The previous definition is directly inspired from [BGS95], where closed forms and currents are defined in the non-Archimedean setting. We choose however to work modulo numerical equivalence instead of rational equivalence. One justification for this choice is Corollary 4.5 below. The fact that each space $N^{1}(\mathcal{X}/S)$ is endowed with a natural topology as a finite dimensional vector space is another reason.