Duality for relative logarithmic de Rham–Witt sheaves and wildly ramified class field theory over finite fields

Duality for relative logarithmic de Rham–Witt sheaves and wildly ramified class field theory over

finite fields

Uwe Jannsen, Shuji Saito and Yigeng Zhao

Compositio Math. 154 (2018), 1306–1331.

doi:10.1112/S0010437X1800711X

Compositio Math. 154 (2018) 1306–1331 doi:10.1112/S0010437X1800711X

Duality for relative logarithmic de Rham–Witt sheaves and wildly ramified class field theory over

finite fields

Uwe Jannsen, Shuji Saito and Yigeng Zhao

Abstract

In order to study p-adic ´etale cohomology of an open subvariety U of a smooth proper variety X over a perfect field of characteristic p > 0, we introduce new p-primary torsion sheaves. It is a modification of the logarithmic de Rham–Witt sheaves of X depending on effective divisors D supported in X −U . Then we establish a perfect duality between cohomology groups of the logarithmic de Rham–Witt cohomology of U and an inverse limit of those of the mentioned modified sheaves. Over a finite field, the duality can be used to study wildly ramified class field theory for the open subvariety U .

Contents

1 Relative logarithmic de Rham–Witt sheaves 1308

2 Filtered de Rham–Witt complexes 1316

3 The pairing on the relative logarithmic de Rham–Witt sheaves 1323

4 Duality over finite fields 1325

5 Duality over perfect fields 1328

Acknowledgements 1330

References 1330

Introduction

Let k be a perfect field of characteristic p > 0 and let X be a smooth proper variety of dimension d over k. The logarithmic de Rham–Witt sheaves W

_m

Ω

^r_X,log

are defined as the subsheaves of the de Rham–Witt sheaves W

m

Ω

^r_X

, which are ´etale locally generated by sections d log[x

1

]

m

∧ · · · ∧ d log[x

r

]

m

with x

ν

∈ O

^×_X

for all ν [Ill79]. By the Gersten resolution [Ros96, Ker10, GS88] and the Bloch–Gabber–Kato theorem [BK86], the d log map induces an isomorphism of ´etale sheaves

d log[−] : K

^M_r,X

/p

^m

−

→^∼=

W

_m

Ω

^r_X,log

; {x

₁

, . . . , x

_r

} 7

→

d log[x

₁

]

_m

∧ · · · ∧ d log[x

_r

]

_m

, (1) where K

^M_r,X

is the sheaf of Milnor K-groups. It is conceived as a p-adic analogue of the `-adic sheaf µ

^⊗r_`m

with ` 6= p. If k is a finite field, there is a non-degenerate pairing of finite groups due to Milne [Mil86]:

H

ⁱ

(X, W

_m

Ω

^r_X,log

) × H

^d+1−i

(X, W

_m

Ω

^d−r_X,log

)

→

H

^d+1

(X, W

_m

Ω

^d_X,log

) −

^Tr→Z

/p

^mZ

.

Received 5 December 2016, accepted in final form 19 December 2017, published online 7 May 2018.

2010 Mathematics Subject Classification14F20 (primary), 14F35, 11R37, 14G17 (secondary).

Keywords:logarithmic de Rham–Witt sheaves, class field theory, wild ramification, ´etale duality, quasi-algebraic

It induces a natural isomorphism

H

^d

(X, W

m

Ω

^d_X,log

) ∼ = H

¹

(X,

Z

/p

^mZ

)

^∨

∼ = π

^ab₁

(X)/p

^m

,

where A

^∨

is the Pontryagin dual of a discrete abelian group and π

^ab₁

(X) is the maximal abelian quotient of Grothendieck’s ´etale fundamental group of X. This gives a description of π

₁^ab

(X)/p

^m

in terms of ´etale cohomology with p-adic coefficient. For `-adic ´etale cohomology, we also have a non-degenerate pairing of finite groups for a smooth non-proper variety U of dimension d over a finite field k [SGA4

¹₂

, Sai89]

H

ⁱ

(U,

Z

/`

^m

(j)) × H

_c^2d+1−i

(U,

Z

/`

^m

(d − j))

→

H

_c^2d+1

(U,

Z

/`

^m

(d)) ∼ =

Z

/`

^mZ

, which can be used to describe π

^ab₁

(U )/`

^m

by `-adic ´etale cohomology:

H

_c^2d

(U,

Z

/`

^m

(d)) ∼ = H

¹

(U,

Z

/`

^m

)

^∨

∼ = π

^ab₁

(U )/`

^m

.

In the p-adic setting there is no obvious analogue of ´etale cohomology with compact support for logarithmic de Rham–Witt sheaves.

In this paper, we propose a new approach. Let X be a proper smooth variety over a perfect field k as before, and let j : U ,

→

X be the complement of an effective divisor D such that Supp(D) has simple normal crossings. We introduce new p-primary torsion sheaves W

_m

Ω

^r_X|D,log

(see Definition 1.1.1), which we call

relative logarithmic de Rham–Witt sheaves. It is defined

as the subsheaf of the de Rham–Witt sheaf W

m

Ω

^r_X

which is ´etale locally generated by sections d log[x

₁

]

_m

∧. . . ∧d log[x

_r

]

_m

with x

₁

∈ Ker(O

^×_X →

O

_D^×

), and x

_ν

∈ j

∗

O

_U^×

for all ν. As in the classical situation, we have the following theorem.

