Timeline for Why does the naive definition of compactly supported étale cohomology give the wrong answer?
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 1, 2011 at 22:08 | comment | added | shenghao | If you would like to have a definition of $f_!$ without using a compactification, you may want to follow the approach of Laszlo-Olsson. As Laumon observed, the dualizing complex is "local", so one may apply the glueing lemma in BBD to define the dualizing complex on a (not even necessarily separated) scheme, say locally of finite type over some nice base, and define $f_!,$ using Poincare duality, to be $Df_*D.$ This applies for non-separated $f$ too. Laszlo-Olsson did this for alg. stacks, most of which are not separated and hence cannot be compactified. | |
| Apr 12, 2010 at 1:49 | vote | accept | Sam Derbyshire | ||
| Apr 12, 2010 at 1:35 | comment | added | BCnrd | It is instructive to prove that for analytification of separated finite-type schemes over the complex numbers, Grothendieck's "alternative" formulation gives "the same" topological theory (respecting all structures). More amusingly, the analogue of the Artin comparison theorem (for proper supports) was proved by Berkovich over non-arch. fields using analytic spaces in his sense, where on his analytic side he uses derived functor of "sections with proper support". Very nice! (He gets P. duality too.) For Huber's adic spaces, there are compactification issues on the analytic side. | |
| Apr 12, 2010 at 1:21 | answer | added | JS Milne | timeline score: 16 | |
| Apr 11, 2010 at 23:59 | history | asked | Sam Derbyshire | CC BY-SA 2.5 |