PERIOD-INDEX PROBLEMS AND WILD RAMIFICATION

By

Yizhen Zhao

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Mathematics—Doctor of Philosophy

2024

ABSTRACT

This thesis consists of two main parts. The first part addresses period-index problems and symbol

length problems of the 𝑝∞-torsion part of Brauer groups of henselian discretely valued fields with

residue fields of characteristic 𝑝 > 0.

In Chapter 1, we provide an overview of the background, progress, and motivation behind

period-index problems of Brauer groups and, more generally, Kato’s groups. Chapter 2 recalls

some properties of Brauer groups, especially the 𝑝-torsion part. In Chapter 3, we survey Kato’s

unit group filtration and Kato’s Swan conductors, which are the main tools in this research area. We

investigate the symbol length problems of certain groups related to absolute logarithmic differential

forms over fields of characteristic 𝑝 > 0. This symbol length problem plays an important role in

the period-index problems of Kato’s groups.

Chapter 4 presents a systematic investigation of period-index problems of the 𝑝-torsion part of

Brauer groups of henselian discretely valued fields with residue fields of characteristic 𝑝 > 0. We

provide positive support for Chipchakov’s conjecture on this topic. Assuming a conjecture on the

symbol length, we offer a complete proof of Chipchakov’s conjecture on the Brauer 𝑝-dimension

of henselian discretely valued fields. We also generalize this idea to investigate the symbol length

problem of higher Kato’s groups, yielding results on the splitting dimension problems.

In Chapter 5, we use Kato’s Swan conductor to investigate the period-index problem of the

𝑝-torsion part of Brauer groups of semiglobal fields. Semiglobal fields are intermediate entities

between local fields and global fields. Using patching methods, we reduce the period-index

problems to two types: period-index problems of henselian discretely valued fields and quotient

fields of a complete local ring of Krull dimension 2. To study the second type, we employ a

Gersten-type exact sequence of logarithmic de Rham cohomology with support, analogous to the

Artin-Mumford ramification sequence. Both sequences are derived from the Bloch-Ogus spectral

sequence. We compute the logarithmic de Rham cohomology with support and their connecting

morphisms in this context. Using these computations, we obtain partial results on the period-index

problem of semiglobal fields in characteristic 𝑝 > 0.

Copyright by
YIZHEN ZHAO
2024

ACKNOWLEDGEMENTS

I would like to express my gratitude to everyone who has supported me over the past six years. It

has been a wonderful journey studying at Michigan State University.

First, I want to thank my advisor, Rajesh Kulkarni. Thank you for suggesting this interesting

problem and motivating me in the area of algebraic geometry. I will always remember how you

visualize algebraic problems as geometric pictures, guiding me throughout my studies. Thank

you for supporting me during difficult times and encouraging me to overcome challenges. Your

guidance was invaluable when I felt uncertain about my direction.

I am deeply grateful to Professor Michael Shapiro for teaching and encouraging me during the

exchange semester in Spring 2017. With your help, I took my first steps into research. Thank you

also for introducing me to my current advisor. I would like to thank Professors Aaron Levin, Igor

Rapinchuk, and Joe Waldron for being on my committee and providing guidance throughout my

studies.

I am thankful to my undergraduate thesis advisor, Guanghao Hong.

I am deeply saddened

that he passed away during the COVID-19 pandemic. Thank you for teaching me and helping me

discover my passion for mathematics.

I also want to thank all my friends who accompanied me on this journey. Special thanks to

Chuangtian Guan, Keping Huang, Zhixin Wang, Zheng Xiao, Chen Zhang, and Zhihao Zhao for

countless conversations about math and life. I want to thank my research group: Mike Annunziata,

Nick Rekuski, Yu Shen, Charlotte Ure, and Shitan Xu for inspiring me and listening to my practice

talks. I also want to thank my friends: Zichun Cao, Xudong Tangtian, Junnuo Yu, Kang Yu, and

Rui Zhang for all the great times we had. Additionally, I am grateful to Qihao Pan and Qinhan

Zhou for the vacations and conversations we shared, which brought me relief and joy.

