Galois closure of essentially finite morphisms

Abstract

a b s t r a c tLet X be a reduced connected k-scheme pointed at a rational point x ∈ X (k). By using tannakian techniques we construct the Galois closure of an essentially finite k-morphismthis Galois closure is a torsor p :X Y → X dominating f by an X -morphism λ :X Y → Y and universal for this property. Moreover, we show that λ :X Y → Y is a torsor under some finite group scheme we describe. Furthermore we prove that the direct image of an essentially finite vector bundle over Y is still an essentially finite vector bundle over X . We develop for torsors and essentially finite morphisms a Galois correspondence similar to the usual one. As an application we show that for any pointed torsor f : Y → X under a finite group scheme satisfying the condition H 0 (Y , O Y ) = k, Y has a fundamental group scheme π 1 (Y , y) fitting in a short exact sequence with π 1 (X, x).

On the “Galois Closure” for Finite Morphisms

“…Our main result, Theorem 2.11, gives necessary and sufficient conditions for the existence of a finite Galois closure Y as in (1). In contrast with theétale case, these conditions are of a global nature, as can be expected from the counterexample above.…”

“…When S is a connected and reduced scheme, proper over k, Antei and Emsalem [1] have introduced another class of finite flat morphisms X → S that can be dominated by a finite torsor. Their construction is based on the tannakian approach to Nori's fundamental group scheme ( [11], chapter I).…”

“…As a matter of fact, the main result of [1] is much more precise: it describes the actual "Galois group" of X/S as the quotient of π(S/k; s) determined by π * O X , as in proposition 3.2.…”

“…In this note, we start with a finite flat morphism π : X → S of schemes of characteristic p > 0 and we try to find a "Galois closure" as in diagram (1), where g and h are torsors under group schemes G and H ≤ G defined over the prime field F p .…”

“…Antei and M. Emsalem have introduced in [1] another class of finite flat morphisms (called essentially finite), admitting a Galois closure. Their construction is based on Nori's tannakian approach to the fundamental group scheme: it is thus restricted to reduced schemes proper over a field, but provides a description of the Galois group.…”