Tests of multivariate independence may rely on asymptotically independent Cramér-von Mises statistics derived from a Möbius 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 asymptotic joint distribution of the terms in the decomposition of the empirical copula process is established under the null hypothesis. Since the latter depends on the unknown copula being tested, we adapt the subsampling procedure to our setting and recall that the multiplier bootstrap as well as the parametric bootstrap also apply to approximate p-values. The benefit in deriving test statistics from a functional decomposition, defined in accordance with the dependence assumption under study, are illustrated and discussed through simulations.