Timeline for How to compute the first derived direct image along an open immersion, for the fppf sheaf represented by a multiplicative group?
Current License: CC BY-SA 3.0
11 events
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| Jul 20, 2012 at 14:29 | vote | accept | Heer | ||
| Jul 20, 2012 at 14:21 | comment | added | Heer | @Jason Starr: however i got confused with something else. I guess the relation between big fppf site and big etale site is: they have the same "opens", but in the fppf site you have more coverings, so it's stronger for a contravariant functor being a fppf sheaf that being an etale sheaf. And if the functor is already a fppf sheaf, then it has the same section as its direct image as an eatle sheaf. | |
| Jul 20, 2012 at 14:18 | comment | added | Heer | @ Jason Starr: I consider the big etale site, and I think now I am convinced. Thanks a lot. | |
| Jul 20, 2012 at 13:23 | comment | added | Jason Starr |
@Heer: Which sites are you working with? If you are working with the big sites, then I believe that $u_{Y*}R^pf_*\mathbb{G}_{m,fppf}$ has the same sections on every $Y$-scheme as $R^pf_*\mathbb{G}_{m,fppf}$. If you are working with the small \'etale site, then do you really care about the value of $R^pf_*\mathbb{G}_{m,fppf}$ except on \'etale $Y$-schemes?
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| Jul 19, 2012 at 20:43 | comment | added | Heer | @Jason Starr: Have I gotten your point? | |
| Jul 19, 2012 at 20:41 | comment | added | Heer | @Jason Starr: I see what you meant. But the natural map should be $R^p f_*\mathbb{G}_{m,\text{et}} \to u_{Y*}R^p f_*\mathbb{G}_{m,\text{fppf}}$, where $u_Y:Y_{\text{fppf}}\rightarrow Y_{\text{et}}$. Then $u_{Y*}R^p f_*\mathbb{G}_{m,\text{fppf}}=0$ provided $R^p f_*\mathbb{G}_{m,\text{et}}$, But this doesn't imply that $R^p f_*\mathbb{G}_{m,\text{fppf}}=0$ | |
| Jul 19, 2012 at 14:26 | comment | added | Jason Starr | ... To go from Grothendieck's theorem to the claim, use Prop. 5.1, Exp. V, SGA 4. | |
| Jul 19, 2012 at 13:23 | comment | added | Jason Starr |
I am not suggesting to apply Theorem 11.7 to $j_*\mathbb{G}_m$, but rather to $\mathbb{G}_m$, where condition (R) is tautological (the open subfunctor is the entire functor). Grothendieck proves that for any scheme $X$, for every integer $p$, the natural map $H^p(X_{\text{\'et}},\mathbb{G}_m) \to H^p(X_{\text{fppf}},\mathbb{G}_m)$ is an isomorphism. I claim that this implies that for every morphism $f:X\to Y$, also the natural map `$R^p f_*\mathbb{G}_{m,\text{\'et}} \to R^p f_*\mathbb{G}_{m,\text{fppf}}$ is an isomorphism, i.e., the derived etale and fppf pushforwards agree ...
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| Jul 18, 2012 at 23:07 | comment | added | Heer | @Jason Starr: what is the definition of an open subfunctor? | |
| Jul 18, 2012 at 23:06 | comment | added | Heer | @Jason Starr: thanks a lot for the reference. The condition (L) is satisfied obviously for $j_*\mathbb{G}_m$. the condition (R) is less obvious, in particular what should be the open subfunctor? Is it $\mathbb{G}_m$. | |
| Jul 17, 2012 at 2:57 | history | answered | Jason Starr | CC BY-SA 3.0 |