Lastly, my deepest thanks go to my parents and elder sister for their unwavering encouragement

and support. Special thanks to my love, Yunlu Zhang. Thank you for always being by my side.

During moments of hesitation, you have always provided direction and strength.

iv

TABLE OF CONTENTS

CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
1.1 Motivation .
.
.
.
. . . . . . . . . . . . . . . . . . . . .
1.2 Case: henselian discretely valued fields
1.3 Case: semi-global fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
1.4 Period-index problems of higher Kato’s groups

.

.

.

.

1
1
5
7
8

CHAPTER 2

REVIEW OF BRAUER GROUPS . . . . . . . . . . . . . . . . . . . . . 10
. . . . . . . . . . . . . . . . . . . . . . . . 10
2.1 Basic properties of Brauer groups
2.2 Structure of 𝑝-primary part of Brauer groups
. . . . . . . . . . . . . . . . . . 12
2.3 Brauer group of a complete discretely valued field . . . . . . . . . . . . . . . . 16

CHAPTER 3

.

.

.

.

KATO’S GROUP AND SWAN CONDUCTOR . . . . . . . . . . . . . . 18
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1 Kato’s group .
3.2 Kato’s Swan conductor
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Equal characteristic case: char(𝐾) = 𝑝 > 0 . . . . . . . . . . . . . . . . . . . 22
3.4 Mixed characteristic case: char(𝐾) = 0 . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Symbol length problem of groups 𝐾 𝑀
𝐹 . . . . . . . . . . . . 28

2 (𝐹)/𝑝 and Ω1

𝐹/𝑍 1

CHAPTER 4

PERIOD-INDEX PROBLEMS OF HENSELIAN DISCRETELY VAL-
UED FIELDS .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1 Reduction to the 𝑝-torsion part of Brauer group . . . . . . . . . . . . . . . . . 32
4.2 Kato’s results in the 𝑝-rank 1 case . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 Equal characteristic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.4 Mixed characteristic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
. . . . . . . . . . . . . . . . 46
4.5 Symbol length problems of higher Kato’s groups

CHAPTER 5

LOGARITHMIC DE RHAM COHOMOLOGY WITH SUPPORT . . . . 50
5.1 Bloch-Ogus spectral sequence in positive characteristic . . . . . . . . . . . . . 51
5.2 Logarithmic de Rham cohomology of affine schemes with support
. . . . . . . 53
5.3 Period-index problems of semi-global fields in positive characteristic . . . . . . 57

BIBLIOGRAPHY .

. .

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

APPENDIX A

SYMBOL LENGTH AND FOLIATION THEORY . . . . . . . . . . . 68

APPENDIX B

RAMIFICATION OF CENTRAL DIVISION ALGEBRAS (𝑝-RANK
1 CASE) .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

.

.

v

CHAPTER 1

INTRODUCTION

1.1 Motivation

Let 𝑘 be a field. For any 𝑘-central simple algebra 𝐴, we denote by per( 𝐴) the order of its class

in the Brauer group Br(𝑘) (called the period) and by ind( 𝐴) its index which is the gcd of all the

degrees of finite splitting fields. It is well-known that

per( 𝐴) | ind( 𝐴),

and these two integers have the same prime factors. Hence the period is bounded by the index and

the index is bounded above by a power of the period. We use notion of the Brauer dimension to

make this relationship precise. For a prime 𝑝, define the Brauer dimension at 𝑝 as follows

Br.dim𝑝 (𝑘) := min

𝑑





ind( 𝐴) | per( 𝐴)𝑑

for any 𝐴 ∈ Br(𝑘) [ 𝑝𝑛] and 𝑛 ∈ N;

∞

otherwise.

Then define the Brauer dimension of 𝑘 to be

Br.dim(𝑘) = sup
𝑝

(cid:8)Br.dim𝑝 (𝑘)(cid:9) .

In general, the Brauer dimension of a field can be finite or infinite. The period-index problem of a

field is to investigate the Brauer dimension of the field.

