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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

We already said in

We already said in Section 4.1 that C is equivalent to the category of topologicalcovers of X. We identify C with C {1} where {1} denotes the trivial group.We recall that an object Y of a category in which coproducts exist is calledconnected if it is not isomorphic to a coproduct Y 1∐Y2 where Y 1 and Y 2 areobjects of this category not isomorphic to the initial object.For an arbitrary discrete group π 1 we define the following category:C(π 1 ) π 1 -setsThe objects are discrete sets with a left or right action of the group π 1 .The morphisms are π 1 -equivariant maps between these sets.We warn the reader that when discussing the fundamental group π 1 in algebraicgeometry one often considers sets with left π 1 -actions whereas in topology oneusually considers sets with right π 1 -actions. Therefore the author decided to berather pedantic about this point, especially after he was trapped when he was notpaying attention to it.We choose a universal cover ˜p : ˜X → X of X in the sense of topology. Wedenote by Aut(˜X) the group of deck transformations of ˜X over X. Then we defineπ top1 (X, ˜X) to be the opposite group to Aut( ˜X). Given a (connected) cover p :Y → X the group Aut(˜X) acts from the right on the set of cover morphismsHom(˜X, Y). Hence there is a left π top1 (X, ˜X)-action on this set. This defines afibre functor from the category C to the category of sets with a left π top1 (X, ˜X)-action and makes C into a Galois category.Here we have to relax Grothendieck’s terminology a little bit: We also allowquotients by discrete groups rather than only finite ones. Also we assume that thefibre functor maps to the category of discrete sets with a group action of a discretegroup rather than only to the category of finite sets together with a continuousaction of a profinite group.Conversely, given a fibre functor F there is always a group π top1 (X, F ) calledthe automorphism group of the functor F . A map between two covers Y 1 and Y 2over X is uniquely determined by the π top1 (X, F )-equivariant map from F (Y 1 )to F (Y 2 ). The main content of Galois theory and the theory of the fundamentalgroup in this setup is that a fibre functor induces an equivalence of categoriesbetween C and C(π 1 ).The connection with the fundamental group defined via loops is as follows:We let F x0 (Y) := p −1 (x 0 ) be the fibre of p : Y → X above a point x 0 ofX. Lifting loops based at x 0 to paths in Y defines a right action of the “loop“-fundamental group π top1 (X, x 0 ) on the set F x0 (Y). Now we fix a point ˜x 0 on thefibre F x0 (˜X) of a universal cover. Then we compare the left Aut( ˜X)-action withthe right π top1 (X, x 0 )-action in this point: For every automorphism ϕ there is aunique element γ in the fundamental group such that ϕ · ˜x 0 = ˜x 0 · γ. This defines32

an isomorphism between these two groups that depends on the choice of ˜x 0 . Infact, given another point ˜x ′ 0 of F x 0(˜X) there is a unique element α ∈ π top1 (X, x 0 )such that ˜x ′ 0 = ˜x 0 · α. We then computeϕ · ˜x ′ 0 = ϕ · (˜x 0 · α) = (ϕ · ˜x 0 ) · α = (˜x 0 · γ) · α = ˜x ′ 0 · (α−1 γα).That is, with respect to ˜x ′ 0 we obtain an isomorphism of π top1 (X, x 0 ) with Aut(˜X)that differs from the isomorphism with respect to ˜x 0 by conjugation with α.If we fix a point ˜x 0 of ˜X in the fibre ˜p −1 (x 0 ) we can identify Hom(˜X, Y) withthe set F x0 (Y) by associating to a morphism ϕ : ˜X → Y the point ϕ(˜x 0 ). Underthis identification the right action of π top1 (X, x 0 ) on F x0 (Y) becomes a left actionon Hom(˜X, Y) and it is this point where the group actions change their side whenpassing from topology to algebraic geometry and vice versa.For a cover p : Y → X we recover the group of its automorphisms as follows:The group Aut(˜X) acts on Hom(˜X, Y). We choose a point on this latter set,i.e. we choose a map from ˜X to Y, and denote by H the subgroup of Aut( ˜X)stabilising this point. This identifies Y with the quotient H\ ˜X. An element ofAut(˜X) induces an automorphism of Y if and only if it normalises H. Since theelements acting trivially on Y are precisely those of H we get an isomorphismbetween the group of cover automorphisms of Y over X and NH/H where NHdenotes the normaliser of H in Aut( ˜X).The same can be done for covers with a G-action. So we assume a finite groupG of automorphisms acts from the left on X. The following constructions werealready sketched in [SGA1, Remarque IX.5.8] and we will fill out some of thedetails:For a connected C-cover p : Y → X we define the following C G -cover:and a left G-action on Y × G viaY × G → X(y, h) ↦→ h · p(y)G × (Y × G) → Y × Gg , (y, h) ↦→ (y, gh)This clearly is a connected object of C G . The object so associated to Y is the sameas the fibre product of Y with (X × G) with G-action as described above over X.Every connected G-cover of X × G is dominated by a G-cover of the formY × X (X × G) where Y → X is a connected topological cover. Indeed, forgettingthe G-action, a connected G-cover of X × G becomes a cover of X consisting ofexactly |G| components. If we choose Y to dominate each of these componentsit is not complicated to obtain a G-morphism from Y × G onto the G-cover ofX × G we started with.33

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