Theorem 1 (see Theorem 1.1.5). The map d log induces an isomorphism d log[−] : K

^M_r,X|D

/(p

^m

K

^M_r,X

∩ K

_r,X^M _|D

) −

→^∼=

W

_m

Ω

^r_X|D,log

{x

1

, . . . , x

r

} 7

→

d log[x

1

]

m

∧ · · · ∧ d log[x

r

]

m

. (2) Here K

^M_r,X|D

is the sheaf of relative Milnor K-groups which has been studied by one of the authors (Saito) and R¨ ulling in [RS18].

If D

1 >

D

2

, we have inclusions (see Proposition 1.1.4)

W

m

Ω

^r_X|D1,log

⊆ W

m

Ω

^r_X|D2,log

⊆ W

m

Ω

^r_X,log

, and thus obtain a pro-system of

Z

/p

^mZ

-sheaves “lim

←

−

D

”W

_m

Ω

^r_X|D,log

, where D runs over the set of all effective divisors with Supp(D) ⊂ X − U .

In case m = 1 these sheaves are related to sheaves of differential forms by the exact sequence (see Theorem 1.2.1)

0

→

Ω

^r_X|D,log→

Ω

^r_X|D ^1−C

−1

−−−−

→

Ω

^r_X|D

/dΩ

^r−1_X|D →

0, (3) where Ω

^r_X|D

= Ω

^r_X

(log D) ⊗

_OX

O

_X

(−D) and C

⁻¹

is the inverse Cartier morphism. In order to extend the above exact sequence to the case m > 1, we need introduce

the filtered relative de Rham–Witt complex

W

m

Ω

^•_X|D

for which we have W

1

Ω

^•_X|D

= Ω

^•_X|D

(see

§

2.3 and Theorem 2.3.1).

Its construction uses the de Rham–Witt complexes in log geometry [HK94], which can be seen as the higher analogue of Ω

^r_X

(log D).

Using the generalization of (3) to the case m > 1, we can define a pairing between W

_m

Ω

^r_U,log

and the pro-system “lim

←

−

D

”W

_m

Ω

^d−r_X|D,log

and obtain the following theorem.

Theorem 2 (see Theorem 4.1.4). Let X, D and U be as above and assume that k is finite. Then the groups H

^j

(X, W

m

Ω

^r_X|D,log

) are finite and there are natural perfect pairings of topological abelian groups

H

ⁱ

(U, W

_m

Ω

^r_U,log

) × lim

←

−

D

H

^d+1−i

(X, W

_m

Ω

^d−r_X|D,log

)

→

H

^d+1

(X, W

_m

Ω

^d_X,log

) −

^Tr→Z

/p

^mZ

, where the first group is endowed with discrete topology, and the second is endowed with profinite topology.

From the case i = 1 and r = 0 of the above theorem, we get a natural isomorphism lim

←

−

D

H

^d

(X, W

_m

Ω

^d_X|D,log

) −

→^∼=

H

¹

(U,

Z

/p

^mZ

)

^∨

∼ = π

₁^ab

(U )/p

^m

,

which gives rise to a series of quotients π

^ab₁

(X, D)/p

^m

of π

₁^ab

(U )/p

^m

using the inverse limit. It is thought of as classifying abelian ´etale covering of U whose degree divides p

^m

and ramification is bounded by the divisor D.

One of the authors (Zhao) [Zha16] has proved a similar duality theorem for a projective semi-stable scheme over an equi-characteristic discrete valuation ring k[[t]] with k finite.

When the base field k is prefect but not necessarily finite, we follow the method of Milne [Mil86] and work in the category

S

(p

^m

) of

Z

/p

^mZ

-sheaves on perfect ´etale site (Pf /k)

_´et

(see

§

5.1). Let D

^b

(

S

(p

^m

)) be the derived category of bounded complexes in

S

(p

^m

). We then get from X, D objects of D

^b

(S (p

^m

)):

Rπ

∗

W

m

Ω

^d−r_X|D,log

and Rπ

∗

Rj

∗

W

m

Ω

^r_U,log

,

where π : X

→

S = Spec(k) is the structure morphism and j : U

→

X is the open immersion.

Then our duality theorem reads as follows.

Theorem 3 (see Theorem 5.2.1). There is a natural isomorphism in D

^b

(

S

(p

^m

)):

R lim

←

−

D

Rπ

∗

W

m

Ω

^d−r_X|D,log

−

→^∼=

RHom

_D^b_(S(p^m₎₎

(Rπ

∗

Rj

∗

W

m

Ω

^r_U,log

,

Z

/p

^mZ

)[−d], where R lim

←

−

D

denotes the homotopy limit over effective Cartier divisors supported on X − U . The paper is organized as follows.

In

§

1, we study the two important results on the relative logarithmic de Rham–Witt sheaves:

the first one is a computation of the kernel of the restriction map R

^m−1

: W

m

Ω

^r_X|D,log →

W

₁

Ω

^r_X|D,log

; the second is the exact sequence (3).

In order to define the desired pairing, we introduce filtered de Rham–Witt complexes in

§

2, and study the behavior of Frobenius and Verschiebung morphisms on these complexes.