An important class of fields for the period-index problem is that of 𝐶𝑚 fields. For any positive in-

teger 𝑚, we say a field 𝑘 satisfies condition 𝐶𝑚 if every homogeneous polynomial 𝑓 ∈ 𝑘 [𝑥1, · · · , 𝑥𝑛]
of degree 𝑑 with 𝑑𝑚 < 𝑛 has a nontrivial zero in 𝑘 𝑛 [27]. Here are some properties of 𝐶𝑚 fields:

1. If a field is 𝐶𝑚, then any finite extension is also 𝐶𝑚.

2. If a field is 𝐶𝑚, then any extension of transcendence degree 𝑛 is 𝐶𝑚+𝑛.

3. If a field 𝑘 is 𝐶𝑚, then 𝑘 ((𝑡)) the field of Laurent series is 𝐶𝑚+1.

4. The Brauer group of a 𝐶1 field is 0. [37]

The examples of 𝐶𝑚 fields include:

𝑪0 fields These are precisely the algebraically closed fields.

1

𝑪1 fields (quasi-algebraically closed fields)

• Finite fields

• The maximal unramified extension of a complete discretely valued field with a perfect

residue field

• Complete discretely valued fields with algebraically closed residue fields.

𝑪𝒎 fields If 𝑉 is a variety of dimension 𝑚 over an algebraically closed field 𝑘, then the function

field 𝑘 (𝑉) is 𝐶𝑚.

Michael Artin [3] conjectured that Br.dim(𝑘) ≤ 1 for a 𝐶2 field 𝑘. This conjecture has been

proved in many cases by several authors. This conjecture has a natural extension to all 𝐶𝑚 fields.

Conjecture 1.1.1

Let 𝑘 be a 𝐶𝑚 field. Then Br.dim(𝑘) ≤ 𝑚 − 1.

It’s important to note that the 𝑝-adic fields are not 𝐶2. Guy Terjanian [42] identified the 𝑝-adic

examples for all 𝑝 that are not 𝐶2. However, the Brauer dimension of 𝑝-adic fields (local fields)

remains 1.

Many recent studies on the period-index problem have been inspired by this conjecture. The

advances in tackling the period-index problems were reviewed [31] by the author. We briefly

mention some of these here.

(i) For 𝐹 a local or global field, Br.dim(𝐹) = 1 (Albert-Brauer-Hasse-Noether [17]).

(ii) For a 𝐶2 field 𝐹, Br.dim2(𝐹) = Br.dim3(𝐹) ≤ 1 (Artin [3]).

(iii) For a finitely generated field 𝐹 of transcendence degree 2 over an algebraically closed field,

Br.dim(𝐹) = 1 (de Jong [16], de Jong-Starr [40], and Lieblich [28]).

(iv) For a finitely generated field 𝐹 of transcendence degree 1 over an 𝑙-adic field, Br.dim𝑝 (𝐹) = 2

for every prime 𝑝 ≠ 𝑙 (Saltman [36]).

(v) If 𝐹 is a henselian discretely valued field with residue field 𝑘 such that Br.dim𝑝 (𝑙) ≤ 𝑑 for

all finite extension 𝑙/𝑘 and all primes 𝑝 ≠ char(𝑘), then Br.dim𝑝 (𝐹) ≤ 𝑑 + 1 for all primes

𝑝 ≠ char(𝑘). (Harbater, Hartmann and Krashen [19]).

By looking at these recent works, we notice that the Brauer 𝑝-dimension of a field 𝐹 is understood

2

systematically when 𝑝 is not equal to the residual characteristic of 𝐹. The main difficulty in the case

that 𝑝 coincides with the residual characteristic of 𝐹 is caused by the wild ramification behavior as

we explain now.

Let 𝐾 be a henselian discretely valued field with the residue field 𝐹 such that char(𝐹) = 𝑝 > 0.

Recall that every central simple algebra 𝐴 over 𝐾 is split by a finite separable extension of 𝐾

with degree ind( 𝐴). We can understand the ramification behavior of a Brauer class through the

ramification behavior of its separable splitting fields. A finite field extension 𝐿 of 𝐾 is called

tame [43], if the residue field extension is separable and the ramification degree is invertible in the

residue field 𝐹.

