Timeline for Can one compute the (etale) cohomology with support at a point for a "big" regular $k$-scheme via limit arguments?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Feb 26, 2017 at 0:18 | comment | added | nfdc23 | @WillSawin: Yeah, I'm now doubtful one can expect to get such smoothness near 0 on $X_j$'s for big $j$ in that way, and it would be best to avoid needing that. I don't think one can take maps to be injective in this Artin-Popescu approximation process, and the ncd condition seems like it will be destroyed when pulled back to $X_j$'s for $j > i$. Hence, I don't quite see how this will bypass the problem. Well, it is a good question to ponder! | |
| Feb 25, 2017 at 23:48 | comment | added | Will Sawin | @nfdc23 The Gysin sequence argument probably doesn't need smoothness to work, so this might be overkill here, but it would still be good to have for other purposes. | |
| Feb 25, 2017 at 23:46 | comment | added | Will Sawin | @nfdc23 Should that be true? I would expect instead that we could modify the $X_i$, preserving the inverse limit, so that for each $i$ the zero-scheme of $t$ is smooth. One could do something like take a resolution to make the zero scheme of $t$ into a divisor with normal crossings, then the map from $\operatorname{Spec} \mathbb C[[t]]$ must intersect the divisor in a smooth component, and you can take an étale neighborhood of that component. The ring of sections still maps to $\mathbb C[[t]]$ and you can take the map to be injective so it seems to me like the inverse limit is preserved. | |
| Feb 25, 2017 at 23:07 | comment | added | nfdc23 | @WillSawin: To show it descends to a smooth subscheme of the appropriate codimension it would suffice to show that if we descend a member of a regular system of parameters in the complete regular local limit $R$ to a function $t$ on some $X_i$ then the zero-scheme of $t$ on $X_j$ is $\mathbf{C}$-smooth (equivalently, regular) near 0 for some $j \ge i$. This is exactly that $\{{\rm{d}}t\}$ freely spans a direct summand of $\Omega^1_{X_j/\mathbf{C},0}$. Initially I thought this would be easy, but even for $R=\mathbf{C}[\![t]\!]$ (using "general" $(X_i)$) it seems not so clear. Huh. | |
| Feb 25, 2017 at 21:45 | comment | added | Will Sawin | To make your updated argument work, is it enough that $0$ descend to a smooth subscheme of the appropriate codimension? Then the Gysin step should still work. Presumably one can rig that up by essentially the same argument... | |
| Feb 25, 2017 at 21:20 | history | edited | Mikhail Bondarko | CC BY-SA 3.0 |
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| Feb 25, 2017 at 20:14 | comment | added | nfdc23 | Exactly, that is the problem. A nice application of descending a closed point to a divisor comes up in Deligne's beautiful "arithmetic" proof of the semistable reduction theorem for abelian varieties in the appendix to Expose I of SGA7 (where he "spreads out" a discrete valuation ring to a finite type Z-algebra, descending the closed point to a divisor). | |
| Feb 25, 2017 at 19:48 | comment | added | Mikhail Bondarko | So, the problem is that one cannot obtain $\mathbb{C}$-points this way? You are probably right; thank you! | |
| Feb 25, 2017 at 19:27 | comment | added | nfdc23 | No, that result in EGA IV$_3$ does not say what you think it says. It only says that (for big enough $i$) one can descend $\{0\}$ to some (finitely presented) closed subscheme (which is rather easy to see directly anyway in your case: just pick $i$ big enough so that the coordinate ring of $X_i$ "contains" the element cutting out $\{0\}$ in the limit). It does not say that this closed subscheme is a $\mathbf{C}$-point, and one cannot expect it to be so (for the reason indicated in my first comment concerning why removing $\{0\}$ at each stage is typically not an inverse system). | |
| Feb 25, 2017 at 18:06 | comment | added | Mikhail Bondarko | Proposition 8.6.3 of EGA4.III appears to say that one can assume that $\{0\}_X\cong \{0_{X_i}\}\times_X_i X$. | |
| Feb 25, 2017 at 15:27 | comment | added | nfdc23 | It doesn't seem that $(X_i - \{0\}, X_i)$ forms an inverse system; plenty of $\mathbf{C}$-points at a later stage can map to 0 at the $i$th stage. To get an inverse system you must remove the entire "divisor" corresponding to a descent to some $X_i$ of the equation cutting out $\{0\}$ in the limit (and remove the preimage of that same divisor in all later stages). In general the formation of etale cohomology commutes with limits for any inverse system of qcqs schemes with affine transition maps. | |
| Feb 25, 2017 at 15:00 | history | asked | Mikhail Bondarko | CC BY-SA 3.0 |