Using two-term complexes, we define the pairing in

§

3 and prove its perfectness when the base field k is finite in

§

4. The last

§

5 is on the duality over a general perfect field.

1. Relative logarithmic de Rham–Witt sheaves

Let X be a smooth proper variety of dimension d over a perfect field k of characteristic p > 0,

let D be an effective divisor such that Supp(D) is a simple normal crossing divisor on X, and

let j : U := X − D ,

→

X be the complement of D.

1.1 Basic properties

Definition 1.1.1. For r ∈

N

let

W

_m

Ω

^r_X|D,log

⊂ j

∗

W

_m

Ω

^r_U,log

be the subsheaf generated ´etale locally by sections

d log[x

₁

]

_m

∧ · · · ∧ d log[x

_r

]

_m

with x

₁

∈ Ker(O

_X^× →

O

^×_D

), x

_ν

∈ j

∗

O

^×_U

for all ν.

For r ∈

N

let K

^M_r,X

be the rth Milnor K-sheaf on X

´et

given by V 7→ Ker

M

η∈V⁽⁰⁾

K

_r^M

(k(η)) −−

^⊕∂→^x M

x∈V⁽¹⁾

K

_r−1^M

(k(x))

for an ´etale V

→

X,

where V

⁽ⁱ⁾

is the set of points of codimension i in V , for i = 0, 1, and ∂

_x

: K

_r^M

(k(η))

→

K

_r^M

(k(x)) is the tame symbol from [BT73,

§

4]. By [Ker10, Proposition 10(8) and Theorem 13], K

^M_r,X

is

´etale locally generated by symbols {x

₁

, . . . , x

_r

} with x

_i

∈ O

_X,x^×

. We have a natural isomorphism of ´etale sheaves

d log[−] : K

_r,X^M

/p

^m

−

→^∼=

W

m

Ω

^r_X,log

{x

1

, . . . , x

r

} 7

→

d log[x

1

]

m

∧ · · · ∧ d log[x

r

]

m

. (1.1.1) This follows from the Gersten resolutions of

∗

K

^M_r,X

and

∗

W

_m

Ω

^r_X,log

from [Ros96] and [GS88]

together with the Bloch–Gabber–Kato theorem [BK86], where : X

_´et→

X

_Zar

is the map of sites.

Definition 1.1.2 [RS18, Definition 2.4]. For r ∈

N

, we define the relative Milnor K-sheaf K

^M_r,X|D

to be image of the map

Ker(O

_X^× →

O

_D^×

) ⊗

_Z

j

∗

K

_r−1,U^M →

j

∗

K

^M_r,U

; x ⊗ {x

₁

, . . . , x

r−1

} 7→ {x, x

₁

, . . . , x

_r

}.

Using some symbol calculations, we get the following proposition.

Proposition 1.1.3 [RS18, Corollary 2.9]. Let D

₁

, D

₂

be two effective divisors on X whose supports are simple normal crossing divisors. Assume D

₁ 6

D

₂

. Then we have the inclusions of sheaves

K

^M_r,X|D

2

⊂ K

^M_r,X|D

1

⊂ K

^M_r,X

.

Corollary 1.1.4. Under the assumption of Proposition

1.1.3, we have inclusions

W

m

Ω

^r_X|D2,log

⊂ W

m

Ω

^r_X|D1,log

⊂ W

m

Ω

^r_X,log

.

Proof.

This follows from the fact that the sheaf W

_m

Ω

^r_X|D,log

is the image of K

^M_r,X|D

under the

map d log[−].

2

The isomorphism (1.1.1) also has the following relative version.

Theorem 1.1.5. The d log map induces an isomorphism of ´ etale sheaves d log[−] : K

^M_r,X|D

/(p

^m

K

^M_r,X

∩ K

_r,X^M _|D

) −

→^∼=

W

m

Ω

^r_X|D,log

{x

₁

, . . . , x

_r

} 7→ d log[x

₁

]

_m

∧ · · · ∧ d log[x

_r

]

_m

. (1.1.2)

Proof.

The assertion follows directly by the following commutative diagram.

K

^M_r,X|D

/(p

^m

K

^M_r,X

∩ K

^M_r,X|D

)

^{ //}

dlog

K

^M_r,X

/p

^m

∼= dlog

W

m

Ω

^r_X|D,log ^{ //}

W

m

Ω

^r_X,log

2

In the rest of this section, we will prove two fundamental results for the relative logarithmic de Rham–Witt sheaves.

Theorem 1.1.6. Write D =

P

λ∈Λ

n

λ

D

λ

, where D

λ

(λ ∈ Λ) are irreducible components of D.

Then we have an exact sequence

0

→

W

m−1

Ω

^r_X|[D/p],log

−

→^p

W

_m

Ω

^r_X|D,log →

W

₁

Ω

^r_X|D,log→

0, where [D/p] =

P

λ∈Λ

[n

_λ

/p]D

_λ

with [n/p] = min{n

⁰

∈

Z

| pn

⁰ >

n}.

Proof.

The claim follows from Theorem 1.1.7 below by the isomorphism (1.1.2).