Let 𝑙 be a prime different from 𝑝 and 𝜔 ∈ Br(𝐾) [𝑙]. Then 𝜔 is split by a separable extension 𝐿

of 𝐾 with degree 𝑙𝑚 for some 𝑚 ∈ N. Since 𝑝 ∤ 𝑙𝑚, the ramification index and the residual degree

of 𝐿/𝐾 are both prime to 𝑝. Hence, the splitting field 𝐿 of 𝜔 is tame. When we work for the Brauer

dimension away from 𝑝, we only need to deal with the tame extensions. When a Brauer class is

split by a tame extension, we call it tamely ramified.

However, when we work with 𝑝-primary torsion Brauer classes over 𝐾, there are Brauer classes

not split by any tame extensions.

Example 1.1.2

Let 𝐾 = F𝑝 ((𝑠))((𝑡)) be the field of iterated Laurent series over F𝑝 in variables (𝑠, 𝑡) with the

complete discrete valuation given by the uniformizer 𝑡. The residue field of 𝐾 is 𝐹 = F𝑝 ((𝑠)).
𝑠
𝑡 𝑝

Consider the 𝐾-central division algebra [

, 𝑡) := {⟨𝑠, 𝑡⟩ | 𝑥 𝑝 − 𝑥 =

, 𝑦 𝑝 = 𝑡, 𝑦−1𝑥𝑦 = 𝑥 + 1}.

𝑠
𝑡 𝑝

It has the maximal order 𝐵 = O𝐾 ⟨1, 𝑡𝑥, 𝑦, 𝑡𝑥𝑦⟩. The residue division ring of 𝐵 is, in fact, a purely

inseparable extension of 𝐹 given by 𝐹 [ ¯𝑡𝑥]. We can show there is no tame extension of 𝐾 which

splits [

𝑠
𝑡 𝑝

, 𝑡).

We will call these Brauer classes wildly ramified. Moreover, the wildly ramified Brauer classes

are general members in the 𝑝-torsion part of Brauer groups, since the tamely ramified Brauer

classes only form a subgroup of the 𝑝-torsion part of the Brauer group of 𝐹.

To investigate the Brauer 𝑝-dimension of a field 𝐾 of residual characteristic 𝑝 > 0, we need

3

to interpret the 𝑝-torsion part of the Brauer group of 𝐾. There are two cases: equal characteristic

case and mixed characteristic case. First, suppose that 𝐾 is a field of characteristic 𝑝 > 0. Then

the 𝑝-torsion part of Br(𝐾) is related to the logarithmic differential form Ω1

𝐾,log. In fact, we have

the following

Br(𝐾) [ 𝑝] (cid:27) 𝐻1

ét(𝐾, Ω1

𝐾,log) (cid:27) Ω1
𝐾/

(cid:16)

P (Ω1

𝐾) + 𝑑 (𝐾)

(cid:17)

,

where P : Ω1

𝐾 → Ω1

𝐾/𝑑 (𝐾), 𝑎dlog(𝑏) ↦→ (𝑎 𝑝 − 𝑎)dlog(𝑏), and 𝑑 : 𝐾 → Ω1

𝐾 is the universal

derivation.

Second, if 𝐾 is a henselian discretely valued field of characteristic 0 containing a primitive 𝑝-th

root of the unity, with the residue field 𝐹 of characteristic 𝑝 > 0, by the Bloch-Kato theorem [7],

we have the following

Br(𝐾) [ 𝑝] = 𝐻2

ét(𝐾, 𝜇𝑝) (cid:27) 𝐻2

ét(𝐾, 𝜇⊗2

𝑝 ) (cid:27) 𝐾 𝑀

2 (𝐾)/𝑝,

where 𝜇 𝑝 is the group of 𝑝-th roots of the unity. This also holds for any field 𝐾 of characteristic

0 containing the primitive 𝑝-th root of the unity, which is implied by the norm residue theorem

proved by Voevodsky [45].

Now, in both cases, we can write a 𝑝-torsion Brauer classes as a sum of symbols. In the equal

characteristic case, symbols are differential forms. In the mixed characteristic case, symbols are

elements in the second Milnor 𝐾-group. Therefore, it leads to understand symbol algebras and

associated symbol length problems. We will talk about them in Chapter 3.

