The K(π,1)-property for marked curves over finite fields

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The K(π,1)-property for marked curves over finite fields

Philippe Lebacque, Alexander Schmidt

To cite this version:

Philippe Lebacque, Alexander Schmidt. The K(π,1)-property for marked curves over finite fields.

2013. �hal-00926253v2�

The K pπ, 1 q -property for marked curves over finite fields

Philippe Lebacque and Alexander Schmidt

Abstract

We investigate theKpπ,1q-property forpof smooth, marked curvespX, Tq defined over finite fields of characteristic p. We prove that pX, Tq has the Kpπ,1q-property if X is affine and give positive and negative examples in the proper case. We also consider the unmarked proper case over a finite field of characteristic different top.

2010 Math. Subj. Class. 11R34, 11R37, 14F20

Key words: Galois cohomology, étale cohomology, restricted ramification

1 Introduction

In [1],[2],[3], the second author investigated theKpπ,1q-property forpof arithmetic curves whose function field is of characteristic different top. As a result, the Galois group of the maximal unramified outside S and T-split pro-p-extension of a global field of characteristic different topis often of cohomological dimension less or equal to two. In this paper we consider the case of a smooth curve over a finite field of characteristic p. We prove that pX, Tq has the Kpπ,1q-property if X is affine and give positive and negative examples in the proper case. We also consider the unmarked proper case over a finite field of characteristic different to p, which was left out in the earlier papers.

The authors would like to thank the referee for his valuable suggestions.

1.1 The marked étale site and the K pπ, 1 q-property

LetX be a regular one-dimensional noetherian scheme defined overFq (withq“p^f) and let T be a finite set of closed points. In [3], the second author defined the marked site pX, Tq of X at T considering finite étale morphisms Y Ñ X inducing isomorphismskpyq – kpxqon the residue fields for any closed pointyPY mapping to xPT. LetM be ap-torsion sheaf. The resulting cohomology groups are denoted by HⁱpX, T, Mqand they satisfy the usual properties we expect from étale cohomology groups. He also proved (see [3] for more details) that these finite marked étale morphisms fit into a Galois theory and (after choosing a base geometric pointsxRT) we denote byπ1pX, Tq the profinite group classifying étale coverings of X in which the points ofT split completely. We denote bypX, TČqppqthe universal pro-p-covering

of pX, Tq. The projection pX, TČqppq Ñ X is Galois with Galois group the maximal pro-p-quotient π1pX, Tqppqof π1pX, Tq.

LetM be a discretep-torsionπ₁pX, Tqppq-module. Consider the Hochschild-Serre spectral sequence:

E₂^ij “Hⁱpπ1pX, Tqppq, H^jppX, TČqppq, T, Mqq ñH^i`jpX, T, Mq.

The edge morphisms provide homomorphisms

φi,M :Hⁱpπ1pX, Tqppq, Mq Ñ HⁱpX, T, Mq.

We say that pX, Tq has the Kpπ,1q-property forp if φi,M is an isomorphism for all M and all i ě 0. The following Lemma 1.1 implies in particular, that pX, Tq has the Kpπ,1q-property for p if φ_i,Fp is an isomorphism foriě2.

Lemma 1.1. (cf. [3] Lemma 2.2) φi,M is an isomorphism for i “ 0,1 and is a monomorphism for i “ 2. Moreover, φi,M is an isomorphism for all i ě 0 if and only if

limÝÑ

pY,T¹q

HⁱpY, T¹, Mq “0 for all iě1,

where the direct limit is taken over all finite intermediate coverings pY, T¹q of the universal pro-p-covering pX, TČqppq Ñ pX, Tq.

1.2 Notation

Unless otherwise stated, we use the following notation:

- pdenotes a prime number.

- F is a finite field, Fs an algebraic closure of F, Fr its maximal pro-p-extension insideFs and GF the Galois group of sF{F.

- X is a smooth projective absolutely irreducible curve defined over F.

- k“FpXqthe function field of X.

- gX the genus ofX.

- Xs “XˆFsF,Xr “XˆFrF.

- S, T are two disjoint sets (possibly empty) of closed points ofX.

