Let $C$ be the category whose objects are Noetherian spectral topological spaces and whose morphisms are homeomorphisms. Let $\mathrm{AffSch}$ be the category of Noetherian affine schemes (morphisms are morphisms of schemes). Let $\mathrm{SpecSpc}$ be the category of Noetherian spectral topological spaces (morphisms are continuous maps).

We have an obvious functor $I:C\rightarrow \mathrm{SpecSpc}$ and the forgetful functor $F:\mathrm{AffSch}\rightarrow \mathrm{SpecSpc}$. Does there exist a functor $G:C\rightarrow \mathrm{AffSch}$ such that $F\circ G=I$?

A slightly less functorial question: is every Noetherian spectral space the underlying space of a Noetherian scheme?


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