Equivariant splitting of the Hodge–de Rham exact sequence

https://doi.org/10.1007/s00209-021-02839-y

Mathematische Zeitschrift

Equivariant splitting of the Hodge–de Rham exact sequence

Je˛drzej Garnek¹

Received: 5 June 2019 / Accepted: 15 July 2021 / Published online: 2 September 2021

© The Author(s) 2021

Abstract

Let X be an algebraic curve with a faithful action of a finite groupG over a fieldk. We show that if the Hodge–de Rham short exact sequence ofXsplitsG-equivariantly then the action ofGonXis weakly ramified. In particular, this generalizes the result of Köck and Tait for hyperelliptic curves. We discuss also converse statements and tie this problem to lifting coverings of curves to the ring of Witt vectors of length 2.

Keywords De Rham cohomology·Equivariant sheaf cohomology

Mathematics Subject Classification Primary 14F40; Secondary 14G17·14H37

1 Introduction

LetXbe a smooth proper algebraic variety over a fieldk. Recall that its de Rham cohomology may be computed in terms of Hodge cohomology via the spectral sequence

E₁^{i j}=H^j(X, Ωⁱ_X/k)⇒H_{d R}^i+j(X/k). (1.1) Suppose that the spectral sequence (1.1) degenerates at the first page. This is automatic if chark =0. For a field of positive characteristic, this happens for instance ifXis a smooth projective curve or an abelian variety, or (by a celebrated result of Deligne and Illusie from [4]) if dimX>charkandXlifts toW₂(k), the ring of Witt vectors of length 2. Under this assumption we obtain the following exact sequence:

0→H⁰(X, ΩX/k)→H_{d R}¹ (X/k)→H¹(X,OX)→0. (1.2)

The author was supported by NCN research grant UMO-2017/27/N/ST1/00497 and by the doctoral scholarship of Adam Mickiewicz University. Part of the work was done during the author’s stay during Simons Semester at IMPAN in Warsaw (November 2018), supported by the National Science Center grant 2017/26/D/ST1/00913, the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund. This paper is part of the author’s Ph.D. thesis.

B

1 Faculty of Mathematics and Computer Science, Graduate School, Adam Mickiewicz University, Umultowska 87, 61-614 Poznan, Poland

If X is equipped with an action of a finite group G, the terms of the sequence (1.2) becomek[G]-modules. In case when chark#G, Maschke theorem allows one to conclude that the sequence (1.2) splits equivariantly. However, this might not be true in case when chark= p>0 andp|#G, as shown in [10]. The goal of this article is to show that for curves the sequence (1.2)usuallydoes not split equivariantly.

LetXbe a curve over an algebraically closed field of characteristicp>0 with an action of a finite groupG. ForP∈X, denote byG_P,nthen-th ramification group ofGatP. Let also:

nP:=max{n:GP,n=0}. (1.3)

Following [9], we say that the action ofGisweakly ramifiedifn_P≤1 for everyP∈X.

Main Theorem Suppose that X is a smooth projective curve over an algebraically closed field k of a finite characteristic p>2with a faithful action of a finite group G. If

H_{d R}¹ (X/k)∼=H⁰(X, ΩX/k)⊕H¹(X,OX) (1.4) as k[G]-modules then the action of G on X is weakly ramified.

The example below is a direct generalization of results proven in [10].

Example 1.1 Suppose thatkis an algebraically closed field of characteristic p. LetX/kbe the smooth projective curve with the affine part given by the equation:

y^m= f(z^p−z),

where f is a separable polynomial andpm. Denote byPthe set of points ofXat infinity.

One checks that #P=δ:=GCD(m,deg f)(cf. [22, Section 1]). The groupG=Z/pacts onXvia the automorphismϕ(z,y)=(z+1,y). In this case

n_P=

m/δ, ifP∈P,

0, otherwise. (1.5)

(cf. Example4.3). Thus if the exact sequence (1.2) splits G-equivariantly, then by Main Theorem eitherp=2, orm|deg f.

The main idea of the proof of Main Theorem is to compareH_{d R}¹ (X/k)^GandH_{d R}¹ (Y/k), whereY := X/G. The discrepancy between those groups is measured by the sheafified version of group cohomology, introduced by Grothendieck in [6]. This allows us to compute the ’defect’

δ(X,G):=dim_kH⁰(X, ΩX/k)^G+dim_kH¹(X,OX)^G

−dimkH_{d R}¹ (X/k)^G

in terms of some local terms connected to Galois cohomology (cf. Proposition3.1). We compute these local terms in case of Artin-Schreier coverings, which leads to the following theorem.

Theorem 1.2 Suppose that X is a smooth projective curve over an algebraically closed field k of characteristic p>0with a faithful action of the group G=Z/p. Then:

δ(X,G)=

P∈X

nP−1−2· nP

p

.