To analyse the wild ramified Brauer classes over a henselian discretely valued field 𝐾, Kato

defined an increasing filtration {𝑀𝑖}𝑖∈N on Br(𝐾) [ 𝑝]. Let 𝛼 ∈ Br(𝐾) [ 𝑝]. Then we can define

Kato’s Swan conductor sw(𝛼) of 𝛼 to be the minimal integer 𝑛 such that 𝛼 ∈ 𝑀𝑛. Kato also

described the consecutive quotients of this filtration.

It is the fundamental tool to analyse the

𝑝-torsion part of Brauer groups of henselian discretely valued fields. We will also review them in

Chapter 3.

4

1.2 Case: henselian discretely valued fields

Let us focus on henselian discretely valued fields first. Let 𝐾 be a henselian discretely valued

field with the residue field 𝐹 of characteristic 𝑝 > 0 and 𝑞 be a prime number. If 𝑞 ≠ 𝑝, it is proved

that Br.dim𝑞 (𝐾) ≤ 𝑑 + 1 if Br.dim𝑞 (𝐹) ≤ 𝑑 for all finite extension 𝐸/𝐹, by Harbater, Hartmann

and Krashen [19]. We hope that similar results also hold when 𝑞 = 𝑝. However, for any 𝑛 ≥ 0,

there are examples of complete discretely valued field 𝐾 with Br.dim𝑝 (𝐾) ≥ 𝑛 and Br.dim𝑝 (𝐹) = 0

by Parimala and Suresh [2014].

In fact, there are bounds for the Brauer 𝑝-dimension of 𝐾 in terms of the 𝑝-rank of 𝐹.

If

the 𝑝-rank of 𝐹 is 𝑛 < ∞, i.e. [𝐹 : 𝐹 𝑝] = 𝑝𝑛, the Brauer 𝑝-dimension of 𝐹 is no more than 𝑛

[11, Corollary 3.4]. Moreover, Chipchakov proved that Br.dim𝑝 (𝐾) ≥ 𝑛 if [𝐹 : 𝐹 𝑝] = 𝑝𝑛 and

Br.dim𝑝 (𝐾) is infinite if and only if 𝐹/𝐹 𝑝 is an infinite extension [13].

Conjecture 1.2.1 (1)

Let 𝐾 be a henselian discretely valued field with residue field 𝐹 of characteristic 𝑝 > 0. Assume

that [𝐹 : 𝐹 𝑝] = 𝑝𝑛. Then

𝑛 ≤ Br.dim𝑝 (𝐹) ≤ 𝑛 + 1.

When 𝑛 = 0, the residue field 𝐹 is perfect. Then there is no purely inseparable extension over 𝐹

and no wildly ramified Brauer class over 𝐾. Therefore, the first nontrivial case of the conjecture is

𝑛 = 1. Kato used the filtration and generalized Swan conductor to give the first result on the wildly

ramified Brauer classes when 𝐹 is complete. We state this elegant result in the following.

Theorem 1.2.2 (Proposition 4.2.1, [24, Section 4, Lemma 5])

Let 𝐾 be a complete discretely valued field with the residue field 𝐹 of characteristic 𝑝 > 0. Suppose

that [𝐹 : 𝐹 𝑝] = 𝑝. Let 𝜔 ∈ Br(𝐾) [ 𝑝] and sw(𝜔) > 0. Then the division algebra 𝐷 which

represents 𝜔 is a degree 𝑝 division algebra whose residue algebra is a purely inseparable field

extension of degree 𝑝 over 𝐹.

When the 𝑝-rank of the residue field is 1, it says that every wildly ramified Brauer class has

equal period and index. It is clear for 𝑝-torsion Brauer classes. For higher 𝑝∞-torsion classes, it

1[6, Conjecture 5.4] [13, Conjecture 1.1]

5

follows from the induction. Kato’s proof can also be applied to the case of henselian discretely

valued fields, since the results on the filtration and Kato’s Swan conductor work in a similar way as

ones in the complete discretely valued field case.

