Wild ramification and a vanishing cycles formula

In [Duke Math. J. 55 (1987) 629–659] K. Kato proved, using techniques from K-theory, a formula which compares the dimensions of the spaces of vanishing cycles in a finite morphism between formal germs of curves over a complete discrete valuation ring. To the best of my knowledge Katoʹs formula is explicit only in the case where this morphism is generically separable on the level of special fibres. In this note we prove, using formal patching techniques à la Harbater, an analogous explicit formula in the case of a Galois cover of degree p between formal germs of curves over a complete discrete valuation ring of unequal characteristic (0,p) which includes the case where we have inseparability on the level of special fibres. The results of this paper play a key role in [math.AG/0106249] where is studied the semi-stable reduction of Galois covers of degree p of curves over a complete discrete valuation ring of unequal characteristics (0,p), as well as the Galois action on these covers.