Timeline for Functorial way of showing that the Segre (or Plucker) morphism is a closed embedding?
Current License: CC BY-SA 2.5
13 events
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| Jan 5, 2014 at 8:23 | comment | added | user41650 | Maybe I will write another answer to this question later.There is a purely functorial way to prove Serge embedding is closed immersion based on the point of view of presheaf which was developed by Kontsevich-Rosenberg in noncommutative setting | |
| Jul 5, 2011 at 22:10 | comment | added | Martin Brandenburg | I just want to mention that now I have a functorial proof that the Plücker embedding is a closed immersion, using the same technique as below, namely constructing the quasi-coherent ideal and showing that the functor of points are isomorphic. | |
| Jan 13, 2011 at 21:37 | vote | accept | Akhil Mathew | ||
| Jan 12, 2011 at 14:27 | answer | added | Martin Brandenburg | timeline score: 5 | |
| Jan 12, 2011 at 8:15 | comment | added | Laurent Moret-Bailly | @Akhil: Unfortunately, even for this special case, every proof I can think of uses some explicit computation. We are essentially trying to recover, say, a $d$-dimensional subspace $F$ of $k^n$ from its determinant line in $\Lambda^d(k^n)$, and I don't see how we can do this just by waving hands. | |
| Jan 11, 2011 at 14:52 | comment | added | Akhil Mathew | @Martin: Yes, your edit improves the question; thanks. | |
| Jan 11, 2011 at 14:52 | comment | added | Akhil Mathew | @Laurent Moret-Bailly: Thanks! (I would be happy to accept that as an answer.) Is there a direct way to check that $X_y$ (for $X$ the Grassmannian, $Y$ the projective space, $X \to Y$ the Plucker map) is isomorphic to $\mathrm{Spec} \kappa(y)$? | |
| Jan 11, 2011 at 12:17 | comment | added | Martin Brandenburg | By the way, as with your other questions: Very interesting, 1+. | |
| Jan 11, 2011 at 12:15 | comment | added | Martin Brandenburg | @Akhil: I hope it's OK for you that I changed the P's into the "projective" P's. For your question, you may write down an ideal $I \subseteq \mathbb{P}(F \otimes G)$ explicitely and verify that $\mathbb{P}(F) \times_S \mathbb{P}(G)$ satisfies the same universal property as $V(I)$. I have not tried it, but this should work. | |
| Jan 11, 2011 at 12:13 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
added 54 characters in body
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| Jan 11, 2011 at 9:31 | comment | added | Laurent Moret-Bailly | One approach to the question (in the Plücker case, say) could be to apply EGA 4 (18.12.6) (or (8.11.5), under finiteness assumptions): every proper monomorphism $X\to Y$ is a closed immersion. In fact it suffices to prove properness plus the fact that for $y\in Y$ the fiber $X_y$ is isomorphic to $\mathrm{Spec}\kappa(y)$. Properness can be deduced from the valuative criterion (if $\mathcal{F}$ is finitely generated you just prove properness of the Grassmannian). The other property is over a field. | |
| Jan 11, 2011 at 9:23 | comment | added | Laurent Moret-Bailly | The advantage of the local method is that it proves more: there is an open covering $(U_i)$ of the Grassmannian and open subsets $(V_i)$ of the projective space such that the Plücker map sends $U_i$to $V_i$ and the induced morphism has a retraction. | |
| Jan 11, 2011 at 6:55 | history | asked | Akhil Mathew | CC BY-SA 2.5 |