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)?

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)?

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 seties), 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.

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)?