2

Let R be the henselization of a local ring of a smooth scheme over a field k of characteristic p > 0. Let (T

1

, . . . , T

_d

) ⊂ R be a part of a system of regular parameters and put T = T

1

· · · T

_d

. We endow

N^d

with a semi-order by

(n

1

, . . . , n

_d

)

6

(n

⁰₁

, . . . , n

⁰_d

) if n

i6

n

⁰_i

for all i and put

1 = (1, . . . , 1).

Following [BK86,

§

4], we define U

ⁿ

K

_r^M

(R) ⊂ K

_r^M

(R) for n = (n

₁

, . . . , n

_d

) ∈

N^d

as the subgroup generated by symbols

{x

₁

, . . . , x

_r

} with x

₁

∈ 1 + T

₁ⁿ¹

· · · T

_d^nd

R, x

_i

∈ R[1/T ]

^×

(2

6

i

6

d).

(Here having the injectivity of K

_r^M

(R)

→

K

_r^M

(K) with the quotient field K of R, the above symbols are considered in K

_r^M

(K).) For an integer m > 0, put

U

ⁿ

k

^M_r

(R)

_m

= Image(U

ⁿ

K

_r^M

(R)

→

K

_r^M

(R)/p

^m

).

Theorem 1.1.7. We have the following exact sequence:

0

→

U

^[n/p]

k

^M_r

(R)

m−1

−

p

→

U

ⁿ

k

_r^M

(R)

m →

U

ⁿ

k

_r^M

(R)

1 →

0, where [n/p] = min{ν ∈

N^d

| pν

>

n} ∈

N^d

.

For the proof we compute

gr

^n,i

k

^M_r

(R)

m

= U

ⁿ

k

^M_r

(R)

m

/U

^n+δi

k

^M_r

(R)

m

with δ

i

= (0, . . . ,

∨i

1, . . . , 0).

We need some preliminaries. For n ∈

N^d

and 1

6

i

6

d and an integer q

>

1 put

ω

_n,i^q

= I

ⁿ

Ω

^q_R

(log T ) ⊗

_R

R

i

with R

i

= R/(T

i

),

where I

ⁿ

= (T

₁ⁿ¹

· · · T

_d^nd

) ⊂ R and Ω

^q_R

(log T) is the sheaf of (absolute) differential q-forms of R with logarithmic poles along T = 0. It is easy to check the exterior derivative induces

d

^q

: ω

^q_n,i→

ω

_n,i^q+1

. Put

Z

_n,i^q

= Ker(ω

^q_n,i

−

^d→^q

ω

_n,i^q+1

), B

_n,i^q

= Image(ω

^q−1_n,i

−

^d

−−

^q−1→

ω

^q_n,i

).

We can easily check the following.

Lemma 1.1.8 [RS18, Theorem 2.16]. Let the notation be as above. Then the inverse Cartier morphism

C

⁻¹

: Ω

^q_R→

Ω

^q_R

/dΩ

^q−1_R

induces an isomorphism

C

_n,i⁻¹

: ω

_[n/p],i^q

−

→^∼=

Z

_n,i^q

/B

_n,i^q

. We define subgroups

B

_n,i^q

= B

^q_1|n,i

⊂ B

_2|n,i^q

⊂ · · · ⊂ Z

_2|n,i^q

⊂ Z

_1|n,i^q

= Z

_n,i^q

⊂ ω

^q_n,i

, by the inductive formulae

B

_s|[n/p],i^q

−−

^'→

C_n,i⁻¹

B

_s+1|n,i^q

/B

_n,i^q

, Z

_s|[n/p],i^q

−−

^'→

C_n,i⁻¹

Z

_s+1|n,i^q

/B

_n,i^q

.

Proposition 1.1.9. Fix n = (n

₁

, . . . , n

_d

) ∈

N^d

and 1

6

i

6

d.

(1) There is a natural map

ρ

n,i

: ω

_n,i^r−1 →

gr

^n,i

k

_r^M

(R)

m

such that for a ∈ R, b

₂

, . . . , b

_d

∈ R[1/T ]

^×

, ρ

n,i

a(T

₁ⁿ¹

· · · T

_d^nd

) db

2

b

₂

∧ · · · ∧ db

r

b

_r

= {1 + aT

₁ⁿ¹

· · · T

_d^nd

, b

2

, . . . , b

r

} ∈ U

ⁿ

K

_r^M

(R).

(2) Write n

i

= p

^s

· n

⁰

with p 6 |n

⁰

. If m > s, ρ

n,i

induces an isomorphism ω

_n,i^r−1

/B

_s|n,i^r−1

−

→^∼=

gr

^n,i

k

^M_r

(R)

m

.

If m

6

s, ρ

n,i

induces an isomorphism

ω

_n,i^r−1

/Z

_m|n,i^r−1

−

→^∼=

gr

^n,i

k

_r^M

(R)

m

.

Proof.

The existence of ρ

n,i

together with the fact that it induces the surjective maps as in (2) is

shown by the same argument as [BK86, (4.5) and (4.6)]. Note that ω

_n,i^r−1

/B

^r−1_s|n,i

and ω

_n,i^r−1

/Z

_m|n,i^r−1

are free R

^p_i^e

-modules, for some e 0. By localization, the injectivity of the maps is reduced to

the case R is a discrete valuation ring, which has been treated in [BK86, (4.8)].

