Weight spectral sequences and independence of l

Let K be a complete discrete valuation field with finite residue field F of order q. We call such a field a local field. The geometric Frobenius FrF is the inverse of the map a 7→ a in the absolute Galois group GF = Gal(F̄ /F ). The Weil group WK is defined as the inverse image of the subgroup 〈FrF 〉 ⊂ GF by the canonical map GK = Gal(K̄/K) → GF . For a scheme XK of finite type over K, the l-adic etale cohomology H(XK̄ ,Ql) is a l-adic representation of the absolute Galois group GK . For σ ∈ GK , the right action σ on XK̄ = X ⊗K K̄ induces the left action σ∗ = (σ ) on H(XK̄ ,Ql).