Theorem1.2shows that if the group action ofG=Z/pon a curve is not weakly ramified andp>2 thenδ(X,G) >0. This immediately implies Main Theorem forG∼=Z/p. The general case may be easily derived from this special one.

The natural question arises:to which extent is the converse of Main Theorem true? We give some partial answers. In particular, we prove the following theorem.

Theorem 1.3 Suppose that the action of G on a smooth projective curve X over an alge- braically closed field k is weakly ramified. Assume also that there exists Q0 ∈Y := X/G such that p#π⁻¹(Q0). Then the sequence

0→H⁰(X, ΩX/k)^G→H_{d R}¹ (X/k)^G→H¹(X,OX)^G→0 is exact also on the right.

To derive Theorem1.3we use the method of proof of Main Theorem and a result of Köck from [9]. We were also able to show the splitting of the Hodge–de Rham exact sequence of a curve with a weakly ramified group action under some additional assumptions.

Theorem 1.4 Keep the above notation. If any of the following conditions is satisfied:

(1) the action of G on X is weakly ramified, a p-Sylow subgroup of G is cyclic and there exists Q0∈Y such that p#π⁻¹(Q0),

(2) the action of G on X lifts to W₂(k), (3) X is ordinary.

then there exists an isomorphism (1.4) ofFp[G]-modules.

Parts (1), (2), (3) of Theorem1.4are proven in Lemma5.3, Theorem5.4and Corollary5.9 respectively. In fact, we prove more precise statements, involving the conjugate spectral sequence. This allows to prove that the conditions (2) and (3) of Theorem1.4imply that the action ofGonXis weakly ramified. In order to prove (1) we use a description of modular representations of cyclic groups. (2) and (3) are easy corollaries of the equivariant version of results of Deligne and Illusie from [4]. The connection of [4] with the splitting of the Hodge–de Rham exact sequence was observed by Piotr Achinger.

Notation.Throughout the paper we will use the following notation (unless stated otherwise):

– kis an algebraically closed field of a finite characteristicp.

– Gis a finite group.

– Xis a smooth projective curve equipped with a faithful action ofG.

– Y :=X/Gis the quotient curve, which is of genusgY. – π:X→Yis the canonical projection.

– R∈Div(X)is the ramification divisor ofπ.

– R :=

π∗R

#G ∈Div(Y), where forδ∈Div(Y)⊗ZQ, we denote by[δ]the integral part taken coefficient by coefficient.

– k(X),k(Y)are the function fields ofXandY.

– ordQ(f)denotes the order of vanishing of a function f at a pointQ.

– A_Xdenotes the constant sheaf onXassociated to a ringA.

Fix now a (closed) pointPinX. Denote:

– G_P,i– theith ramification group ofπatP, i.e.

G_P,i := {g∈G:g(f)≡ f (modmⁱ⁺¹_X,P) for all f ∈OX,P}.

Note in particular that (sincekis algebraically closed) the inertia groupGP,0coincides with the decomposition group atP, i.e. the stabilizer ofPinG.

– d_P– the different exponent atP, i.e.

dP:=

i≥0

(#GP,i−1).

Recall thatR=

P∈Xd_P·(P)(cf. [18, IV §1, Proposition 4]).

– eP– the ramification index ofπatP, i.e.eP=#G_P,0. – nPis given by the formula (1.3).

Also, by abuse of notation, forQ ∈Y we writeeQ :=e_P,d_Q := d_P,n_Q :=n_Pfor any P∈π⁻¹(Q). Note that these quantities don’t depend on the choice ofP.

Let for any abelian category A,C₊(A)denote the category of non-negative cochain complexes inA. We denote byhⁱ(F^•)theith cohomology of a complexF^•.

Outline of the paper Section2presents some preliminaries on the group cohomology of sheaves. We focus on the sheaves coming from Galois coverings of a curve. We use this theory to express the ’defect’δ(X,G) as a sum of local terms coming from Galois cohomology of certain modules in Sect.3. In Sect.4we compute these local terms for Artin- Schreier coverings, which allows us to prove of Main Theorem and Theorem1.2. In the final section we discuss the converse statements to Main Theorem and its relation to the problem of lifting curves with a given group action.

2 Review of group cohomology

Recall that our goal is to compareH_{d R}¹ (X/k)^GandH_{d R}¹ (Y/k), whereY := X/G. To this end, we need to work in theG-equivariant setting.

2.1 Group cohomology of sheaves

LetAbe any commutative ring andGa finite group. We define thei-th group cohomology, H_Aⁱ(G,−), as thei-th derived functor of the functor

(−)^G:A[G]-mod→A-mod, M→M^G:= {m∈M:g·m=m}.