- if x is a closed point of a X, Xx denotes the henselization of X at x and Tx “ txu if xPT and Hotherwise.

- k^T_S denotes the maximal pro-p-extension of k which is unramified outside S and in which all places ofT split completely. If empty, we omit S (or T) from the notation.

- G^T_Spkq “Galpk^T_S{kq “π1pX´S, Tqppq.

- HⁱpX ´S, Tq denotes the i-th étale cohomology group Hetⁱ pX ´S, T,Fpq of the marked curve pX´S, Tq.

- for a pro-p-group Gwe set HⁱpGq “ HⁱpG,Fpq.

- for an abelian groupA and an integer m we write Arms “kerpAÑ^¨m Aq

1.3 New results

Let X be a smooth projective absolutely irreducible curve defined over the finite field F and let k “ FpXq be the function field of X. Let S and T be finite disjoint sets of closed points ofX. In this paper, we prove the following result:

Theorem 1.2. Assume that p“charpFq.

(i) If S ‰ H, then pX´S, Tq has the Kpπ,1q-property for p and cdG^T_Spkq “1.

(ii) If T “ H, then pX´Sq has the Kpπ,1q-property for p and cdGSpkq ď2.

In the remaining cases, we have the following results.

Theorem 1.3. Assume that p“charpFq, S“ H and T ‰ H.

(i) If PicpXqrps “ 0, then pX, Tq has the Kpπ,1q-property for p if and only if T “ txuconsists of a single point withp∤degx. In this caseπ1pX, Tqppq “1.

(ii) If PicpXqrps ‰0 and ÿ

xPT

degpxq

p#Fq^degpxq{2´1 ągX ´1,

then π1pX, Tqppq is finite and pX, Tq has not the Kpπ,1q-property for p.

Finally, we consider the unmarked proper case over a finite field of characteristic different top, which was left out in the earlier papers.

Theorem 1.4. Assume thatp‰charpFq. Then X has the Kpπ,1q-property for p if and only if µppFq “ 1 or PicpXqrps ‰0.

In the remaining case µp ĂF and PicpXqrps “ 0 we have π₁^etpXqppq –π₁^etpFqppq –Zp.

In particular, Hⁱpπ₁^etpXqppqq is always finite and vanishes for ią3.

2 Computation of étale cohomology groups

Proposition 2.1(Local computation). LetK be a nonarchimedean local (or henselian) field of characteristic p. Let Y “ SpecO_K, y P Y the closed point and let T be H or tyu. Then the local cohomology groups H_yⁱpY, Tq vanish fori‰2 and

H_y²pY, Tq “

#H_{nr¹ pKq if T “ H

H¹pKq if T “ tyu, where H_{nr¹ “H¹pKq{H_nr¹ pKq.

Proof: We use the excision sequence:

¨ ¨ ¨ ÑH_yⁱpY, Tq ÑHⁱpY, Tq ÑHⁱpY ´ tyuq ÑH_y^i`1pY, Tq Ñ ¨ ¨ ¨ .

SinceY is henselian, HⁱpYq –Hⁱpyq “ H_nrⁱ pKq, henceHⁱpYq “0foriě2. SinceY is normal, H¹pY, Tq Ñ H¹pY ´ tyuqis injective, hence H_y¹pY, Tq “0. Furthermore,

HⁱpY ´ tyuq “ HⁱpKq and this group vanishes for i ě 2 since cdpK “ 1 (see [4], Cor. 6.1.3). It follows that H_yⁱpY, Tq –HⁱpY, Tqfor iě3.

ForT “ H we obtain H_yⁱpYq “0 for iě3 and the short exact sequence 0ÑH¹pYq Ñ H¹pY ´ tyuq ÑH_y²pYq Ñ 0

implies the result forH_y²pYq.

IfT “ tyu, the identity of pY, Tqis cofinal among the covering families ofpY, Tq, henceHⁱpY,tyuq “ 0for iě1. We obtainH_y²pY,tyuq –H¹pKq and H_yⁱpY,tyuq “ 0

foriě3. l

Proposition 2.2. (Global computation) LetX be a smooth projective and geomet- rically irreducible curve overF, k“FpXqand S and T finite, disjoint sets of closed points of X.

