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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 $H^*(X_j, F_j) \to H^*(X_i, p_{ij}^*F_j) \to H^*(X_i, F_i)$).


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I don't know the answer to your question. But when I have seen this sort of situation come up in practice -- for example in work of Carayol/Langlands/Deligne on how Galois representations and automorphic forms on GL(2) are inter-related [the $X_i$ are then modular curves or Shimura curves of higher and higher levels] then here is what I have done: (1) I have not even cared about whether the projective limit of the schemes exists. (2) Whenever I have wanted an exact sequence of the sort that a cohomology theory spits out, I have used the fact that direct limits are exact and... – Kevin Buzzard May 11 '11 at 5:51
...just taken direct limits of long exact sequences of cohomology coming from each $X_i$. This has always sufficed for me. I don't know if it will suffice for you, and probably someone will answer your question anyway, but it's another way of thinking about things which seems to me to be useful in practice. – Kevin Buzzard May 11 '11 at 5:53
Note that even in the case when $X=X_i$ for all $i$ conditions are needed. It is true when $X$ is quasi-compact and may be false for an infinite disjoint union of spectra of fields. – Torsten Ekedahl May 11 '11 at 6:04

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