  In this note we study the geometry of torsors under flat and finite
commutative group schemes of rank p above curves in characteristic p and above
relative curves over a complete discrete valuation ring of inequal
characteristics. In bothe cases we study the Galois action of the Galois group
of the base field on these torsors. We also study degeneration of
$\mu_p$-torsors from characteristic 0 to characteristic p and show that this
degeneration is compatible with the Galois action. We then discuss the lifting
of torsors under flat and commutative group schemes of rank p from positive to
0 characteristics. finally, for a proper and smooth curve X over a complete
discrete valuation field of inequal characteristics we show the existence of a
canonical Galois equivariant filtration on the first etale cohomology group of
the geometric fibre of X with values in $\mu_p$. This apaper is the first of a
series of papers [Sa] and [Sa-1] where we compute the vanishing cycles arising
from degeneration of $\mu_p$-torsors above curves as well as the semi-stable
reduction of these torsors and the Galois action on them.
