Torsors under finite and flat group schemes of rank <Emphasis Type="Italic">p</Emphasis> with Galois action

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 characteristic. In both cases we study the Galois action of the Galois group of the base field on these torsors. We also study the degeneration of _p -torsors, from characteritic 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 zero characteristics. Finally, for a proper and smooth curve X over a complete discrete valuation field, of inequal characteristic, which has good reduction, we show the existence of a canonical Galois equivariant filtration, on the first étale cohomology group of the geometric fibre of X, with values in _p .