2

Now we prove Theorem 1.1.7. It is easy to see that we have a complex as in the theorem. Its exactness on the left follows from the fact that K

_r^M

(R) is p-torsion free (cf. [GL00, Theorem 8.1]

and [Ros96, Theorem 6.1]). It remains to show the exactness in the middle. For this it suffices to show the injectivity of the map induced by multiplication by p:

K

_r^M

(R)/U

^[n/p]

K

_r^M

(R) + p

^m−1

K

_r^M

(R) −

→^p

K

_r^M

(R)/U

ⁿ

K

_r^M

(R) + p

^m

K

_r^M

(R).

This follows from the following claims.

Claim 1.1.10. The multiplication by p induces an injective map:

K

_r^M

(R)/U

¹

K

_r^M

(R) + p

^m−1

K

_r^M

(R)

→

K

_r^M

(R)/U

¹

K

_r^M

(R) + p

^m

K

_r^M

(R).

Proof.

We have a map (cf. [RS18, the first displayed formula in the proof of Proposition 2.10]) K

_r^M

(R)/U

¹

K

_r^M

(R)

→ M

16i6d

K

_r^M

(R

_i

); {a

₁

, . . . , a

_r

} 7→ ⊕

_i

{a

₁

mod T

_i

, . . . , a

_r

mod T

_i

}, where (a mod T

i

) ∈ R

i

is the image of a ∈ R. By [RS18, Proposition 2.10] and Proposition 1.1.3, we see that this map is injective. Combining with the fact that

L

16i6d

K

_r^M

(R

_i

) is p-torsion free,

we conclude this claim.

2

Claim 1.1.11. For n and i as in Proposition

1.1.9, the multiplication by

p induces an injective map:

gr

^[n/p],i

k

_r^M

(R)

m−1 →

gr

^n,i

k

_r^M

(R)

_m

.

Proof.

It is easy to check that the multiplication by p induces such a map. Its injectivity follows from the commutative diagram

ω

^r−1_[n/p],i

/B

^r−1s−1|[n/p],i C_n,i⁻¹

−−−

→

ω

_n,i^r−1

/B

^r−1_s|n,i



 y^'



 y^'

gr

^[n/p],i

k

_r^M

(R)

m−1

−−−

→

gr

^n,i

k

^M_r

(R)

m

if m > s,

and the commutative diagram

ω

_[n/p],i^r−1

/Z

m−1|[n/p],i^r−1 C_n,i⁻¹

−−−

→

ω

_n,i^r−1

/Z

_m|n,i^r−1



 y^'



 y^'

gr

^[n/p],i

k

_r^M

(R)

m−1

−−−

→

gr

^n,i

k

_r^M

(R)

m

if m

6

s,

where the vertical isomorphisms are from Proposition 1.1.9.

2 1.2 Relation with differential forms

The sheaf Ω

^r_X|D,log

relates to coherent sheaves as follows.

Theorem 1.2.1. We have an exact sequence

0

→

Ω

^r_X|D,log →

Ω

^r_X|D

−−−−

^1−C−1→

Ω

^r_X|D

/dΩ

^r−1_X|D →

0,

where Ω

^r_X|D

= Ω

^r_X/k

(log D) ⊗

O_X

O

_X

(−D).

Proof.

For the exactness on the right, it suffices to show the surjectivity of 1 − C

⁻¹

on sections over the strict henselization of a local ring of X. In fact, by the argument in the classical case where D =

∅

[Mil76, Lemma 1.3], it suffices to show the following claim.

Claim 1.2.2. Let A be a strictly henselian regular local ring of equi-characteristic p > 0 and

m

⊂ A be the maximal ideal. Let π ∈

m

and a ∈ A. If a ∈ πA, then there exists b ∈ A, such that b ∈ πA and b

^p

− b = a.

Proof of Claim1.2.2.

Let k be the residue field of A. Since φ : A

→

A is surjective, there exists

˜ b ∈ A such that ˜ b

^p

− ˜ b = a. Letting β ∈ k be the image of ˜ b, β

^p

− β = 0 ∈ k by the assumption a ∈ πA ⊂

m. Hence

β ∈

F^p

⊂ A and we put b = ˜ b − β ∈ A. Then

b(b

^p−1

− 1) = b

^p

− b = ˜ b

^p

− ˜ b = a ∈ πA.

Since b ∈

m_A

by the construction, b

^p−1

− 1 ∈ A

^×

and we get b ∈ πA.

2

It remains to show the exactness in the middle, i.e., to show that Ω

^r_X|D

∩ Ω

^r_X,log

= Ω

^r_X|D,log

. This is a ´etale local question, which is a consequence of Proposition 1.2.3 below, which is a

refinement of [Kat82, Proposition 1].

2

Let R be the henselization of a local ring of X and choose a system T

1

, . . . , T

d

of regular parameters of R such that Supp(D) = Spec(R/(T

₁

· · · T

_e

)) ⊂ Spec(R) for some e

6

d = dim(R).

