Homeomorphism onto a closed subset of a scheme that isn't a closed immersion - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T06:58:48Z http://mathoverflow.net/feeds/question/15235 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/15235/homeomorphism-onto-a-closed-subset-of-a-scheme-that-isnt-a-closed-immersion Homeomorphism onto a closed subset of a scheme that isn't a closed immersion solbap 2010-02-14T05:36:39Z 2010-02-14T13:54:13Z <p>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?</p> http://mathoverflow.net/questions/15235/homeomorphism-onto-a-closed-subset-of-a-scheme-that-isnt-a-closed-immersion/15236#15236 Answer by Emerton for Homeomorphism onto a closed subset of a scheme that isn't a closed immersion Emerton 2010-02-14T05:49:47Z 2010-02-14T13:54:13Z <p>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 $K = L$.</p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/15235/homeomorphism-onto-a-closed-subset-of-a-scheme-that-isnt-a-closed-immersion/15237#15237 Answer by Rebecca Bellovin for Homeomorphism onto a closed subset of a scheme that isn't a closed immersion Rebecca Bellovin 2010-02-14T05:50:37Z 2010-02-14T05:50:37Z <p>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.</p>