Hello,

Suppose that $X_i$ is a projective system of schemes and $F_i$ is a compatible projective system of abelian sheaves on the $X_i$ (i.e. if $p_{ij} : X_i \to X_j$ is the transition map, then we are given maps $F_j \to {p_{ij}}_*F_i$ satisfying some cocycle condition.

Suppose that $X = \varprojlim X_i$ and $F = \varprojlim F_i$ exist.

Question: Under what conditions do we have that $H^\*(X, F) = \varinjlim H^*(X_i, F_i)$?

(the maps above are given by $H^*(X_j, F_j) \to H^*(X_i, p_{ij}^*F_j) \to H^*(X_i, F_i)$).

Thanks