One checks that ifA→Bis a morphism of rings andMis aB[G]-module thenH_Bⁱ(G,M) andH_Aⁱ(G,M)are isomorphicA-modules for alli ≥0 (cf. [19, Lemma 0DVD]). In partic- ular,H_Aⁱ(G,M)is isomorphic as aZ-module to the usual group cohomology (H_Zⁱ(G,M)in our notation). Thus without ambiguity we will denote it byHⁱ(G,M). For a future use we note the following properties of group cohomology:

– IfM =Ind^G_HN is an induced module (which for finite groups is equivalent to being a coinduced module) then

Hⁱ(G,M)∼=Hⁱ(H,N), (2.1)

(this property is known asShapiro’s lemma, cf. [18, Proposition VIII.2.1.]).

– IfMis aFp[G]-module andGhas a normalp-Sylow subgroupPthen:

Hⁱ(G,M)∼=Hⁱ(P,M)^G/P (2.2) (for a proof observe thatHⁱ(G/P,N)is killed by multiplication bypfor anyFp[G/P]- moduleN and use [18, Theorem IX.2.4.] to obtainHⁱ(G/P,N)=0 fori ≥1. Then use Lyndon–Hochschild–Serre spectral sequence).

– Suppose that Ais a local ring with the maximal idealm. IfM is a finitely generated A-module then

Hⁱ(G,M)∼= Hⁱ(G,M_m), (2.3)

where M_m denotes the completion of M with respect to m (see e.g. proof of [1, Lemme 3.3.1] for a brief justification).

The above theory extends to sheaves, as explained e.g. in [1,6]. We briefly recall this theory.

Let(Y,O)be a ringed space and letGbe a finite group. By anO[G]-sheaf on(Y,O)we understand a sheafFequipped with anO-linear action ofGonF(U)for every open subset U ⊂ Y, compatible with respect to the restrictions. TheO[G]-sheaves form a category O[G]-mod, which is abelian and has enough injectives. For anyO[G]-sheafF one may define a sheafF^Gby the formula

U →F(U)^G:= {f ∈F(U): ∀g∈G g· f = f}.

We denote thei-th derived functor of

(−)^G:O[G]-mod→O-mod

byH_(Yⁱ_,O)(G,−). Similarly as in the case of modules, one may neglect the dependence on the sheaf_O and write simply_Hⁱ(G,M). If_F = Mis a quasicoherent_O[G]-module coming from aO(Y)[G]-module M, one may compute the group cohomology of sheaves via the standard group cohomology:

Hⁱ(G,F)∼=Hⁱ(G,M).

In particular, group cohomology of a quasicoherent_O[G]-sheaf is a quasicoherent_O-module.

Moreover for anyQ∈Y:

Hⁱ(G,F)Q∼=Hⁱ(G,FQ). (2.4) The sheaf group cohomology may be also computed using ˇCech complex (cf. [1, section 3.1]).

However, we will not use this fact in any way.

2.2 Galois coverings of curves

We now turn to the case of curves over a fieldk. LetX/kbe a smooth projective curve with a faithful action of a finite groupG, i.e. a homomorphismG →Autk(X). In this case one can define the quotientY:= X/GofXby theG-action. It is a smooth projective curve. Its underlying space is the topological quotientX/Gand the structure sheaf is given byπ_∗^G(OX), whereπ:X→Y is the quotient morphism. We say thatXis aG-covering ofY.

In this section we will investigate theG-sheaves on Y coming from its G-coverings.

Suppose thatπ: X→Y is aG-covering ofY. LetF be aOX-sheaf onXwith aG-action lifting that on X. Thenπ∗F is anOY[G]-module. It is natural to try to relate the group cohomology ofπ∗Fto the ramification ofπ. Suppose for a while that the action ofGonX is free, i.e. thatπ:X→Y is unramified. In this case the functors

F →π_∗^G(F) π^∗(G)←G

are exact and provide an equivalence between the category of coherentOY-modules and coherentOX[G]-modules (cf. [13, Proposition II.7.2, p. 70]). In particular,Hⁱ(G, π_∗F)=0

for alli ≥ 1 and every coherentOX[G]-moduleF. The following Proposition treats the general case.

Proposition 2.1 Keep the notation introduced in Sect.1. LetF be a coherentOX-module, which is G-equivariant. Then for every i≥1

Hⁱ(G, π_∗F)

is a torsion sheaf, supported on the wild ramification locus ofπ.

To prove Proposition2.1we shall need the following lemma involving group cohomology of modules over Dedekind domains.

Lemma 2.2 Let k be an algebraically closed field. Let B be a k-algebra, which is a Dedekind domain equipped with a k-linear action of the group G. Suppose that A:=B^Gis a discrete valuation ring with a maximal idealq. Let G_p,idenote the i -th higher ramification group of a prime idealp∈SpecB overq. Then for every B-module M we have an isomorphism of B-modules:

Hⁱ(G,M)∼=Hⁱ(Gp,1,M_p)^Gp,0/Gp,1 (here M_pdenotes the localisation of M atp).

