On the dual of a flat module in TOP

Let k be a field, and A a k-algebra. In the category of A-modules, the dual of a (faithfully) flat module is a (cogenerating) injective module. Theorems of Malgrange and Palamodov suggest that this might be true also in the category of topologicalA-modules when A is the C-algebra C[∂1,…,∂n] (Bourlès and Oberst [1,2]). This note presents a counter-example, namely of a flat topological A-module whose topological dual is not injective, but which again is flat.