Let Ω

¹_R

(log D) denotes the module of differentials with logarithmic poles along D and put Ω

^q_R

(log D) = ∧

^q

Ω

¹_R

(log D). For a tuple of integers n = (n

1

, . . . , n

e

) with n

i >

1, put

G

ⁿ

Ω

^q_R

= (T

₁ⁿ¹

· · · T

_e^ne

) · Ω

^q_R

(log D) ⊂ Ω

^q_R

, G

ⁿ

ν

_R

(q) = Ker G

ⁿ

Ω

^q_R

−−−−

^1−C−1→

Ω

^q_R

(log D)/dΩ

^q−1_R

(log D)

. Proposition 1.2.3. G

ⁿ

ν

R

(q) is generated by elements of the form

dx

1

x

₁

∧ · · · ∧ dx

q

x

_q

with x

1

∈ 1 + (T

₁ⁿ¹

· · · T

_e^ne

), x

i

∈ R

1

T

₁

· · · T

_e ×

(2

6

i

6

q).

Proof.

The following argument is a variant of Part (B) of the proof of [Kat82, Proposition 1 (see p. 224)]. By [Art69], we may replace R by R = k[[T

1

, . . . , T

_d

]]. Indeed, to use Artin approximation we have to equip any R-algebra with the log structure coming via pullback from the canonical one on (R, D) to extend the group G

ⁿ

ν

_R

(q) to a functor on R-algebras S 7→ G

ⁿ

ν

_S

(q). Put A = k[[T

1

, . . . , T

d−1

]] and T = T

d

so that R = A[[T ]]. Let Ω

^q_A

(log E) be the module of differential q-forms on Spec(A) with logarithmic poles along E = Spec(A/(T

1

· · · T

d−1

)) ⊂ Spec A. By [Kat82, Proposition 1], we have an isomorphism

(R ⊗

_A

Ω

^q_A

(log E)) ⊕ (R ⊗

_A

Ω

^q−1_A

(log E)) ' Ω

^q_R

(log D); (a ⊗ w, b ⊗ v)

→

aw + bv ∧ dT

T . (1.2.1) For each n

>

1, let V

_n

⊂ Ω

^q_R

(log D) be the image of

(T

ⁿ

A[[T ]] ⊗ Ω

^q_A

(log E)) ⊕ (T

ⁿ

A[[T ]] ⊗

_A

Ω

^q−1_A

(log E)).

We easily check the following.

Claim 1.2.4. For a tuple of integers n = (n

1

, . . . , n

d−1

, n) with n, n

i >

1, we have G

ⁿ

Ω

^q_R

⊂ V

n

and it coincides with the image of

(T

ⁿ

A[[T]] ⊗

_A

(T

₁ⁿ¹

· · · T

_d−1^nd−1

) · Ω

^q_A

(log E)) ⊕ (T

ⁿ

A[[T ]] ⊗

_A

(T

₁ⁿ¹

· · · T

_d−1^nd−1

) · Ω

^q−1_A

(log E)).

The map (1.2.1) restricted on V

_n

induces an isomorphism

(T

₁ⁿ¹

· · · T

_d−1^nd−1

) · Ω

^q_A

(log E) ⊕ (T

₁ⁿ¹

· · · T

_d−1^nd−1

) · Ω

^q−1_A

(log E) −

→^∼=

G

ⁿ

Ω

^q_R

/G

ⁿ⁰

Ω

^q_R

, (w, v)

→

T

ⁿ

w + v ∧ dT T

, where n

⁰

= (n

1

, . . . , n

d−1

, n + 1).

Let I

q

be the set of strictly increasing functions {1, . . . , q}

→

{1, . . . , d − 1}. For s ∈ I

q

write ω

_s

= dT

_s(1)

T

_s(1)

∧ · · · ∧ dT

_s(q)

T

_s(q)

∈ Ω

^q_A

(log E).

Then ω

s

(s ∈ I

q

) form a basis of Ω

^q_A

(log E) over A. Put U

n

= V

n

∩ Ker Ω

^q_R

(log D)

^1−C

−1

−−−−

→

Ω

^q_R

(log D)/dΩ

^q−1_R

(log D) .

We have the following description of U

_n

/U

_n+1

(see Part (B) of the proof of [Kat82, Proposition 1]).

If (p, n) = 1, we have an isomorphism

ρ

_n

: Ω

^q−1_A

(log E) −

→^∼=

U

_n

/U

_n+1

,

X

s∈Iq−1

a

_s

ω

_s

7→

X

s∈Iq−1

d(1 + a

_s

T

ⁿ

)

(1 + a

s

T

ⁿ

) ∧ ω

_s

(a

_s

∈ A). (1.2.2) If p|n, we have an isomorphism

ρ

_n

: Ω

^q−1_A

(log E)/Ω

^q−1_A

(log E)

_d=0

⊕ Ω

^q−2_A

(log E)/Ω

^q−2_A

(log E)

_d=0

−

→^∼=

U

_n

/U

_n+1

,

X

s∈Iq−1

a

s

ω

s

,

X

t∈Iq−2

b

t

ω

t

7

→ X

s∈Iq−1

d(1 + a

s

T

ⁿ

)

(1 + a

s

T

ⁿ

) ∧ ω

s

+

X

t∈Iq−2

d(1 + b

t

T

ⁿ

) (1 + b

t

T

ⁿ

) ∧ dT

T ∧ ω

t

, (1.2.3) where a

s

, b

t

∈ A.

Claim 1.2.5. Fix a tuple of integers n = (n

1

, . . . , n

d−1

, n) with n

i >

1.

