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. http://mathoverflow.net/questions/164980/%C3%89tale-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.