Correction to “Vanishing cycles and nonclassical parabolic cohomology”

Stefan Wevers has kindly pointed out to me that the deduction of Corollary 2.16 from Theorem 2.15 is unjustified. Here is an argument which fills this gap. (Wevers has also found a proof of a much more general result than this one.) I claim that in the diagram in the statement of Theorem 2.15

0 ⟶ H^{0}(\overline{Z}, ℚ_{ℓ}) ⟶ H_{c}^{1}(\overline{U}, Sym^{k} F) ⟶ H^{1}(\overline{X}, g_{∗} Sym^{k} F) ⟶ 0
↓  ↓^{β}  ↓^{γ}
0 ⟶  A  ⟶  B  ⟶  C  ⟶ 0

the right-hand surjection γ admits an inertia-equivariant splitting δ: C → H^{1}(\overline{X}, g_{∗}G). Granted this, the corollary is immediate: if σ: H^{1}(\overline{X}, g_{∗}G) → H_{c}^{1}(\overline{U}, G) is a splitting of the middle row, then β ∘ σ ∘ δ splits the map B → C, contradicting the fact that B is indecomposable. To prove the claim, we first replace K by its completion at v. Put Xv̄ = X ⊗ \overline{k(v)} and G = Sym^{k} F. Consider the maps (with notation as in the proof of 2.15)

H_{x}^{1}(Xv̄, RΨg_{∗}G) → H^{1}(Xv̄, RΨg_{∗}G) = H^{1}(\overline{X}, g_{∗}G) → (R^{1}Φg_{∗}G)x̄ = C

which are evidently Gal(K̄_{v}/K_{v})-equivariant. To obtain the desired splitting, I shall show that their composite is an isomorphism. As this is local around x we can replace X by a neighbourhood of x, and therefore assume that X is smooth and that F is lisse on U. From 1.7 and 1.9 we know R^{1}Φg_{∗}G ≃ x_{∗}(Gξ̄). We need to compute Hq ; x̄(R^{0}Ψg_{∗}G). From the definition of the nearby cycles functor and the assumptions just made, it is the subsheaf of g_{∗}(G_{v}) whose restriction to U_{v} is G_{v} and whose fibre at x is the space of invariants of ℤ_{ℓ}(1)^{2} acting on Gξ̄. But as in 2.19 this is the same as the space of local invariants g_{∗}(G_{v})_{x}; thus R^{0}Ψg_{∗}G = g_{∗}(G_{v}). Now recall that for any constructible ℚ_{ℓ}-sheaf F on a strictly henselian trait S of residue characteristic prime to ℓ we have (in the usual notation) H_{s}^{1}(S, F) = coker[F_{s} → F I ; η̄] and H_{s}^{2}(S, F) = Fη̄(−1)_{I}. In the present case this gives

H1 ; x̄(R^{0}Ψg_{∗}G) = 0 and H2 ; x̄(R^{0}Ψg_{∗}G) = Gξ̄(−1)_{ℤ_{ℓ}(1)}.

The “Leray” spectral sequence E_{2}^{ab} = H_{x}^{a}(R^{b}Ψ) ⇒ H_{x}^{∗}(RΨ) then becomes a short exact sequence

0 → H1 ; x̄(Xv̄, RΨg_{∗}G) → (R^{1}Ψg_{∗}G)x̄ →^{τ} Gξ̄(−1)_{ℤ_{ℓ}(1)}

Now by the theory of the Tate curve, the representation Gξ̄ = Sym^{k} Fξ̄ of the local Galois group at x is indecomposable, and its monodromy filtration has successive quotients ℚ_{ℓ}(−i), i ∈ {0, 1, …, k}. Therefore Gξ̄(−1)_{ℤ_{ℓ}(1)} ≃ ℚ_{ℓ}(−k − 1). Therefore τ must vanish, as required.