(1) Assume (p, n) = 1 and ρ

n

(ω) ∈ G

ⁿ

Ω

^q_R

mod U

n+1

for ω =

X

s∈Iq−1

a

s

ω

s

∈ Ω

^q−1_A

(log E).

Then we have a

s

∈ (T

₁ⁿ¹

· · · T

_d−1^nd−1

) for all s ∈ I

q−1

.

(2) Assume p|n and ρ

n

(ω) ∈ G

ⁿ

Ω

^q_R

mod U

n+1

for

ω = (ω

1

, ω

2

) ∈ Ω

^q−1_A

(log E)/Ω

^q−1_A

(log E)

_d=0

⊕ Ω

^q−2_A

(log E )/Ω

^q−2_A

(log E)

_d=0

. Then one can write

ω

1

=

X

s∈Iq−1

a

s

ω

s

mod Ω

^q−1_A

(log E)

d=0

, ω

₂

=

X

t∈Iq−2

b

_t

ω

_t

mod Ω

^q−2_A

(log E)

_d=0

,

with a

s

, b

t

∈ (T

₁ⁿ¹

· · · T

_d−1^nd−1

) for all s ∈ I

q−1

and t ∈ I

q−2

.

Proof of Claim1.2.5.

Assume (p, n) = 1. From (1.2.2) we get

ρ

n

X

s∈Iq−1

a

s

ω

s

= T

ⁿ X

s∈Iq−1

da

s

∧ ω

s

± nT

ⁿ X

s∈Iq−1

a

s

ω

s

∧ dT

T mod U

n+1

.

Hence (1) follows from Claim 1.2.4 noting da

_s

∧ ω

_s

∈ Ω

^q_A

(log E). Next assume p|n. From (1.2.3) we get

ρ

_n

X

s∈Iq−1

a

_s

ω

_s

,

X

t∈Iq−2

b

_t

ω

_t

= T

ⁿ X

s∈Iq−1

da

_s

∧ ω

_s

± T

ⁿ X

t∈Iq−2

db

_t

∧ ω

_t

∧ dT T . By Claim 1.2.4, if the left-hand side lies in G

ⁿ

Ω

^q_R

mod U

n+1

, we get

da

s

∧ ω

s

∈ (T

₁ⁿ¹

· · · T

_d−1^nd−1

) · Ω

^q_A

(log E), db

t

∧ ω

t

∈ (T

₁ⁿ¹

· · · T

_d−1^nd−1

) · Ω

^q−1_A

(log E).

Thus the desired assertion follows from the following.

Claim 1.2.6. Assume dη ∈ (T

₁ⁿ¹

· · · T

_d−1^nd−1

) · Ω

^q_A

(log E) for η =

P

s∈Iq−1

a

_s

ω

_s

∈ Ω

^q−1_A

(log E). Then there exist α

s

∈ A for s ∈ I

q−1

such that a

s

− α

s

∈ (T

₁ⁿ¹

· · · T

_d−1^nd−1

) for all s and that dξ = 0 for ξ =

P

s∈Iq−1

α

_s

ω

_s

.

Indeed write a

s

= α

s

+ a

⁰_s

where a

⁰_s

∈ (T

₁ⁿ¹

· · · T

_d−1^nd−1

) and α

s

are expanded as

X

i1,...,id−1

α

s,i1,...,id−1

T

₁ⁱ¹

· · · T

_d−1^id−1

(α

s,i1,...,id−1

∈ k),

where i

₁

, . . . , i

d−1

range over non-negative integers such that there exists 1

6

ν

6

d − 1 with i

ν

< n

ν

. Then one easily check that α

s

satisfy the desired condition.

2

Now we can finish the proof of Proposition 1.2.3. In the following we fix a tuple of integers n = (n

₁

, . . . , n

d−1

, n

_d

) with n

_i >

1 and take ω ∈ G

ⁿ

Ω

^q_R

. By Claim 1.2.5 there exist a series of elements

a

s,n

∈ (T

₁ⁿ¹

· · · T

_d−1^nd−1

) (s ∈ I

q−1

, n

>

n

d

),

b

t,pm

∈ (T

₁ⁿ¹

· · · T

_d−1^nd−1

) (t ∈ I

q−2

, m

>

n

_d

/p),

such that ω =

X

n>nd

X

s∈Iq−1

d(1 + a

s,n

T

ⁿ

)

(1 + a

_s,n

T

ⁿ

) ∧ ω

s

+

X

pm>nd

X

t∈Iq−2

d(1 + b

t,m

T

^pm

) (1 + b

_t,m

T

^pm

) ∧ dT

T ∧ ω

t

=

X

s∈Iq−1

X

n>n_d

d(1 + a

s,n

T

ⁿ

) (1 + a

_s,n

T

ⁿ

)

∧ ω

_s

+

X

t∈Iq−2

X

pm>n_d

d(1 + b

t,m

T

^pm

) (1 + b

_t,m

T

^pm

)

∧ dT T ∧ ω

_t

. The products

x =

Y

n>nd

(1 + a

s,n

T

ⁿ

), y =

Y

pm>nd

(1 + b

t,m

T

^pm

) converge in 1 + (T

₁ⁿ¹

· · · T

_d^nd

) ⊂ R

^×

and we get

ω =

X

s∈Iq−1

dx

x ∧ ω

_s

+

X

t∈Iq−2

dy y ∧ dT

T ∧ ω

_t

.

