  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.
