RELATIVE PURITY IN LOG E´ TALE COHOMOLOGY

K. HIGASHIYAMA AND T. KAMIYA KODAI MATH. J.

40 (2017), 178–183

RELATIVE PURITY IN LOG E´ TALE COHOMOLOGY

Kazumi Higashiyama and Takashi Kamiya

Abstract

We study log e´tale cohomology. The goal is to prove relative purity in log e´tale cohomology.

1. Main results

In the present paper, we prove relative purity in log e´tale cohomology, i.e., Theorem 1.1. Let S be an fs log scheme; N an fs monoid such that the natural morphism f :X^def¼S½N !S is a log smooth morphism; j:U^def¼S½N^gp ,!X the open immersion; nAZ_>0 which is invertible on S; F0 a sheaf of Z=nZ- modules on the Kummer log e´tale site of S; F^def¼fF₀. Then

R^qjjF¼ F ðq¼0Þ 0 ðq>0Þ.

The Theorem was conjectured by L. Illusie in [3]. It is known that [5], Lemma 7.6.5 proved the case of Theorem where S¼Speck with log. str. byN^r0 (r⁰AN) with k being a separably closed field; X¼S½N^r ðrANÞ; F ¼Z=nZ.

We outline the proof. First, we prove the case N¼N. By replacing the category of log e´tale sheaves with the category of e´tale sheaves on which the Galois group acts, we calculate the higher direct images using the group cohomology.

Next, by induction, we prove the case of N¼N^r (rAN).

Finally, by log-blow-up, the general case reduces to the case in which N¼N^r.

2. The case N¼N

Let xAX and xðlogÞ be a log geometric point of x on X. It su‰ces to prove that ðR^qjjFÞ_xðlogÞ¼0 ðq>0Þ and ðjjFÞ_xðlogÞ¼F_xðlogÞ. Since the

178

2010 Mathematics Subject Classification. Primary 14F20; Secondary 14A20, 14M25.

Key words and phrases. Log e´tale cohomology; relative purity.

Received June 27, 2016; revised August 3, 2016.

problem is local, we may assume that S is a quasi-compact and quasi-separated fs log scheme.

First, we prove the caseN¼N. It su‰ces to show it at a point xAX such that M_X;x=O_X;x is isomorphic to P^def¼ ðMS;s=O_S;s ÞlN, where s^def¼ fðxÞ.

Strict e´tale locally on X, fix a chart X!SpecZ½P aroundx. Let mAZ_>0 be invertible at x. For each m, Xm^def¼X_SpecZ½PSpecZ½P^1=m; XX~ ^def¼SpecOX;x

!X; XX~m^def¼XX~_SpecZ½PSpecZ½P^1=m; UU~^def¼UXXX;~ UU~m^def¼UU~XX~XX~m. Let pm: ~UUm!U be the projection morphism.

Lemma 2.1. It holds that

ðR^qjjFÞ_xðlogÞ¼lim_!

m

H^qðUU~m;p_m jFÞ;

where mAZ_>0 runs over the set of integers invertible at x.

Proof. It follows immediately from the definitions that ðR^qjjFÞ_xðlogÞ¼lim

!V

H^qðUXV;ðUXV!UÞjFÞ;

where V!X is a ke´t (i.e., log e´tale and of Kummer type) neighborhood at xðlogÞ. Then

¼lim

!Q

lim! V

H^qðUXV;ðUXV!UÞjFÞ;

where Qis a sharpP-fs monoid such that Q^gpP^gp and such that there exists an integer m being invertible at x and satisfying ma¼0 for any aAQ^gp=P^gp, and V !X_SpecZ½PSpecZ½Q is a strict e´tale neighborhood at the lift of xðlogÞ.

Since there exists an mAZ_>0 such that m is invertible at x and the natural homomorphism P!P^1=m factors through Q, we have

¼lim

!m

lim! V

H^qðUXV;ðUXV!UÞjFÞ;

where V !Xm is a strict e´tale neighborhood at the lift of xðlogÞ. Thus,

¼lim

!m

H^qðUX SpecOXm;x;ðUX SpecOXm;x!UÞjFÞ:

Since UX SpecOXm;xFUU~_m,

¼lim

!m

H^qðUU~m;p_mjFÞ: r

Definition 2.2. For a log scheme Y, Y^cl denotes the log scheme (the underlying scheme of Y, the trivial log structure). The natural morphism e:Y !Y^cl is called the forgetting-log morphism.

Let UU~_y^cldef¼lim_mUU~_m^cl and r_m: ~UU_y^cl!UU~_m^cl. For any m;nAZ_>0, which are invertible at x, we consider the diagram

U~

Umn !

emn

U~

U_mn^{cl rmn} UU~y^cl

??

?y

??

?y U ^pm UU~m !

em

U~ U_m^cl:

pmn

r_m

We consider

lim! m

r_mR^pemp_m jF:

Lemma 2.3. It holds that lim_!

m

r_mR^pemp_mjF ¼0 ðp>0Þ:

Proof. By proper base change theorem (cf. [5], Theorem 5.1), we reduce to the case in which S is the spectrum of a separably closed field k. Let

I_m^def¼ lim

n:prime to chðkÞ

HomððP^1=mÞ^gp;Kerðn:k!kÞÞ:

By [5], Proposition 4.6, let G_mAI_m-Z=nZ-Mod₌UU~m (cf. [5], Notation 4.5) which corresponds to p_m jFAS_UU~^Z=nZ

m (cf. [5], Notation 2.3), then H^pðIm;G_mÞ corre- sponds to r_mR^pemp_m jF. Since lim

!mH^pðIm;GmÞ ¼0 (p>0), it holds that

lim_!m r_mR^pemp_m jF ¼0 (p>0). r

Lemma 2.4. It holds that lim!