This completes the proof of Proposition 1.2.3.

2

Remark

1.2.7. In fact, the above proof shows that the exactness in the middle of the complex in Theorem 1.2.1 already holds in the Nisnevich topology.

2. Filtered de Rham–Witt complexes

Let X, D, j : : U ,

→

X be as before. Let {D

_λ

}

_λ∈Λ

be the (smooth) irreducible components of D.

We endow

Z^Λ

with a semi-order by

n := (n

_λ

)

λ∈Λ>

n

⁰

:= (n

⁰_λ

)

λ∈Λ

if n

_λ >

n

⁰_λ

for all λ ∈ Λ. (2.0.1) For n = (n

_λ

)

λ∈Λ

∈

Z^Λ

let

D

_n

=

X

λ∈Λ

n

_λ

D

_λ

be the associated divisor.

2.1 Definition and basic properties

Let E be a Cartier divisor on X. It is given by {V

_i

, f

_i

}, where {V

_i

}

_i

is an open cover of X and f

i

∈ Γ(V

i

, M

^×_X

) is a section of the sheaf of total fractional ring.

Definition 2.1.1. We define an invertible W

m

O

_X

-module W

m

O

_X

(E) associated to E as W

m

O

_X

(E)

|V_i

:= W

m

O

_Vi

·

1 f

i

m

⊂ W

m

M

_Vi

, where [·]

m

: O

→

W

m

O the Teichm¨ uller lifting.

This definition gives us an invertible sheaf W

m

O

_X

(D

n

) for any D

n

as above.

Lemma 2.1.2. We have:

(i) F (W

m+1

O

_X

(D

n

)) ⊂ W

m

O

_X

(D

pn

);

(ii) V (W

m

O

_X

(D

pn

)) ⊂ W

m+1

O

_X

(D

n

);

(iii) R(W

_m+1

O

_X

(D

_n

)) ⊂ W

_m

O

_X

(D

_n

).

Proof.

The claims (i) and (iii) are clear by the definition. Claim (ii) follows from the equalities V (x · F y) = V (x) · y and F [y]

_m+1

= [y

^p

]

_m

.

2

Let W

_m

Ω

^∗_X

(log D) be the de Rham–Witt complex with respect to the canonical log structure (X, j

∗

O

_U^×

∩ O

_X

) defined in [HK94,

§

4].

Definition 2.1.3. For n = (n

_λ

)

λ∈Λ

∈

Z^Λ

, we define the filtered de Rham–Witt complex as W

m

Ω

^∗_X|Dn

:= W

m

O

_X

(−D

n

) · W

m

Ω

^∗_X

(log D) ⊂ j

∗

W

m

Ω

^∗_U

,

where W

m

Ω

^∗_X

(log D) is canonically viewed as a subsheaf of j

∗

W

m

Ω

^∗_U

(cf. [HK94, (4.20)]).

Note that

W

m

Ω

^∗_X|Dn

∼ = W

m

Ω

^∗_X

(log D) ⊗

_WmOX

W

m

O

_X

(−D

n

).

In particular, W

₁

Ω

^∗_X|D

n

= Ω

^∗_X

(log D) ⊗ O

_X

(−D

_n

) = Ω

^∗_X|D

n

(cf. notation in Theorem 1.2.1).

Lemma 2.1.4. We have the following inclusions:

(i) F(W

m+1

Ω

^∗_X|D

n

) ⊂ W

m

Ω

^∗_X|D

pn

; (ii) V (W

m

Ω

^∗_X|D

pn

) ⊂ W

m+1

Ω

^∗_X|D

n

; (iii) R(W

m+1

Ω

^∗_X|D

n

) ⊂ W

m

Ω

^∗_X|D

n

.

Proof.

This follows from Lemma 2.1.2 and the basic properties of de Rham–Witt complex [HK94,

§

4.1] [Lor02, Proposition 1.5].

2

2.2 Canonical filtration

On W

_m

Ω

^∗_X

(log D), we can define the canonical filtration as in [Ill79, I (3.1.1)]:

Fil

^s

W

m

Ω

^r_X

(log D) :=







W

m

Ω

^r_X

(log D) if s

6

0 or r

6

0, Ker(R

^m−s

: W

_m

Ω

^r_X

(log D)

→

W

_s

Ω

^r_X

(log D)) if 1

6

s

6

m,

0 if s

>

m.

For 1

6

s

6

m, we have [Lor02, Proposition 1.16]

Fil

^s

W

m

Ω

^r_X

(log D) = V

^s

W

m−s

Ω

^r_X

(log D) + dV

^s

W

m−s

Ω

^r−1_X

(log D).

Definition 2.2.1. For 1

6

s

6

m, we define

Fil

^s

W

m

Ω

^r_X|Dn

:=







W

_m

Ω

^r_X|D

n

if s

6

0 or r

6

0,

Ker(R

^m−s

: W

_m

Ω

^r_X|D

n →

W

_s

Ω

^r_X|D

n

) if 1

6

s

6

m,

0 if s

>

m.

Theorem 2.2.2. We have

Fil

^s

W

m

Ω

^r_X|Dn

= V

^s

W

m−s

Ω

^r_X|D

psn

+ dV

^s

W

m−s

Ω

^r−1_X|D

psn

.