Modos de transmisión

group

In this section, we shall show that for X =_An_k orX ⊂_Pn

k an open subvariety, any

finite open subgroupH <CX whose index is prime tochar(k) = pis realisable, and

thus a preimage. Together with the results of the previous chapter, this proves the Key Lemma 4.4.4.

We note that our construction will not use any theorems on geometric finiteness, contrary to the proof given for the tame variety case in [5].

Theorem 6.2.1. Let k be a finite field of characteristic p, and let X ⊂ _Pn k be

an open subvariety. If D is a covering datum of cyclic index m on X such that

(m, p) = 1, then D is realisable with a finite realisation.

Proof. We already showed in the previous chapter that it suffices to trivialise D over an affine open subset of the form D(G)⊂_An

of the fibrations Φi :Ank −→A1k, letω denote a closed point ofA1k with residue field k(ω)'k[xi]/(hi(xi)), and Cω,i the fiber of Φi overω.

As before, we denote X ∩Cω,i by Cω,i0 , and assume by induction that Dω,i is

realisable by a Galois coverY_[f

ω,i/gω,iaω,i]

whose equivalence classe [fω,i/g aω,i

ω,i ]∈ Km(AG)

has a representative such thataω,i ≤Dand deg_xjfω,i ≤adeg_xjgω,i for allj 6=i. As before, the base case is given by our assumption that D is a covering datum that is defined on the projective closure D+(G), and thus ´etale at the point at infinity of a

curve D(g)⊂_A1

k. This ensures that degxf ≤adegxg in this case. Furthermore, as

D is tame and of index bounded bym, it is geometrically bounded by Lemma 4.4.6, so we may assume that a ≤ D universally for some D. Recalling the canonical surjections AAω,i, whereAω,i =k(ω)[x1, . . . ,xi, . . . , xnˆ ], and AGAω,i_gω,i. They clearly induce surjections of Kummer spaces analogously to those of Artin-Schreier space:

pω,i :Km(A)Km(Aω,i) and πω,i :Km(AG)Km(Aω,ig ).

Then the lemma below is proven entirely analously to Lemma 5.2.9 of the Artin- Schreier case:

Lemma 6.2.2. Given[f /ga_{]∈ K}

m(Aω,ig ), there exists a preimage[F/Ga]∈π

−1

ω,i([f /ga])

which has a representative such that deg_xjF = deg_xjf for all j 6=i.

So for each ω, i, we let Fω,i ∈ A denote this representative, and set Mi = h[Fω,i/Gaω,i] :ω ∈ A1ki. Then as before, Mi corresponds to a pro-´etale cover

of X trivialising all the fibers of Φi. As the fibers are regular, every closed point

of X is a regular point of a fiber. So D is trivialised at all closed points, i.e. YMi

weakly trivialisesD onX. Combining Theorem 5.0.8 and Property 4) of the Kum- mer correspondence 6.1.3, this implies that the cover YM −→ X corresponding to M = ∩iMi also weakly trivialises D. M ⊂ MD(AG) is finite, so YM is an ´etale cover. By Proposition 2.3.10, it gives a full trivialisation of D, and is thus re- alised by an element [F/Ga_{] ∈ M ⊂ M}

D(AG). In particular, we have a ≤ D and

deg_xjF ≤adeg_xjG for all j, so all the induction assumptions are satisfied.

Remark 6.2.3. As remarked in Section 5.5.2, this includes the case of affine n-space

X =_An

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