Next, we want to investigate the period-index bound when the 𝑝-rank of the residue field is

greater than 1. We are going to state our main results in this direction. The proof generalizes Kato’s

ideas in the 𝑝-rank 1 case.

Theorem 1.2.3

Let 𝐾 be a henselian discretely valued field with the residue field 𝐹 of characteristic 𝑝 > 0 and
[𝐹 : 𝐹 𝑝] = 𝑝𝑛, 𝑛 ∈ N>0. Suppose that 𝛼 ∈ Br(𝐾) [ 𝑝] and 𝑝 ∤ sw(𝛼) > 0. Then ind(𝛼) | per(𝛼)𝑛.

Notice that we have a restriction on the Swan conductors of Brauer classes. To remove this

restriction, however, we need estimates of the symbol length of two groups: Ω1

2 (𝐹)/𝑝.
The first group is the quotient of the absolute differential 1-forms modulo the closed differential

𝐹/𝑍 1

𝐹 and 𝐾 𝑀

1-forms. The second group is the second Milnor 𝐾-group of 𝐹 modulo by 𝑝. We propose the

following conjecture on the symbol length of both groups.

Conjecture 1.2.4 (Theorem 3.5.3, Conjecture 3.5.7)

Let 𝐹 be a field of characteristic 𝑝 > 0 and [𝐹 : 𝐹 𝑝] = 𝑝𝑛, 𝑛 ∈ N. Assume that 𝐹 does not admit

any finite extension of degree prime to 𝑝. Then both of the symbol length of the group Ω1

𝐹/𝑍 1

𝐹 and

𝐾 𝑀
2 (𝐹)/𝑝 are no more than 𝑛 − 1.

For this conjecture, the known case is ( 𝑝, 𝑛) = (2, 2). While we discuss henselian discretely

valued fields in both equal characteristic and mixed characteristic separately, these two groups

appear differently in the two cases. In the equal characteristic case, we only need the symbol length

result for the first group. However, in the mixed characteristic case, we need the symbol length

results for both groups. These requirements follow from Kato’s description of the consecutive

quotients of the filtration of Brauer groups.

6

Proposition 1.2.5 (( 𝑝, 𝑛) = (2, 2))

Let 𝐹 be a field of characteristic 𝑝 = 2 and [𝐹 : 𝐹 𝑝] = 𝑝2. Then

len(Ω1

𝐹/𝑍 1

𝐹) = len(𝐾 𝑀

2 (𝐹)/𝑝) = 1.

The conjecture regarding the symbol length problem implies the conjecture about the Brauer

𝑝-dimensions.

Theorem 1.2.6 (Smaller Bounds for Wildly Ramified Brauer Classes)

Let 𝐾 be a henselian discretely valued field with residue field 𝐹 of characteristic 𝑝 > 0 and

[𝐹 : 𝐹 𝑝] = 𝑝𝑛, 𝑛 ∈ N. Suppose that 𝐹 does not admit any finite extension of degree prime to 𝑝.

Let 𝛼 ∈ Br(𝐾) [ 𝑝] and sw(𝛼) > 0. Then Conjecture 1.2.4 implies ind(𝛼) | per(𝛼)𝑛.

To approach the symbol length of Ω1

𝐹/𝑍 1

𝐹, we will give several possible methods including the

brutal force way and the Galois correspondence of purely inseparable extensions of height 1.

1.3 Case: semi-global fields

Next, we consider semi-global fields. A semi-global field is one-variable function field 𝐹 over

a complete discretely valued field 𝐾, i.e. the function field of a curve over 𝐾. Examples include

𝐹 = Q𝑝 (𝑥), 𝐹 = 𝑘 ((𝑡))(𝑥), and any finite extension of these. These fields can be thought of as

intermediate objects between global fields and local fields. A natural question to ask is what their

Brauer 𝑝-dimensions are. Here is a list of known results regarding this question,

semi-global fields 𝐹 Br.dim𝑝 (𝐹)

F𝑝 ((𝑡)) (𝑥)

Q𝑝 (𝑥)

¯F𝑝 ((𝑡)) (𝑥)

Frac(𝑊 ( ¯F𝑝)) (𝑥)

2

2

?

