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I am trying to understand the coniveau spectral sequence for the cohomology of a "big" regular scheme over a field. This involves cohomology with support at points, and I am getting some strange results when trying to compute this.

To fix ideas, assume that $X$ is a regular local scheme over the spectrum of complex numbers (say, something like the spectrum of a ring of formal power series; so, excellent noetherian of finite dimension), and the residue field of $X$ at the closed point $0$ is $\mathbb{C}$. I would like to compute the relative $\mathbb{Z}/l\mathbb{Z}$-etale cohomology for the pair $(X-\{0\},X)$.

Now, $X$ can be presented as the inverse limit of smooth $\mathbb{C}$-varieties $X_i$. Moreover, we can assume that $0$ lifts to $X_i$. Is it true that the relative ($\mathbb{Z}/l\mathbb{Z}$-etale) cohomology $(X-\{0\},X)$ is the limit of that for $(X_i-\{0\},X_i)$? Anyway, the latter pairs form an inductive system, and so one can pass to the limit in the corresponding long exact sequences (cf. Étale cohomology with support and functoriality). However, the relative cohomology of $(X_i-\{0\},X_i)$ is concentrated in the degree $2codim_{X_i}{0}$ according to the Gysin long exact sequence; hence the limit appears to be zero if the codimensions do not "stabilize" and to be $\mathbb{Z}/l\mathbb{Z}$ in the degree $2codim$ if they do. This conclusion seems to be very strange to me, so I would like to know whether my argument contains any errors and what is the correct answer.

Upd. So, it is probably impossible to present $(X-\{0_X\},X)$ as a limit of $(X_i-\{0_{X_i}\},X_i)$ for $0_{X_i}$ being $\mathbb{C}$-points (see the comments of nfds23). Still, does there exist a method for "computing" this relative cohomology (preferably by relating to the cohomology of some $\mathbb{C}$-varieties)?

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  • $\begingroup$ 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. $\endgroup$ – nfdc23 Feb 25 '17 at 15:27
  • $\begingroup$ Proposition 8.6.3 of EGA4.III appears to say that one can assume that $\{0\}_X\cong \{0_{X_i}\}\times_X_i X$. $\endgroup$ – Mikhail Bondarko Feb 25 '17 at 18:06
  • $\begingroup$ 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). $\endgroup$ – nfdc23 Feb 25 '17 at 19:27
  • $\begingroup$ So, the problem is that one cannot obtain $\mathbb{C}$-points this way? You are probably right; thank you! $\endgroup$ – Mikhail Bondarko Feb 25 '17 at 19:48
  • $\begingroup$ 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). $\endgroup$ – nfdc23 Feb 25 '17 at 20:14

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