The connection between amenability of a locally compact group G and injectivity of its group von Neumann algebra L(G) has been studied by many of the world's leading operator algebraists for decades. In this work we clarify this connection by showing the equivalence between amenability of G and 1-injectivity of L(G) as an operator module of the Fourier algebra A(G). In fact, we prove a corresponding result for all locally compact quantum groups \G, establishing at the same time new homological manifestations of quantum group duality, and a novel tool for the development of abstract harmonic analysis on locally compact quantum groups. We give several applications of our general duality result, including a proof that closed quantum subgroups of amenable quantum groups are amenable, and a simplified proof that co-amenability and amenability of the dual are equivalent for compact quantum groups which avoids the use of modular theory, suggesting a potential avenue for generalization beyond the compact setting. We also introduce a notion of inner amenability for locally compact quantum groups and study its connection to relative injectivity, establishing further homological manifestations of duality which not only help to elucidate previously known results, but provide new approaches to open problems concerning the operator homology of A(G).