m

H^qðUU~_m;p_m jFÞ ¼lim

!m

H^qðUU~_m^cl;emp_m jFÞ:

Proof. We consider the spectral sequence E₂^s;t¼lim

!m

H^sðUU~_m^cl;R^temp_m jFÞ )lim

!m

H^sþtðUU~m;p_mjFÞ:

E₂-terms are lim!

m

H^sðUU~_m^cl;R^temp_m jFÞ ¼H^s UU~y^cl;lim

!m

r_mR^temp_mjF

! :

By Lemma 2.3,

lim! m

r_mR^temp_m jF ¼0 ðt>0Þ:

Thus, E₂^s;t¼0 (t>0) and lim_!

m

H^sðUU~_m^cl;emp_m jFÞ ¼lim_!

m

H^sðUU~m;p_mjFÞ: r

Lemma 2.5. It holds that lim!

m

H^qðUU~_m^cl;emp_m jFÞ ¼0 ðq>0Þ; lim!

m

H⁰ðUU~_m^cl;emp_mjFÞ ¼FxðlogÞ:

Proof. We consider the spectral sequence E₂^p;q ¼lim

!m

H^pðXX~_m^cl;R^qje_mp_mjFÞ )lim

!m

H^pþqðUU~_m^cl;e_mp_m jFÞ:

Since XX~_y^cldef¼lim

m

X~

X_m^cl is strictly henselian, it holds that E₂^p;q¼0 (p>0). In particular,

lim! m

H^qðUU~_m^cl;emp_mjFÞ ¼lim

!m

GðXX~_m^cl;R^qjemp_m jFÞ:

Let ZZ~_m^cldef¼XX~_m^clnUU~_m^cl and give ZZ~_m^cl the reduced induced subscheme structure;

i: ~ZZ_m^cl ,!XX~_m^clthe closed immersion;S_m¼SSpecZ½M_S;s=O_S;sSpecZ½ðMS;s=O_S;sÞ^1=m. By proper base change theorem (cf. [5], Theorem 5.1),

U~

Um ! UU~_m^cl

??

?y

??

?y

S _proper! S^cl; it holds that

lim! m

GðXX~_m^cl;R^qjjðXX~_m^cl!S_m^cl!S^clÞeSF₀Þ !^@ lim

!m

GðXX~_m^cl;R^qjemp_m jFÞ:

By relative purity (cf. [2], XVI 3.7), lim!

m

GðXX~_m^cl;R^qjjðXX~_m^cl !S^clÞeSF₀Þ ¼0 ðq>1Þ;

lim! m

GðXX~_m^cl;jjðXX~_m^cl!S^clÞeSF0Þ ¼FxðlogÞ:

If q¼1, lim_!

m

GðXX~_m^cl;R¹jjðXX~_m^cl!S^clÞeSF0Þ ¼lim_!

m

H²ðZZ~_m^cl;ðZZ~_m^cl!S^clÞeSF0Þ:

Since the transition map

H²ðZZ~_m^cl;ðZZ~_m^cl!S^clÞeSF0Þ !H²ðZZ~_mn^cl ;ðZZ~_mn^cl !S^clÞeSF0Þ is 0 map, it holds that

lim_!

m

H²ðZZ~_m^cl;ðZZ~_m^cl!S^clÞeSF0Þ ¼0: r

This completes the proof of Theorem 1.1 in the case where N¼N.

3. General case

Next, in the case of N¼N^r (rAZ_b1), we consider the decomposition U¼S½Z^r ¼S½Z½Z^r1 !^j1 S½Z½N^r1 ¼S½N^r1½Z !^j2 S½N^r1½N ¼S½N^r ¼X:

By Theorem 1.1 in the case where N¼N and by induction, RjjFFRj2Rj1j₁j₂FFRj2j₂F F^qis F:

Finally, in the general case, we consider a log-blow-up p:X⁰!S½N;

where X⁰ is covered by S½N^slZ^t with various s and t. Thus, we obtain a commutative diagram

S½N^gp S½NX⁰ !

j⁰ X⁰

p⁰

??

?y ^p

??

?y S½N^gp

j S½N:

! By the previous case,

pF F^qisRj⁰j⁰pF:

Then it holds that

RppF F^qis RpRj⁰j⁰pF F^qisRjRp⁰p⁰jF: By [1], (2.7) (cf. [4], §6),

Rp⁰p⁰¼id; Rpp¼id:

Therefore,

F F^qisRjjF:

We complete the proof of Theorem 1.1.

Acknowledgements. We would like to thank Professor Chikara Nakayama for suggesting the topics, helpful discussions, warm encouragement, and valuable advice.

References

[ 1 ] K. Fujiwara and K. Kato, Logarithmic e´tale topology theory, preprint.

[ 2 ] A. Grothendieck, with M. Artin and J.-L. Verdier, The´orie des topos et cohomologie e´tale des sche´mas, Lect. notes math. 269, 270, 305, 1972–73.

[ 3 ] L. Illusie, Quelques proble`mes en cohomologie log e´tale, unpublished notes, 14avril 1998.

[ 4 ] L. Illusie, An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic e´tale cohomology, Cohomologies p-adiques et applications Arithme´tiques (II) (P. Berthelot, J. M. Fontaine, L. Illusie, K. Kato and M. Rapoport., e´d.), Aste´risque 279 (2002), 271–322.

[ 5 ] C. Nakayama, Logarithmic e´tale cohomology, Math. Ann. 308 (1997), 365–404.

Kazumi Higashiyama

Research Institute for Mathematical Sciences Kyoto University

Kyoto 606-8502 Japan

E-mail: [email protected] Takashi Kamiya

E-mail: [email protected]