Proof One easily sees that we have an isomorphism ofB[G]-modules M_q∼=Ind^G_Gp,0M_p

(see [18, II §3, Proposition 4] for a proof forM= B. The general case follows by tensoring both sides byM). Thus by (2.3) and (2.1) Hⁱ(G,M) ∼= Hⁱ(G_p,0,M_p). Moreover,G_p,1 is a normal p-Sylow subgroup ofG_p,0 (cf. [18, Corollary 4.2.3., p. 67]). Hence the proof follows by (2.2).

Proof of Proposition2.1 Denote byξthe generic point ofY. Recall that by the Normal Base Theorem (cf. [8, sec. 4.14]),k(X)=Ind^Gk(Y)is an inducedG-module. Therefore(π_∗F)_ξ is also an inducedG-module (since it is ak(X)-vector space of finite dimension) and by (2.1):

Hⁱ(G, π∗F)ξ =Hⁱ(G, (π∗F)ξ)=0.

Thus, since the sheafHⁱ(G, π_∗F)is coherent, it must be a torsion sheaf. Note that if a point Q∈Yis tamely ramified thenG_P,1=0 for anyP∈π⁻¹(Q)and thusHⁱ(G, π_∗F)Q=0 by Lemma2.2. This concludes the proof.

We will recall now a standard formula describingG-invariants of aOY[G]-module coming from an invertibleOX-module. For a reference see e.g. the proof of [1, Proposition 5.3.2].

Lemma 2.3 For any G-invariant divisor D∈Div(X):

π_∗^G(OX(D))=OY

π_∗D

#G

,

where forδ∈Div(Y)⊗_ZQ, we denote by[δ]the integral part taken coefficient by coefficient.

Corollary 2.4 Keep the notation of Sect.1. Let:

R = π∗R

#G

∈Div(Y).

Then:

π_∗^GΩX/k=ΩY/k⊗OY(R).

In particular:

dimkH⁰(X, Ω_X¹_/k)^G=

g_Y, if R =0,

gY−1+degR, otherwise.

Proof The first claim follows by Lemma2.3by taking Dto be the canonical divisor ofX and using the Riemann-Hurwitz formula. To prove the second claim, observe that

H⁰(X, ΩX/k)^G= H⁰(Y, π_∗^GΩX/k)=H⁰(Y, ΩY/k⊗OY(R)) and apply the Riemann–Roch theorem (cf. [7, Theorem IV.1.3]).

We end this section with one more elementary observation.

Lemma 2.5 R (given as above) vanishes if and only if the morphismπ: X→Y is tamely ramified.

Proof Recall thatR=

P∈XdP·(P). Hence

R =

Q∈Y

d_Q·#π⁻¹(Q)

#G

(Q)

=

Q∈Y

d_Q e_Q

(Q).

Note however thatd_Q ≥e_Q−1 with an equality if and only ifπis tamely ramified atQ.

This completes the proof.

3 Computing the defect

The goal of this section is to prove the following Proposition.

Proposition 3.1 We follow the notation introduced in Sect.1. Suppose that there exists Q₀ ∈ Y such that p#π⁻¹(Q₀).

Then:

δ(X,G)=

Q∈Y

dim_kim

H¹(G, (π_∗OX)Q)→H¹(G, (π_∗Ω_X/k)Q)

,

where

H¹(G, (π∗OX)Q)→H¹(G, (π∗ΩX/k)Q) is the map induced by the differentialOX→ΩX/k.

3.1 Proof: Preparation

Recall that thei-th hypercohomologyHⁱ(Y,−)can be defined as thei-th derived functor of H⁰(Y,−):C₊(k_Y-mod)→k-mod, H⁰(Y,F^•):=H⁰(Y,h⁰(F^•))

(cf. [24, ex. 5.7.4 (2)]). The hypercohomology may be computed in terms of the usual cohomology using the spectral sequences:

E₁^{i j} =H^j(Y,Fⁱ)⇒H^i+j(Y,F^•), (3.1)

I IE₂^{i j} =Hⁱ(Y,h^j(F^•))⇒H^i+j(Y,F^•). (3.2) The de Rham cohomology ofX is defined as the hypercohomology of the de Rham com- plexΩ^•_X/k. Note thatπis an affine morphism. Thereforeπ∗is an exact functor on the category of quasicoherent sheaves. Thus using the spectral sequence (3.1) we obtain:

H_{d R}ⁱ (X/k)=Hⁱ X, Ω^•X/k

=Hⁱ

Y, π∗Ω^•X/k

.

We start with the following observation.

Lemma 3.2 The spectral sequence E^{i j}₁ =H^j

Y, π_∗^GΩⁱ_X/k

⇒H^i+j

Y, π_∗^GΩ^•_X/k degenerates at the first page.

