The geometry of projective, injective, and flat Banach modules

In this paper, we prove general facts on metrically and topologically projective, injective, and flat Banach modules. We prove theorems pointing to the close connection between metric, topological Banach homology with the geometry of Banach spaces. For example, in geometric terms we give a complete description of projective, injective, and flat annihilator modules. We also show that for an algebra with the geometric structure of an $\mathscr{L}_1$- or $\mathscr{L}_\infty$-space all its homologically trivial modules possess the Dunford–Pettis property.