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Let $K$ be a local field of characteristic $p\>0$. The aim of this paper is to describe the ramification groups for the pro-$p$ abelian extensions over $K$ with regards to the Artin-Schreier-Witt theory. We shall carry out this investigation entirely in the usual framework of local class field theory. This leads to a certain non-degenerate pairing that we shall define in detail, generalizing in this way the Schmid formula to Witt vectors of length $n$. Along the way, we recover a result of Brylinski but with...