Weight spectral sequences and independence of

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 → 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 an element σ ∈ WK , let n(σ) denote the integer such that the image of σ in GF is Fr n(σ) F . For a scheme XK of finite type over K, the -adic etale cohomology H(XK̄ ,Q ) is an -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 ,Q ). For an algebraic correspondence Γ ∈ CH(XK ×K XK) on XK , the endomorphism Γ ∗ of H(XK̄ ,Q ) is defined as pr1∗ ◦ ([Γ]∪ ) ◦ pr∗ 2.