Hi,

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).

Thanks!

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.