Proof We have a morphism of complexesΩ_Y^•_/k →π_∗^GΩ^•_X/k, which is an isomorphism on the zeroth term. Thus forj=0,1 we obtain a commutative diagram:

H^j(Y,OY) H^j(Y, π_∗^GOX)

H^j(Y, ΩY/k) H^j(Y, π_∗^GΩX/k),

∼=

where the upper arrow is an isomorphism. Note also that the left arrow in the above diagram is zero for j =0,1, since the Hodge–de Rham spectral sequence forY degenerates on the first page. Therefore forj =0,1 the maps

H^j(Y, π_∗^GOX)→H^j(Y, π_∗^GΩ_X/k) are zero. This is the desired conclusion.

Corollary 3.3

δ(X,G)=

dimkH¹

Y, π_∗^GΩ^•_X/k

−dimkH¹

Y, π_∗Ω^•_X/kG

−

dimkH¹

Y, π_∗^GOX

−dimkH¹(Y, π_∗OX)^G

. Proof. By Lemma3.2we obtain an exact sequence:

0→H⁰(Y, π_∗^GΩ_X/k)→H¹(Y, π_∗^GΩ^•_X/k)→H¹(Y, π_∗^GOX)→0.

Hence:

δ(X,G)=dim_kH⁰(X, Ω_X/k)^G+dim_kH¹(X,OX)^G−dim_kH_{d R}¹ (X/k)^G

=

dim_kH¹

Y, π_∗^GΩ^•_X/k

−dim_kH¹

Y, π_∗^GOX

+dimkH¹(X,OX)^G−dimkH_{d R}¹ (X/k)^G

=(dimkH¹(Y, π_∗^GΩ^•_X/k)−dimkH¹(Y, π_∗Ω_X^•_/k)^G)

−(dim_kH¹(Y, π_∗^GOX)−dim_kH¹(Y, π∗OX)^G).

Corollary3.3implies that we need to compare the hypercohomology groups Hⁱ(Y, (F^•)^G)andHⁱ(Y,F^•)^G.

forF^• =π∗OX (treated as a complex concentrated in degree 0) andF^• =π∗Ω_X^•_/k(note that it is a complex ofk_Y[G]-modules rather than_OY-modules, since the differentials in the de Rham complex are notOY-linear).

Consider the commutative diagram:

C₊(k_Y[G]-mod) C₊(k_Y-mod) k[G]-mod k-mod.

(−)^G

H⁰(Y,−) H⁰(Y,−) (−)^G

(3.3)

Note that the categories in the diagram (3.3) are abelian and have enough injectives (cf. [17, Theorem 10.43. and the following Remark]). By applying the Grothendieck spectral sequence to compositions of the functors in the diagram (3.3), we obtain two spectral sequences:

IE₂^{i j} =Hⁱ(Y,H ^j(G,F^•))⇒R^i+jΓ^G(F^•) (3.4)

I IE₂^{i j} = Hⁱ(G,H^j(Y,F^•))⇒R^i+jΓ^G(F^•), (3.5) (note that here H ^j(G,F^•) denotes a complex of k_Y-modules with lth term being H ^j(G,F^l)). For motivation, suppose at first that the ’obstructions’

Hⁱ(G,F^l) and Hⁱ(G,H^l(Y,F^•))

vanish for all i ≥ 1 and l ≥ 0 (this happens e.g. if chark = 0). Then the spectral sequences (3.4) and (3.5) lead us to the isomorphisms:

Hⁱ(Y, (F^•)^G)∼=RⁱΓ^G(F^•)∼=(Hⁱ(Y,F^•))^G.

In general case the relation betweenHⁱ(Y, (F^•)^G)andHⁱ(Y,F^•)^Gis more complicated.

However, in the case of the first hypercohomology group, one can extract some information from the low-degree exact sequences of (3.4) and (3.5):

0→H¹(Y, (F^•)^G)→R¹Γ^G(F^•)→

→H⁰(Y,H¹(G,F^•))→H²(Y, (F^•)^G)→

→R²Γ^G(F^•) (3.6)

and

0→H¹(G,H⁰(Y,F^•))→R¹Γ^G(F^•)→

→H¹(Y,F^•)^G→H²(G,H⁰(Y,F^•))→

→R²Γ^G(F^•). (3.7)

This will be done in the Sect.3.2.

3.2 Proof: Low-degree exact sequences

Note that if p #G, then Proposition3.1is immediate. Thus, we may assume thatπ is wildly ramified and by Lemma2.5we haveR =0. Then, as one easily sees by Lemma3.2, Corollary2.4and the Riemann–Roch theorem (cf. [7, Theorem IV.1.3]):

H²(Y, π_∗^GΩ^•X/k)=H¹(Y, π_∗^GΩX/k)

=H¹(Y, ΩY/k⊗OY(R))

=0.

