Timeline for Closed points on a variety
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Aug 22, 2011 at 23:38 | comment | added | Allen Knutson | There's also the non-geometric Frobenius to worry about, that is the identity on points but not on the structure sheaf, an example that's still tricky even for reduced schemes over an algebraically closed field (of characteristic $p$). | |
| Aug 22, 2011 at 13:34 | comment | added | A. Pascal | If $X=Y=\mathbb{A}^{1}$ over $k=\mathbb{F}_{2}$, then the identity map $x \mapsto x$ and the the squaring map $x \mapsto x^{2}$ have the same values on $k$-points, but are different as maps of schemes. Of course, you can distinguish them by looking at maps on points over field extensions of $k$. Such points do correspond to closed points of the underlying scheme, but it seems to me a little strange to talk about old-fashioned non-schemey varieties unless you are over an algebraically closed field. I believe Hartshorne presumes his ground field is algebraically closed for classical case. | |
| Aug 22, 2011 at 11:50 | comment | added | Descartes | Good question... | |
| Aug 22, 2011 at 11:12 | comment | added | Qfwfq | @A.Pascal: but $X$ and $Y$ being reduced, isn't it the same even if they coincide just as maps of sets? | |
| Aug 22, 2011 at 9:05 | comment | added | A. Pascal | It's important that the maps $X^0 \rightarrow Y^0$ agree as morphisms, not just as maps of sets. | |
| Aug 22, 2011 at 8:01 | comment | added | Martin Brandenburg | The formal statement is that the category of varieties in the scheme-sense is equivalent to the category of varieties in the classical sense (where each point is closed). You can find this everywhere, e.g. in Hartshorne chapter II. | |
| Aug 22, 2011 at 7:24 | history | asked | Descartes | CC BY-SA 3.0 |