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Given two schemes $X$ and $Y$, and a continuous map $f$ between the underlying spaces, what conditions are necessary for the existence of a map of sheaves $g:\mathcal{O}_Y\rightarrow f_*\mathcal{O}_X$ such that $(f, g)$ defines a morphism of schemes?

Assume $X$ and $Y$ are locally of finite type over $\mathrm{Spec}\:k$ for $k$ an algebraically closed field, if that makes life any easier.

Some conditions are necessary because there is no morphism of schemes $\mathrm{Spec}\:\mathbb{Q}\rightarrow \mathrm{Spec}\:\mathbb{R}$ (cardinality). Alternatively, there is no morphism of schemes from the $k$-affine line with double origin to the $k$-affine line that restricts to a homeomorphism on the underlying spaces (albeit there is a homeomorphism).

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  • $\begingroup$ According to wikipedia, a morphism of schemes is just a morphism of the underlying locally ringed spaces. So I guess you just want your map g of sheaves to induce a map of local rings (i.e. preserving the maximal ideal) on each stalk. en.wikipedia.org/wiki/Morphism_of_schemes $\endgroup$ May 22, 2019 at 11:15
  • $\begingroup$ @SamGunningham note that OP does not have "your map g of sheaves". So I am not sure how your comment addresses the question. $\endgroup$
    – user138661
    May 22, 2019 at 12:24
  • $\begingroup$ Ah, I misread the question... $\endgroup$ May 22, 2019 at 12:34

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