By (3.6) we see that

dim_kR¹Γ^G(π∗Ω^•_X/k)=dim_kH¹(Y, (π∗Ω_X/k^• )^G)

+dimkH⁰(Y,H¹(G, π∗Ω^•_X/k)). (3.8) On the other hand, (3.7) yields:

dimkR¹Γ^G(π_∗Ω^•_X/k)=dimkH¹(G,H⁰(Y, π_∗Ω^•_X/k))

+dim_kH¹(Y, π∗Ω^•_X/k)^G−c₁, (3.9) where

c1=dimkker

H²(G,H⁰(Y, π∗Ω_X/k^• ))→R²Γ^G(π∗Ω^•_X/k)

. (3.10)

Thus by comparing (3.8) and (3.9):

dim_kH¹(Y, π_∗Ω^•_X/k)^G =dim_kH¹(Y, (π_∗Ω^•_X/k)^G) +dim_kH⁰(Y,H¹(G, π∗Ω^•X/k))

−dim_kH¹(G,H⁰(Y, π∗Ω^•_X/k))+c₁. (3.11) By repeating the same argument forπ_∗OX, we obtain:

dimkH¹(Y, π_∗OX)^G=dimkH¹(Y, (π_∗OX)^G) +dim_kH⁰(Y,H¹(G, π∗OX))

−dim_kH¹(G,H⁰(Y, π_∗OX))+c₂, (3.12) where:

c₂ =dim_kker

H²(G,H⁰(Y, π∗OX))→R²Γ^G(π∗OX))

. (3.13)

By combining (3.11), (3.12) and Corollary3.3we obtain:

δ(X,G)=dimkim

H⁰(Y,H¹(G, π_∗OX))→H⁰(Y,H¹(G, π_∗ΩX/k)) +(c2−c₁).

Note that sinceH¹(G, π∗OX),H¹(G, π∗ΩX/k)are torsion sheaves, we can compute their sections by taking stalks and using (2.4):

dim_kim

H⁰(Y,H¹(G, π∗OX))→H⁰(Y,H¹(G, π∗ΩX/k))

=

Q∈Y

dim_kim

H¹(G, (π∗OX)Q)→H¹(G, (π∗ΩX/k)Q)

Thus we are left with showing thatc₁ =c₂. This will be done in Sect.3.3.

3.3 Proof: The end

Recall that in order to prove Proposition3.1we have to investigate the map

H²(G,H⁰(Y,F^•))→R²Γ^G(F^•) (3.14) arising from the exact sequence (3.7). Note that for any complexF^•ofO[G]-sheaves onY there exists a natural map:

R²Γ^G(F^•)I E⁰²_∞→I E₂⁰²=H⁰(Y,H²(G,F^•)). (3.15) We will investigate this map forF^•=π∗OX(treated as a complex concentrated in degree 0).

Note that by Proposition2.1, the support of the quasicoherent sheafH²(G, π∗OX)is finite.

Therefore:

H²(G, π∗OX)∼=

Q∈Y

i_Q,∗

H²(G, (π∗OX)Q)

,

whereiQ:Spec(OY,Q)→Yis the natural morphism.

Lemma 3.4 Keep the above notation. The map (3.15) forF^• =π_∗OX is an isomorphism.

Moreover, the map:

H²(G,k)→H⁰(Y,H²(G, π_∗OX))∼=

Q∈Y

H²(G, (π_∗OX)Q)

(composition of maps(3.14)and(3.15)forF^• = π∗OX) is induced by the natural maps k→(π∗OX)Q.

Proof. Observe that for_F^•=π∗OXone has_IE^{i j}₂ =0 fori,j ≥1 and fori≥2. Therefore

IE¹¹_∞=IE₂¹¹=0 andIE_∞²⁰=IE₂²⁰=0, which leads to the proof of the first claim.

The morphism of sheaves:

π∗OX→G :=

Q∈Y

i_Q,∗((π∗OX)Q) induces by functoriality the commutative diagram:

H²(G,H⁰(Y, π_∗OX)) R²Γ^G(π_∗OX) H⁰(Y,H²(G, π_∗OX))

H²(G,H⁰(Y,G)) R²Γ^G(G) H⁰(Y,H²(G,G)).

∼

∼ (3.16)

Note thatHⁱ(Y,G)=

Q∈YHⁱ(Y,i_Q,∗((π∗OX)Q))=0 fori≥1. Therefore forF^•:=G we have_IE₂^{i j} =0 fori ≥1 and_{I I}E₂^{i j} =0 forj≥1. This implies that the two lower arrows in the diagram (3.16) are isomorphisms. Moreover, the composition of those arrows is given by the composition of the natural maps:

H²(G,H⁰(Y,G))=H²

⎛

⎝G,

Q∈Y

(π_∗OX)Q

⎞

⎠∼=

Q∈Y

H²(G, (π_∗OX)Q)

∼=H⁰

⎛

⎝Y,

Q∈Y

i_Q,∗(H²(G,GQ))

⎞

⎠∼=H⁰(Y,H²(G,G)).

Thus the second claim follows by the diagram (3.16).

We are now ready to finish the proof of Proposition3.1. Recall that we are left with showing thatc₁=c₂(wherec₁andc₂are given by (3.10) and (3.13) respectively).

