Suppose that I have a nice variety *X* over ℚ_{p}, with good reduction if you like, and a nice sheaf on *X*, say coming from a smooth group scheme *G*. I can cover *X* by some p-adic open sets *U _{α}*, for example the mod-p neighbourhoods coming from some model $\mathcal{X}$ of

*X*. Clearly I can't expect to use Čech cohomology in a naïve way to compute the

*H*in terms of the cohomology of the

^{i}(X,G)*U*, because they don't overlap. But the information about how they fit together to make

_{α}*X*is instead contained in the geometry of the special fibre of $\mathcal{X}$.

Is there a spectral sequence which calculates

Hin terms of some sort of cohomology of the^{i}(X,G)U, and some information about how they fit together?_{α}

## Motivation

In the situation described above, the Leray spectral sequence gives $$ 0 \to H^1(\mathcal{X},j_*G) \to H^1(X,G) \to H^0(\mathcal{X}, R^1 j_* G)$$ where $j\colon X \to \mathcal{X}$ is the inclusion of the generic fibre.

So in this situation I can compute *H ^{1}(X,G)* in terms of: $R^1 j_* G$, which I want to think of as somehow being the cohomology of the p-adic discs covering

*X*; and the cohomology of $\mathcal{X}$, which I want to think of as saying how those discs fit together.

I would like to see how to generalise this to smaller p-adic neighbourhoods.

Supplementary question:

Should I just go away and read a book on rigid cohomology?