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PRIME TO p ETALE COVERS OF ALGEBRAIC GROUPS AND HOMOGENEOUS SPACES

14 pages
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


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IMIFA-OH-p AO2EA4HVAMNHB2ECA3M2I4 CMHPIN 2G5 DHFHCAGAHPN NI24AN MICHEL BRION AND TAM AS SZAMUELY
1. IUfdaGgFfiaU ByaclassicalresultofSchreier[23],thefundamentalgroupofa connected and locally connected topological group G is commutative. If moreover G is(semi-)locallysimplyconnected,theneveryGaloiscover ϕ : Y ! G carries a group structure for which ϕ is a homomorphism. Thus G is the quotient of Y by an abelian normal subgroup. Therstpartofthefollowingpropositionstatesananalogueofthis 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 G a (see [3], [4], [5], [21).] Denote by g the dimension of A and by r the rank of G a is by definition the dimension of (which amaximaltorus.)Furthermore,denoteby Z ( p 0 ) the direct product of the rings Z for ̸ = p . IZVpVa i tVU´.´. Let G be a connected algebraic group over an alge-braicallyclosed eldofcharacteristic p  0 . a) EveryetaleGaloiscover Y ! G of degree prime to p carries the structure of a central isogeny. In particular, the maximal prime-to-p quotient  ) ) of theetale fundamental group of (1 p 0 ( G 0 G is commuta p tive.ntof Z 2 gp 0 +) r . ( b) The group  1) ( G ) is a quotie ( By convention, all Galois covers are assumed to be connected. State-ment a ) was known in the special cases where G is an abelian variety (Lang{Serre [11)] or G isane(Miyanishi[15],Magid[13.)]Notethat restricting to prime-to-p covers is crucial because if one allows p -covers in characteristic p > 0, the statement fails disastrously already for the additive group C a (Raynaud[20.]) In part b ) the bound is sharp (take the direct product of an abelian varietyandatorus.)Inthecasewhen G is commutative it follows from ( p 0 ) ( Theorem 1.1 and ([25], x 6.4,Cor.4)that  1 G ) is moreover a free Z ( p 0 ) -module. However, in the non-commutative case it may contain torsion. For instance,  1 (SO( n ) in characteristic 0 is of order 2. (Of 9LkR : April 18, 2012. ´
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