In fact, we will show thatc₁ =c₂=0. We start by proving that the map:

H²(G,k)= H²

G,H⁰(Y, π∗OX)

→R²Γ^G(π∗OX)

∼= H⁰

Y,H²(G, π∗OX)∼=

Q∈Y

H²(G, (π∗OX)Q) (3.17) is injective. It suffices to show that the map:

H²(G,k)→H²(G, (π∗OX)Q0) (3.18) is injective. Choose anyP₀∈π⁻¹(Q₀). Observe that by Lemma2.2we have:

H²(G, (π∗OX)Q)∼= H²(G_P0,1,O_X,P0)^GP0,0/GP0,1. ButO_X,P0∼=k⊕mX,P0as ak[G_P0,1]-module and therefore

H²(GP₀,1,OX,P₀)∼= H²(GP₀,1,k)⊕H²(GP₀,1,mX,P₀).

Note thatGacts trivially onk. Hence the map (3.18) factors as

H²(G,k)→H²(GP0,1,k) →H²(GP0,1,k)⊕H²(GP0,1,mX,P0)^GP0,0/GP0,1. where the first map is the restriction map res^G_G

P0,1. Now note thatp#π⁻¹(Q0)= [G:GP0] and thusG_P0,1is a p-Sylow subgroup ofGby [18, Corollary 4.2.3., p. 67]. Thus by [18, Theorem IX.4, p. 140] res^G_G

P0,1is injective. This shows the injectivity of (3.17). Hencec2=0.

Consider now the diagram:

H²(G,H⁰(Y, π∗OX)) R²Γ^G(π∗OX)

H²(G,H⁰(Y, π_∗Ω^•_X/k)) R²Γ^G(π_∗Ω^•_X/k),

∼=

which is obtained by functoriality for the map of complexesπ∗Ω^•_X/k→π∗OX(whereπ∗OX

is treated as a complex concentrated in degree zero). Note that the left arrow is an isomorphism and the upper arrow is injective, as shown above. Therefore also the lower arrow must be injective, which proves thatc1 =0. This concludes the proof of Proposition3.1. Note that we obtain by (3.12) the following Corollary:

Corollary 3.5 In the notation of Sect. 1 suppose that there exists Q₀ ∈ Y such that p#π⁻¹(Q₀). Then:

dimkH¹(X,OX)^G=gY+

Q∈Y

H¹(G, (π_∗OX)Q)−dimkH¹(G,k).

4 Computation of local terms 4.1 Proofs of main results

The main goal of this section is to compute the local terms occuring in Proposition3.1. This is achieved in the following proposition.

Proposition 4.1 Keep the notation introduced in Sect.1and suppose that G ∼=Z/p. Then for any Q∈Y the dimension of

im

H¹(G, (π_∗OX)Q)→H¹(G, (π_∗ΩX/k)Q)

equals

nQ−1−2· nQ

p

.

Proposition4.1will be proven in the Sect.4.2. We now show how the Proposition4.1 implies the Theorems announced in the Introduction.

Proof of Theorem1.2 Ifπis unramified, the action ofGonXlifts toW2(k), cf. [21, Theorem 5.7.9]. Henceδ(X,G) = 0 by Theorem5.4. Suppose now thatπ is ramified. In this case Theorem1.2follows by combining Propositions3.1and4.1.

Proof of Main Theorem We consider first the caseG = Z/p. An easy computation shows that for anyn≥1,p≥3 one has:

n−1−2· n

p

≥0

with an equality only forn = 1 (here is where we use the assumption p > 2). Thus by Theorem1.2,δ(X,G)=0 holds if and only ifπis weakly ramified.

Suppose now thatG is arbitrary andGP,2 = 0 for some P ∈ X. Note thatGP,2 is a p-group (cf. [18, Corollary 4.2.3., p. 67]) and thus contains a subgroup H of order p.

Observe thatπ : X → X/H is an Artin-Schreier covering and it is non-weakly ramified, sinceH_P,2=H =0. Therefore by the first paragraph of the proof, the sequence (1.2) does not splitH-equivariantly and therefore it cannot split as a sequence ofk[G]-modules.

4.2 Galois cohomology of sheaves on Artin–Schreier coverings

We start by recalling the most important facts concerning Artin–Schreier coverings. For a reference see e.g. [16, sec. 2.2]. LetXbe a smooth algebraic curve with a faithful action of

G=Z/pover an algebraically closed fieldkof characteristicp. By Artin–Schreier theory, the function field ofXis given by the equation

z^p−z= f (4.1)

for some f ∈k(Y), whereY := X/G. The action ofG = σ ∼= Z/pis then given by σ (z):=z+1. Let_P ⊂Ydenote the set of points at whichπis ramified. Note that_Pis contained in the set of poles of f and moreover for anyQ∈Y:

#π⁻¹(Q)=

p, forQ∈/P, 1, otherwise.

