  We introduce notions of finite presentation and co-exactness which serve as
qualitative and quantitative analogues of finite-dimensionality for operator
modules over completely contractive Banach algebras. With these notions we
begin the development of a local theory of operator modules by introducing
analogues of the local lifting property, nuclearity, and semi-discreteness. For
a large class of operator modules we prove that the local lifting property is
equivalent to flatness, generalizing the operator space result of Kye and Ruan.
We pursue applications to abstract harmonic analysis, where, for a locally
compact quantum group $\mathbb{G}$, we show that $L^1(\mathbb{G})$-nuclearity
of $\mathrm{LUC}(\mathbb{G})$ and $L^1(\mathbb{G})$-semi-discreteness of
$L^\infty(\mathbb{G})$ are both equivalent to co-amenability of $\mathbb{G}$.
We establish the equivalence between $A(G)$-injectivity of $G\bar{\ltimes}M$,
$A(G)$-semi-discreteness of $G\bar{\ltimes} M$, and amenability of
$W^*$-dynamical systems $(M,G,\alpha)$ with $M$ injective. We end with remarks
on future directions.
