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We generalize to webs of any codimension results already known in codimensionone. Given a holomorphic $d$-web $\cal W$ of codimension $q$ $(q\leq n-1)$ inan ambiant $n$-dimensional holomorphic manifold $U$, we define for any integer$p$ $(1\leq p\leq q)$ the condition for such a web to be \emph{$p$-ordinary}$($resp. \emph{strongly $p$-ordinary}$)$. If this condition is satisfied, wethen prove that its $p$-rank $r_p({\cal W})$ $\bigl($resp. its closed $p$-rank$\widetilde r_p({\cal W})\bigr)$, i.e. the maximal dimension of the vectorspace of the germs of $p$-abelian relations $($resp. of closed $p$-abelianrelations$)$ at a point $m$ of $U$, is finite. We then give an upper-bound$\pi_p^0(n,d,q)$ $\bigl($resp. $\pi'_p(n,d,q)\bigr)$ for these ranks.  Moreover, for some values of $d$, and we then say then that the web is\emph{$p$-calibrated} $($resp. \emph{strongly $p$-calibrated}$)$, we define atautological holomorphic connection on a holomorphic vector bundle of rank$\pi_p^0(n,d,q)$ $\bigl($resp. $\pi'_p(n,d,q)\bigr)$, for which the sectionswith vanishing covariant derivative may be identified with $p$-abelianrelations $($resp. closed $p$-abelian relations$)$. The curvature of thisconnection is then an obstruction for the rank $r_p({\cal W})$ $\bigl($resp.$\widetilde r_p({\cal W})\bigr)$ to be maximal.  The main change is the correction of a mistake $($proposition 4, section6-5$)$ in the first version : the 1-rank of the concerned web is not 0 as weclaimed, but 1. However, the important corollary remains true : even at thelevel of germs, some 2-abelian relation exhibited by Goldberg in $ [G]$ on someweb of codimension 2 in an ambiant space of dimension 4, is the coboundary ofnone 1-abelian relation. The section 7, devoted to this correction, is selfcontent, not depending on the previous results of the paper.

Relations ab\'eliennes des tissus ordinaires de codimension arbitraire