to S$ is also quasicompact, then $S$ must be locally noetherian: this boils down to the wellknown fact that if $A\to B$ is a faithfully flat ring homomorphism and $B$ is noetherian, then so is $A$. This proves that Jason's example above is indeed a counterexample. More generally, any nonnoetherian local scheme $S$ is a counterexample: if $T\to S$ is faithfully flat, there is an open affine $U\subset T$ which covers $S$. EDIT: In fact, here is a complete answer ($S$ is any given scheme): (1) The following are equivalent: (2) The following are equivalent: Proof: exercise. To show that (2b) implies (2a), take $T=\coprod_{s\in S}\mathrm{Spec}(\mathcal{O}_{S,s})$. 