Then HⁱpX´S, Tq “0 for iě3 and H²pX´S, Tq “0 if S ‰ H. We have an exact sequence

0ÑH¹pX´S, Tq ÑH¹pX´Sq Ñ à

xPT

H_nr¹ pkxq Ñ H²pX´S, Tq ÑH²pX´Sq Ñ 0.

Proof: In the case T “ H we have HⁱpX´Sq “0 for iě3 and H²pX´Sq “0if S ‰ H by [5] exp. 10, Thm. 5.1 and Cor. 5.2. Moreover, the sequence is exact for trivial reasons.

Now assume T ‰ H. Consider the excision sequence for pX ´S, Tq and pX ´ pSYTqq:

¨ ¨ ¨ Ñà

xPT

H_xⁱppX´Sqx, Txq ÑHⁱpX´S, Tq ÑHⁱpX´ pSYTqq Ñ ¨ ¨ ¨. Proposition 2.1 shows that HⁱpX´S, Tq – HⁱpX´ pSYTqq “0 for iě3 and the exactness of the sequence

0ÑH¹pX´S, Tq ÑH¹pX´ pSYTqq Ñà

xPT

H¹pkxq Ñ H²pX´S, Tq Ñ0. p˚q

Comparing this with the excision sequence forpX´Sqand pX´ pSYTqq 0ÑH¹pX´Sq Ñ H¹pX´ pSYTqq Ñà

xPT

H_{nr¹ pkxq ÑH²pX´Sq Ñ0, we obtain the exact sequence of the proposition.

IfS ‰ H, the Strong Approximation Theorem implies that H¹pX´ pSYTqq ÝÑà

xPT

H¹pkxq

is surjective (see [4] Thm. 9.2.5). Using p˚q this shows that H²pX ´S, Tq “ 0 in

this case. l

Corollary 2.3. If G^T_Spkqis finite and nontrivial, then pX´S, Tq does not have the Kpπ,1q-property for p.

Proof. In this case we havecdG^T_Spkq “ 8but HⁱpX´S, Tq “0for iě3.

Corollary 2.4. We have the Euler-Poincaré characteristic formula ÿ2

i“0

p´1qⁱdim_FpHⁱpX, Tq “ #T.

Proof. If S “ H, all groups in the exact sequence of Proposition 2.2 are finite and we obtain

ÿ2 i“0

p´1qⁱdim_FpHⁱpX, Tq “#T ` ÿ2 i“0

p´1qⁱdim_FpHⁱpXq.

Recall that H¹pXq “s HompPicpXqrps,s Fpq (every connected étale covering of Xs comes by base change from an isogeny of the Jacobian of X). Hences

H²pXq “ H¹pF, H¹pXqq “s H¹pF,HompPicpXqrps,s Fpqq “HompPicpXqrps,s FpqG_F.

Furthermore, we have an exact sequence

0ÑH¹pFq ÑH¹pXq Ñ HompPicpXqrps,s Fpq^GF Ñ0.

Thus Lemma 2.5 below shows ÿ2

i“0

p´1qⁱdim_FpHⁱpXq “1´dim_FpH¹pFq “ 0.

Lemma 2.5. We have PicpXqrpss ^GF “PicpXqrps and

dim_FpPicpXqrpss G_F “dim_FpPicpXqrps.

Proof. The first equality follows from the Leray spectral sequence E₂^ij “HⁱpF, H^jpX,s Gmqq ñ H^i`jpX,Gmq

and the vanishing of the Brauer group of a finite field:

H²pF, H⁰pX,s Gmqq “H²pF,sF^ˆq “0.

The equality of dimensions follows from the exact sequence of finite-dimensional Fp-vector spaces

0ÑPicpXqrpss ^GF ÑPicpXqrpss ^1´FrobÝÑ PicpXqrps Ñs PicpXqrpss G_F Ñ0.