Lemma 4.2 Keep the above setting. Fix a point Q ∈ Pand letπ⁻¹(Q) = {P}. Suppose that pn:= −ord_Q(f). Then for some t∈_OX,Pand x∈_OY,Q:

– _OX,P=k[[t]],_OY,Q=k[[x]], – t^−np−t⁻ⁿ =x⁻ⁿ,

– the action of G∼=Z/p on t is given by an automorphism:

σ (t)= t

(1+tⁿ)^1/n =t−1

ntⁿ⁺¹+ (terms of order ≥n+2). (4.2) In particular, n is equal to n_Qas defined by(1.5).

Proof Let x,t be arbitrary uniformizers at Q and P respectively. Then _OY,Q = k[[x]]

and_OX,P = k[[t]]. Before the proof observe that an equationu^m = h(x)has a solution u ∈k[[x]], whenever p mandm|ord(h)(this follows easily from Hensel’s lemma). We will denote any solution byh(x)^1/m. Note that:

f⁻¹= z^−p 1−z^1−p.

By comparing the valuations we see that ord_P(z)= −n. Thus we may replacetbyz^−1/nto ensure thatz=t⁻ⁿ. Then:

σ (t)ⁿ =σ (tⁿ)=σ 1

z

= 1

z+1 = 1

t⁻ⁿ+1 = tⁿ 1+tⁿ

and thus we can assume without loss of generality (by replacingσby its power if necessary) thatσ (t)= _(1+t^tn)^1/n. Finally, we replacexby f(x)^−1/nto ensure thatt^−np−t⁻ⁿ=x⁻ⁿ. Example 4.3 LetXbe the curve considered in Example1.1. ThenXis aZ/p-covering of a curveYwith the affine equation:

y^m= f(x).

The function field ofXis given by the equationz^p−z=x. As proven in [22] the function x ∈k(Y)hasδ :=GCD(m,deg f)poles, each of them of orderm/δ. This establishes the formula (1.5).

Remark 4.4 Suppose thatπ:X→Yis an Artin-Schreier covering. For every pointQ∈P we can find functions f_Q∈k(Y),z_Q∈k(X)such that the function field ofXis given by the equationz_Q^p −z_Q= f_Qand either f_Q∈OY,Q, orpord_Q(f_Q). Indeed, in order to obtain fQone can repeatedly subtract from f a function of the formh^p−h, wherehis a power of a uniformizer atQ.

Example 4.5 It might not be possible to find a function f such the function field ofXis given by (4.1) and for any poleQof f one haspord_Q(f). Take for example an ordinary elliptic curveX/Fp. Letτ ∈Aut(X)be a translation by a p-torsion point. Consider the action of G = τ ∼=Z/ponX. This group action is free and hencenP =0 for allP∈ X. Thus, if k(X)would have an equation of the formz^p−z= f, wherepordQ(f)for allQ ∈P, then f would have no poles. This easily leads to a contradiction.

Keep the notation of Lemma4.2. Fix an integera∈Zand denote:

– B:=_OX,P=k[[t]],L:=k((t)),I:=t^aB, – A:=_OY,Q=k[[x]],K :=k((x)).

In the below Lemma we will computeH¹(G,I). The dimension ofH¹(G,I)is computed also in [1, Théorème 4.1.1] (see also [11, formula (18)]). However, we need an explicit description of a basis ofH¹(G,I).

Lemma 4.6 1. H¹(G,I)may be identified with

M:=coker(L^G→(L/I)^G).

2. A basis of H¹(G,I)is given by the images of the elements(tⁱ)i∈Jin M, where J := {a−n≤i≤a−1:pi}.

3. dimkH¹(G,I)=n−

a−1

p +

a−1−n

p .

4. The images of the elements:

tⁱ for a−n≤i≤a−1, p|i are zero in M.

Proof. For anyh∈L, we will denote its image inL/Iby[h]L/I. Analogously, ifh∈ L^G, we denote its image inMby[h]M.

1. The proof follows by taking the long exact sequence of cohomology for the short exact sequence ofk[G]-modules:

0→I →L→L/I →0 and using the Normal Base Theorem (cf. [8, sec. 4.14]).

2. Note that for anya−n≤i≤a−1, we have[tⁱ]L/I ∈(L/I)^G, since σ ([tⁱ]L/I)= [σ (tⁱ)]L/I = [(t−1

ntⁿ⁺¹+O(t²ⁿ))ⁱ]L/I

= [tⁱ− i

ntⁱ⁺ⁿ+O(tⁱ⁺²ⁿ)]L/I= [tⁱ]L/I.

We’ll show now that the set([tⁱ]M)i∈JspansM. Note thatL^G =K. Therefore it suffices to show that for any[h]L/I ∈(L/I)^G, one has

h∈K+

i∈J

k·tⁱ. (4.3)