Injectivity and flatness of semitopological modules

The spaces D, S and E’ over $\mathbb{R}$^(n) are known to be flat modules over A=$\mathbb{C}$[\partial_{1},…,\partial_{n}], whereas their duals D’, S’ and E are known to be injective modules over the same ring. Let A be a Noetherian k-algebra (k=$\mathbb{R}$ or $\mathbb{C}$). The above observation leads us to study in this paper the link existing between the flatness of an A-module E which is a locally convex topological k-vector space and the injectivity of its dual. We show that, for dual pairs (E,E’) which are (K) over A–a notion which is explained in the paper–, injectivity of E’ is a stronger condition than flatness of E. A preprint of this paper (dated September 2009) has been quoted and discussed by Shankar.