3 Proof of Theorem 1.2

Assume S ‰ H. From the computations in the last section, we know that HⁱpX´ S, Tq “0 for i ě2. By Lemma 1.1, pX´S, Tq has the Kpπ,1q-property for p and cdG^T_Spkq ď1. But G^T_Spkq is nontrivial, which follows from the exact sequence

0ÑH¹pX´S, Tq ÑH¹pX´Sq Ñ à

xPT

H_nr¹ pkxq Ñ 0 together with the fact thatÀ

xPTH_nr¹ pkxqhas finiteFp-dimension whereasH¹pX´Sq is infinite dimensional.

Now assume that S “ H and T “ H. Let rF be the maximal p-extension of F in sF. Then H²pX_Frq “ H²pXFsq^GalpsF{rFq “ 0. Hence X_rF is a Kpπ,1q for p and the Hochschild-Serre spectral sequence for X_rF{X shows the same for X. This finishes the proof of Theorem 1.2.

4 Proof of Theorem 1.3

Proposition 4.1. Assume thatPicpXqrps “0andT ‰ Hand let p^rbe the maximal p-power dividing gcdpdegx, x PTq. Then

G^Tpkq “π1pX, Tqppq –GalpF¹{Fq,

where F¹ is the unique extension of F of degree p^r.

Proof. LetrF be the maximal p-extension of F inF. Using Lemma 2.5, we haves H²pXq “ H¹pF, H¹pXqq –s HompPicpXqrps,Fpq “ 0

and Corollary 2.4 shows that H¹pXq is 1-dimensional. Hence π1pXqppq is free of rank1 and therefore the surjection

π1pXqppq։GalprF{Fq

is an isomorphism (cf. [4], Prop. 1.6.15). The maximal subextension F¹{F of rF{F such that all points inT split completely in the base changeXbFF¹ ÑX is exactly the unique extension of degreep^r of F.

Corollary 4.2. Assume that PicpXqrps “0 and T ‰ H. Then pX, Tq is a Kpπ,1q for p if and only if T “ txu consists of a single point with p ∤ degx. In this case the fundamental group π1pX, Tqppq is trivial.

Proof. By Proposition 4.1,π1pX, Tqppqis finite cyclic. Ifp|gcdpdegx, xPTq, then π1pX, Tqppqis nontrivial and pX, Tq is not a Kpπ,1q forp by Corollary 2.3.

Assumep∤gcdpdegx|xPTq. Thenπ1pX, Tqppqis the trivial group,H¹pX, Tq “ 0 and pX, Tq is a Kpπ,1q if and only if H²pX, Tq “ 0. By Corollary 2.4 this is equivalent to#T “1.

Lemma 4.3. Assume that π1pX, Tqppq is finite and PicpXqrps ‰ 0. Then pX, Tq is not a Kpπ,1q for p.

Proof. By Corollary 2.3, pX, Tq is not a Kpπ,1q for p if π1pX, Tqppq is nontriv- ial. Assume that π1pX, Tqppq “ 1. Then pX, Tq is a Kpπ,1q for p if and only if H²pX, Tq “ 0. But by Proposition 2.2, H²pXq – HompPicpXqrps,Fpq ‰ 0 is a quotient of H²pX, Tq.

The following theorem is due to Ihara, see [6], Thm. 1 (FF).

Theorem 4.4. Assume that T ‰∅ and let q“#F. If ÿ

xPT

degpxq

q^degpxq{2 ´1 ąmaxpgX ´1,0q,

then π1pX, Tq is finite. In particular, π1pX, Tqppq is finite.

Summing up, we obtain Theorem 1.3.

5 Proof of Theorem 1.4

LetrF be the maximal p-extension ofF inFs and Xr “XˆFF. Thenr X is a Kpπ,1q for pðñX˜ is a Kpπ,1q forp and we have

Hetⁱ pXq –r Hetⁱ pXqs ^GpsF{rFq

for all i. Hence Hetⁱ pXqr vanishes for i ě 3 and Het²pXq “r µppFqr ^˚ “ µppFq^˚. We conclude that X has the Kpπ,1q-property for p if µppFq “ 1. In the following we assume that F contains all p-th roots of unity. For every tower of finite connected étale p-coverings Z ÑY ÑX the natural map

Z{pZ“Het²pY , µr pq ÝÑHet²pZ, µr pq “Z{pZ is multiplication by the degreerZr :Yrs. Hence, by Lemma 1.1,

Xr is a Kpπ,1qfor pðñ#`

π₁^etpXqppqr ˘

“ 8.

