Wild ramification and a vanishing cycles formula

In this paper we prove an explicit formula which compares the dimensions of the spaces of vanishing cycles in a Galois cover of degree p between formal germ of curves over a complete discrete valuation ring of inequal characteristics (0,p). This formula can be easily generalised to the case of a Galois cover with group which is nilpotent or which has a normal p-Sylow subgroup. The results of this paper play a key role in [Sa-1] where is studied the semi-stable reduction of Galois covers of degree p above semi-stable curves over a complete discrete valuation ring of inequal characteristics (0,p), as well as the Galois action on these covers.