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We introduce notions of finite presentation and co-exactness which serve asqualitative and quantitative analogues of finite-dimensionality for operatormodules over completely contractive Banach algebras. With these notions webegin the development of a local theory of operator modules by introducinganalogues of the local lifting property, nuclearity, and semi-discreteness. Fora large class of operator modules we prove that the local lifting property isequivalent to flatness, generalizing the operator space result of Kye and Ruan.We pursue applications to abstract harmonic analysis, where, for a locallycompact quantum group $\mathbb{G}$, we show that $L^1(\mathbb{G})$-nuclearityof $\mathrm{LUC}(\mathbb{G})$ 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 remarkson future directions.

Finite presentation, the local lifting property, and local approximation properties of operator modules