Tests of multivariate independence may rely on asymptotically independent Cram\'er-von Mises statistics derived from a M\"obius decomposition of the empirical copula process. We generalize this approach to some other copula-based assumptions, with the help of a functional decomposition based on commuting idempotent maps. As soon as the null hypothesis reflects the stability of the copula under the action of the composition of such operators, the methodology applies. The empirical testing process, which depends on the decomposition, allows the derivation of a new family of test statistics. The asymptotic distributions are obtained. Since the latter depend on the unknown copula being tested, we adapt the subsampling procedure and the multiplier bootstrap to our setting and recall that the parametric bootstrap also applies to approximate p-values. The benefits in deriving test statistics from a functional decomposition are illustrated and discussed through simulations.