?

where Frac(𝑊 ( ¯F𝑝)) is the fraction field of the Witt ring of ¯F𝑝. For the field 𝐹 = ¯F𝑝 ((𝑡)) (𝑥), as a

𝐶2 field, we expect that Br.dim𝑝 (𝐹) = 1. Indeed, we will demonstrate that there is a subgroup of

𝑝-torsion Brauer classes over 𝐹 that satisfies this period-index bound.

7

Theorem 1.3.1 (Theorem 5.3.1)

Let 𝑋 be a smooth projective curve over 𝑘 ((𝑡)) where 𝑘 is an algebraically closed fields of

characteristic 𝑝 > 0. Suppose that there is a model X over 𝑘 [[𝑡]] with good reduction. Suppose

that 𝜔 ∈ Br(𝑋) [ 𝑝] satisfies swX (𝜔) < 𝑝. Then per(𝜔) = ind(𝜔).

We have Br(𝑋) ↩→ Br(𝐹) by the purity of Brauer groups, where 𝐹 is the function field of 𝑋.

We use Kato’s Swan conductor to define a X-Swan conductor for elements in Br(𝑋) (Definition

5.3.4). The definition is based on the model X at the beginning. We do not know if the definition

depends on the choice of the model with good reduction.

We use the patching methods to reduce the period-index problem of the semi-global field to

two types of local period-index problems. The first type of period-index problem is addressed by

considering period-index problems of complete discretely valued with residual 𝑝-rank 1. The sec-

ond type of local period-index problem is analysed using a Gersten-type exact sequence (Theorem

5.1.4). We will discuss this Gersten-type exact sequence in detail in Chapter 5.

1.4 Period-index problems of higher Kato’s groups

Brauer groups are special cases of Kato’s groups. In the 1980s, Kato used differential forms to

define groups 𝐻𝑛+1(𝐹, Z/𝑚(𝑛)) for a field 𝐹 and a prime number 𝑚, even when 𝑚 is not invertible

in 𝐹. These groups generalize many arithmetical cohomological groups. For example,

• 𝐻1

• 𝐻2

ét(𝐹, Z/𝑚(0)) (cid:27) 𝐻1
ét(𝐹, Z/𝑚(1)) (cid:27) Br(𝐹) [𝑚]: the 𝑚-torsion subgroup of the Brauer group of 𝐹.

ét(𝐹, Z/𝑚): the group classifying cyclic Z/𝑚-extensions of 𝐹.

These higher cohomology groups have already been investigated from various perspectives.

We could also discuss the period-index bounds for these groups.

Definition 1.4.1 ([21])

Let 𝐹 be a field and 𝑝 be a prime number. A field extension 𝐸/𝐹 is called a splitting field for a class

𝛼 ∈ 𝐻𝑖
ét

(𝐹, (Z/𝑝)(𝑖 − 1)), if the image of 𝛼𝐸 of 𝛼 under the natural map 𝐻𝑖
ét

(𝐹, (Z/𝑝) (𝑖 − 1)) →

𝐻𝑖
ét

(𝐸, (Z/𝑝)(𝑖 − 1)) is trivial.

The index of a class 𝛼 ∈ 𝐻𝑖
ét
divisor of the degrees of splitting fields of 𝛼 that are finite over 𝐹.

(𝐹, (Z/𝑝) (𝑖 − 1)), denoted by ind(𝛼), is the greatest common

8

It is clear that the definitions of period-index for higher Kato’s groups are direct generalizations

of those for torsion parts of Brauer groups.

Since Kato’s filtration and Kato’s Swan conductor can be defined for these higher cohomology

groups, we also investigate the period-index bounds for these groups using a symbol length ap-

proach. More concretely, we prove that any wildly ramified element in 𝐻3
ét

(𝐾, (Z/𝑝) (2)) is split

by a purely inseparable extension of degree 𝑝, when 𝐾 is a henselian discretely valued field with a

residue field 𝐹 of characteristic 𝑝 > 0 and [𝐹 : 𝐹 𝑝] = 𝑝2.

