Flat topological modules. Surprises of the nonmetrizable case

Flat Banach modules, which were introduced by A. Ya. Helemskii in 1971, are by now classical and relatively well-studied objects. They are important, in particular, because of their relation to amenability. In contrast, very little is known about flatness in the more general context of locally convex topological modules. We begin by showing that the "naive" generalization of the notion of a flat Banach module, while being reasonable for Fréchet modules, is no longer convenient in the nonmetrizable case. In particular, we give an example of a nonflat (in the "naive" sense) topological module over an amenable Banach algebra, a situation impossible in the context of Banach (or Fréchet) modules. Next we suggest a modified definition of flatness, and we show how it works in concrete situations. As an application, we give a characterization of amenable Köthe co-echelon algebras obtained in our recent paper with Krzysztof Piszczek. If time permits, we also briefly discuss an abelian extension of the Tor functor that is compatible with our new notion of a flat module.