As far as I understand these matters, for a regular $\mathfrak{X}$ that is proper flat of finite type over $\operatorname{Spec}\mathbb{Z}_p$, the Rappoport-Zink spectral sequence relates the etale cohomology of the special fiber of $\mathfrak{X}$ with the one of its generic fibre $X$ in the case when the former is a normal crossing divisor in $\mathfrak{X}$.

On the other hand, $X$ is just the complement of the special fibre to the whole $\mathfrak{X}$. Hence quite a similar spectral sequence exists for quite 'elementary' reasons. Yet it has a striking distinction from the Rappoport-Zink one: it 'has one more term'. It takes into account the etale cohomology of $\mathfrak{X}$ itself (more precisely, the cohomology of the base change of $\mathfrak{X}$ to the spectrum of the integral closure of $\mathbb{Z}_p$ in the algebraic closure of $\mathbb{Q}_p$; one may say that $\mathfrak{X}$ is a compactification of $X$). By proper base change, this is isomorphic to the cohomology of the special fibre (I was silly not to notice this when I wrote the original question); so this is the cohomology of the whole divisor and not of its smooth components. So, my question is: what is the relation between these spectral sequences; could they be isomorphic (possibly, starting from $E_2$ or after some 'modification that kills the extra term')? Are there any easy references or examples for this matter?

PS. Most probably, instead of $\mathbb{Z}_p$ one could consider $\mathbb{C}[[t]]$ here.

Upd. It seems that one should add the vanishing cycles functor to this picture (somehow). Any hints or references would be very welcome!


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