Torsors under finite and flat group schemes of rank p with Galois action

Abstract

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 $μ_p$-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 $μ_p$. This apaper is the first of a series of papers [Sa] and [Sa-1] where we compute the vanishing cycles arising from degeneration of $μ_p$-torsors above curves as well as the semi-stable reduction of these torsors and the Galois action on them.