ALMOST ASTERISK-MODULES NEED NOT BE FINITELY GENERATED

We construct examples of infinitely generated almost -modules, i.e. o f infinitely generated small modules satisfying the Colpi's condition C(kappa) for all kappa < N0. We also prove that the examples are shrin kable (in the sense of Facchini and Salce). Hence, they do not satisfy C(N0), and they are not -modules. This implies that the possible sol ution of the open problem whether each -module is finitely generated depends substantially on the validity of C(kappa) for infinite cardina ls kappa.