PRIME TO p ETALE COVERS OF ALGEBRAIC GROUPS AND HOMOGENEOUS SPACES

PRIME-TO-p ETALE COVERS OF ALGEBRAIC GROUPS AND HOMOGENEOUS SPACES MICHEL BRION AND TAMAS SZAMUELY 1. Introduction By a classical result of Schreier [23], the fundamental group of a connected and locally connected topological groupG is commutative. If moreover G is (semi-)locally simply connected, then every Galois cover ? : Y ? G carries a group structure for which ? is a homomorphism. Thus G is the quotient of Y by an abelian normal subgroup. The first part of the following proposition states an analogue of this result in algebraic geometry. The second part gives a bound on the number of topological generators of the prime-to-p fundamental group. To state it, we need to introduce some notation. Recall that by Cheval- ley's theorem G is an extension of an abelian variety A by a linear algebraic group Gaff (see [3], [4], [5], [21]). Denote by g the dimension of A and by r the rank of Gaff (which is by definition the dimension of a maximal torus). Furthermore, denote by Z(p?) the direct product of the rings Z for 6= p. Proposition 1.1. Let G be a connected algebraic group over an alge- braically closed field of characteristic p ≥ 0. a) Every etale Galois cover Y ? G of degree prime to p carries the structure of a central isogeny.

• abelian varieties suffices


• every etale


• h˜ ?


• group


• induced map


• primary torsion


• commutative algebraic