Note that

π₁^abpXq{pr – Het¹pXqr ^˚ – `

Het¹pXqs ^GpsF{rFqq^˚ – PicpXqrpss _GpsF{rFq and by Lemma 2.5

PicpXqrpss _GpsF{rFq “0ðñPicpXqrps “r 0.

Furthermore, sinceGprF{Fq is a pro-p-group:

PicpXqrps “r 0ðñPicpXqrps “PicpXqrpsr ^GprF{Fq “0,

and therefore, PicpXqrps ‰ 0 ðñ π₁^abpXq{pr ‰ 0. Hence it suffices to show the equivalences

#`

π^et₁ pXqppqr ˘

“ 8 ô#`

π₁^abpXqppqr ˘

“ 8 ô#π₁^abpXq{pr ‰0.

Elementary theory of pro-p-groups shows that it remains to show the implication π₁^abpXq{p˜ ‰0ùñ#`

π₁^abpXqppq˜ ˘

“ 8.

SettingT :“TppXq “s π^ab₁ pXqppq, we can write this implication in the forms pT_GpsF{rFqq{p‰0ùñ #pT_GpsF{rFqq “ 8.

The group GpsF{Fq˜ is pro-cyclic of supernatural order prime to p. Furthermore, T –Z^2g_p and the kernel of the reduction map Gl_2gpZpq ÑGl_2gpFpq is a pro-p-group.

Hence the action ofGpsF{Fqr onT factors through a finite cyclic group of order prime top. We conclude that Theorem 1.4 follows from Lemma 5.1 below. l The following Lemma 5.1 and its application in the proof of Theorem 1.4 were proposed to us by J. Stix. We thank the referee for suggesting the short proof given below.

Lemma 5.1. Let G be a finite group of order n, p a prime number with p ∤n and T a finitely generated freeZp-module with a G-action. Then

#TG“ 8 ðñ pT{pqG ‰0.

Proof. Since the Tate cohomology of ZprGs-modules vanishes, we obtain the split exact sequence of ZprGs-modules

0ÝÑkerpNq ÝÑ T ÝÑ^N T^G ÝÑ0, where N “ ř

gPGg. For B “ kerpNq, Hˆ^´1pG, Bq “ 0 implies BG “0. We obtain TG –T^G and pT{pqG – pT^Gq{p. Hence both assertions of the lemma are equivalent toT^G‰0.

References

[1] A. Schmidt, “Rings of integers of type Kpπ,1q”, Doc. Math. 12 (2007), 441–471.

[2] A. Schmidt, “On the Kpπ,1q-property for rings of integers in the mixed case”, Algebraic number theory and related topics, 2007, 91–100, RIMS Kôkyûroku Bessatsu, B12, Res. Inst. Math. Sci. (RIMS), Kyoto, 2009.

[3] A. Schmidt, Über Pro-p-Fundamentalgruppen markierter arithmetischer Kurven, J. Reine Angew. Math. 640 (2010), 203–235.

[4] J. Neukirch, A. Schmidt, K. Wingberg,Cohomology of Number Fields, 2nd ed., 2nd corr. print., Grundlehren der math. Wiss. 323, Springer 2013.

[5] M. Artin, A. Grothendieck et J. L. Verdier, “Théorie des topos et coho- mologie étale des schémas”, Tome 3”, Lecture Notes in Mathematics, Vol.

305, Springer-Verlag 1973.

[6] Y. Ihara, “How many primes decompose completely in an infinite unram- ified Galois extension of a global field?”, J. Math. Soc. Japan 35 (1983), no. 4, 693–709.

Laboratoire de Mathématiques de Besançon, 16 route de Gray, 25030 Besançon, France

INRIA Saclay - Ile-de-France, Equipe-projet GRACE email: philippe.lebacque@univ-fcomte.fr

Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, D-69120 Heidelberg, Deutschland

email: schmidt@mathi.uni-heidelberg.de