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Suppose that $F$ is a local field, $G_F$ its Galois group, $I$ the inertia subgroup, $k$ its residue field. Let $X$ be a finite type scheme over $k$. Let $C$ be a constructible $l$-adic sheaf on $X$ equipped with an action of $G_F$ compatibly with the action on $X_{\overline k}$ via $G_F\to G_k$.

Is it true that there exists a filtration $0 = C_0 \subset \dots \subset C_n = C$ such that $I$ acts through a finite quotient on the associated graded?

The statement when $X = Spec k$ is well-known (Grothendieck monodromy theorem).


Answer to Olivier's comment: As you mention, the object of interest is $H^*_{et}(X\otimes_k\overline k,C)^I$. I would like to have the result on the level of sheaves for some intermediate manipulations in my argument.

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This question looks very interesting to me, but there are a few things I don't quite understand. Are you looking at the $G_{F}$ action the étale cohomology of $X$ with coefficients in $C$? Would you give an example of the typical situation you have in mind? – Olivier Feb 20 '13 at 8:57

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