Timeline for Universally Injective Morphisms
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| May 27, 2013 at 23:01 | comment | added | Peter Crooks | This solution definitely seems to work! The key point seems to be that if $f:X\rightarrow Y$ is a $k$-morphism of schemes of finite type over an algebraically closed field $k$, then $f$ is surjective if and only if $f(k):X(k)\rightarrow Y(k)$ is surjective. | |
| May 27, 2013 at 0:24 | comment | added | user29283 | If $f$ is a $k$-map between locally finite type $k$-schemes with $k$ an arbitrary field (not assumed algebraically closed) then $f$ is universally injective if and only if it is injective and $k(x)$ is purely inseparable over $k(f(x))$ for all closed points $x \in X$. The idea is that one looks for failure of injectivity on $\overline{k}$-points and unravels its consequences at the level of closed points over $k$, using that $f$ is injective. | |
| May 27, 2013 at 0:18 | comment | added | user29283 | A map of schemes $f:X \rightarrow Y$ is universally injective if and only if the diagonal $\Delta_f:X \rightarrow X \times_Y X$ is bijective on points valued in any field. But $\Delta_f$ is a locally closed immersion, so it is equivalent to say that $\Delta_f$ is surjective. For schemes locally of finite type over an algebraically closed field $k$, a locally closed subscheme is the entire space if and only if it contains all $k$-valued points, so it follows that when $f$ is a $k$-map between such $k$-schemes then it is universally injective if and only if it is injective on $k$-valued points. | |
| May 26, 2013 at 21:55 | comment | added | Jérémy Blanc | Not really, and it is true that it would be good to have a reference. | |
| May 26, 2013 at 20:04 | comment | added | Peter Crooks | Hi Jeremy, did you happen to find a definitive statement in the literature to the effect that for varieties over an algebraically closed field, an injective etale morphism is universally injective? | |
| May 26, 2013 at 19:04 | comment | added | Jérémy Blanc | it seems to be a duplicate of mathoverflow.net/questions/114400/… | |
| May 26, 2013 at 16:48 | answer | added | user19475 | timeline score: 2 | |
| May 26, 2013 at 15:36 | history | edited | Peter Crooks | CC BY-SA 3.0 |
added 209 characters in body
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| May 26, 2013 at 15:31 | history | asked | Peter Crooks | CC BY-SA 3.0 |