Timeline for How do you define the Euler Characteristic of a scheme?
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Aug 21, 2010 at 22:11 | vote | accept | babubba | ||
| Aug 19, 2010 at 1:01 | answer | added | Dave Anderson | timeline score: 8 | |
| Aug 18, 2010 at 23:02 | answer | added | algori | timeline score: 6 | |
| Aug 12, 2010 at 4:36 | answer | added | JME | timeline score: 17 | |
| Aug 12, 2010 at 4:21 | comment | added | Torsten Ekedahl | @angoleirovero: There is a little bit more than (classical) GAGA. First there is Grothendieck's result that for any (proper or not) smooth variety the algebraic de Rham cohomology is equal to the analytic one. Second for singular ones one embeds the variety (when it is possible) into a smooth and takes the de Rham cohomology of the formal completion of the variety in the ambient one. | |
| Aug 11, 2010 at 23:25 | comment | added | babubba | @Torsten Ekedahl and Kevin Lin: Is using the algebraic de Rham complex to obtain the Euler characteristic just an application of GAGA or is there something I'm missing? | |
| Aug 11, 2010 at 23:21 | history | edited | babubba | CC BY-SA 2.5 |
corrected typo
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| Aug 11, 2010 at 7:58 | comment | added | Simon Pepin Lehalleur | Yet another approach to the finiteness of the Betti numbers (which proves much more). Complex algebraic varieties can be compactified in the following way : let $X$ be such a variety, then there exists $\bar{X}$ smooth projective in which $X$ is a dense Zariski open and $D=\bar{X}-X$ is a strict normal crossing divisor in $\bar{X}$. You can then relate the cohomology groups of $X$ to those of $\bar{X}$ and $D$ (which are finite dimensional) in De Rham theory by using logarithmic differential forms : see e.g. the treatment of this in Claire Voisin's book on Hodge theory. | |
| Aug 10, 2010 at 19:07 | comment | added | Torsten Ekedahl | 1) You can use étale cohomology to get a purely algebraic definition. (You can also use algebraic de Rham cohomology as Kevin suggests which has an extension to the singular non-proper case). 2) On way of seeing that the Euler characteristic is well-defined is to use the fact that $X$ is homotopy equivalent to a finite complex. (You can also use the finiteness theorem for cohomology of constructible sheaves.) | |
| Aug 10, 2010 at 18:46 | comment | added | Kevin H. Lin | If X is e.g. smooth projective you can use Hodge decomposition. | |
| Aug 10, 2010 at 18:26 | history | asked | babubba | CC BY-SA 2.5 |