A unital commutative Banach algebra with a maximal ideal M of codimension 1 and a Banach A-module X such that H^{2}(A, X) = 0 but H^{2}(M, X) ≠ 0.

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Let A = C^{2} with the product (a, b)(a′, b′) = (aa′, ab′ + a′b). M = {0} ⊕ C, being the kernel of the character ϕ : A ⟶ C defined by ϕ(z, w) = z, is a maximal ideal of codimension 1. Regard X = C as an annihilator A-module. By [B&D&L, Proposition 2.2],

H^{2}(A, X) = {0}. If μ((0, w_{1}), (0, w_{2})) = w_{1}w_{2}, then μ ∈ Z^{2}(M, X), but μ ∉ N^{2}(M, X) ( otherwise w_{1}w_{2} = μ((0, w_{1}), (0, w_{2})) = (δ^{1}λ)((0, w_{1}), (0, w_{2})) = (0, w_{1}).λ((0, w_{2})) − λ((0, w_{1}).(0, w_{2})) + λ(0, w_{1}).(0, w_{2}) = 0 − λ(0) + 0 = 0, for all w_{1}, w_{2} ∈ C, a contradiction). Thus H^{2}(M, X) ≠ 0.

Ref. [B&D&L] W.G. Bode, H.G. Dales and Z. Lykova, Algebraic and strany splittings of extensions of Banach algebras, mem. Amer. Math. Soc. 137 (1999).