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More precisely, is there a map of schemes $X$ --> $Y$ such that $f$ gives a homeomorphism between $X$ and a closed subset of $Y$, but the corresponding map on sheaves is not surjective?

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The topology is not enough to determine the scheme structure in general as show the counteramples of Rebecca and of Matt. On the positive direction, you could have a look at… – Qing Liu Feb 14 '10 at 22:27
up vote 7 down vote accepted

Yes, for example if $K \subset L$ is an inclusion fields, then the induced map Spec $L \to $ Spec $K$ is a homeomorphism (both source and target are single points), but the induced map on sheaves is the given inclusion of $K$ into $L$, which is surjective only if $K = L$.

For another example, let $X'\to Y$ be a closed immersion of schemes over ${\bar{\mathbb F}}\_p$, and let $X \to X'$ be the relative Frobenius morphism. Then $X\to X'$ is a homeomorphism on underlying topological spaces but is not an isomorphism of schemes, and so the composite $X\to Y$ is a closed embedding on underlying spaces but not a closed immersion of schemes.

As one last example, let $X'$ be the cuspidal cubic given by $y^2 = x^3$ in the affine plane $Y$ (over $\mathbb C$, say), and let $X$ be the normalization of $X'$ (which is just the affine line). Then $X \to X'$ is a homeomorphism on underlying spaces, but is not an isomorphism of schemes. The composite $X \to Y$ is thus not a closed immersion, but induces a closed embedding of underlying topological spaces.

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In the second example, is $X$ the normalisation of $X'$ (as in the third example)? – jnewton Feb 14 '10 at 8:46
I suspect that $X \to X'$ is meant to be the relative Frobenius map. But I hope we'll get a confirmation soon. – David Speyer Feb 14 '10 at 12:50
David is correct; I've added this. – Emerton Feb 14 '10 at 13:59

Here's an example: Consider the morphism $f:\text{Spec} k[\varepsilon]/(\varepsilon^2)\rightarrow \text{Spec} k$ corresponding to the inclusion $k\hookrightarrow k[\varepsilon]/(\varepsilon^2)$. More generally, you can get lots of examples from non-reducedness.

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