It is a jointwork with I. Mondello (Creteil) and D. Tewodrose (Bruxelles). J. Cheeger and A. Naber have shown that if \((X, d)\) is a Gromov-Hausdorff limit of a sequence of complete Riemannian manifold with uniformly bounded Ricci curvature and a non collapsing hypothesis, then \((X, d)\) is a \(C^{1,\alpha}\) Riemannian manifold outside an open set of codimension 4. We have obtained a generalisation of this result under a Kato condition on the Ricci tensor.