Theorem 1.4.2 (Theorem 4.5.1, Theorem 4.5.2)

Let 𝐾 be a henselian discretely valued field with the residue field 𝐹 of characteristic 𝑝 > 0.

Suppose that [𝐹 : 𝐹 𝑝] = 𝑝2 and 𝐹 does not admit any finite extension of degree prime to 𝑝. Let

(𝐾, (Z/𝑝)(2)) such that sw(𝛼) > 0. Then Conjecture 3.5.7 implies that 𝛼 = 𝜔 ∧

𝛼 ∈ 𝐻3
ét
some 𝜔 ∈ Ω1

𝐾 and 𝑐 ∈ 𝐾 ×.

𝑑𝑐
𝑐 for

9

CHAPTER 2

REVIEW OF BRAUER GROUPS

2.1 Basic properties of Brauer groups

A central simple algebra 𝐴 (CSA) over a field 𝐾 is a finite-dimensional associative 𝐾-algebra

𝐴 that is simple with center 𝐾.

Two central simple algebras 𝐴, 𝐴′ are called Morita equivalent if there exist integers 𝑟, 𝑠 ∈ N

such that 𝐴 ⊗ 𝑀𝑟 (𝐾) ≃ 𝐴′ ⊗ 𝑀𝑠 (𝐾) as 𝐾-algebras. By the Artin-Wedderburn theorem, a finite-

dimensional simple algebra 𝐴 is isomorphic to the matrix algebra 𝑀𝑛 (𝐷) for a 𝐾-central division

algebra 𝐷. Moreover, such a division algebra is uniquely determined by a central simple algebra.

The Brauer group of a field 𝐾 is a torsion abelian group whose element are Morita equivalence

classes of central simple algebras over 𝐾. The addition in the Brauer group is given by the tensor

product of algebras.

As mentioned above, there is a unique central division algebra in each Brauer class. The degree

deg( 𝐴) of a central simple algebra 𝐴 is the integer 𝑛 such that dim𝐾 ( 𝐴) = 𝑛2. Then we define the

index ind( 𝐴) of a central simple algebra 𝐴 to be the degree of the division algebra 𝐷 associated to

𝐴 by the Artin-Wedderburn theorem. In particular, note that the index is well-defined for a Brauer

class. Also, for a Brauer class [ 𝐴] associated to a central simple algebra 𝐴, the period per( 𝐴) is its

order in the Brauer group Br(𝐾).

It is well-known that the period divides the index of a central simple algebra, and these two

integers have the same prime factors. So the index divides a power of the period. The period-index

problem asks if one can bound the index in terms of the power of the period. Here are the relevant

definitions from the introduction.

Definition 2.1.1 (Brauer dimension [31])

• Let 𝐾 be a field. For a prime 𝑝, the Brauer dimension at 𝑝, Br. dim𝑝 (𝐾), is the smallest

integer 𝑑 such that for any 𝐴 ∈ Br(𝐾) [ 𝑝𝑛], ind( 𝐴) | per( 𝐴)𝑑, and ∞ if no such number

exists.

10

• The Brauer dimension of 𝐾 is

Br.dim(𝐾) = sup
𝑝

(cid:8)Br.dim𝑝 (𝐾)(cid:9) .

The period-index problem asks if Br.dim(𝐾) is finite, and the local period-index problem asks

if Br.dim𝑝 (𝐾) is finite for an arbitrary prime 𝑝.

The Brauer group can also be defined in terms of Galois (étale) cohomology. We have

Br(𝐾) = 𝐻2

ét(𝐾, G𝑚),

where G𝑚 denotes the sheaf of units in the structure sheaf.

In general, the Brauer group of a scheme is defined in terms of Azumaya algebras. An Azumaya

algebra is a generalization of central simple algebras to 𝑅-algebras where 𝑅 may not be a field. For a

scheme 𝑋 with structure sheaf O𝑋, an Azumaya algebra on 𝑋 is a coherent sheaf A of O𝑋-algebras

that is étale locally isomorphic to the sheaf of matrices over the structure sheaf. The Brauer group

Br(𝑋) is an abelian group of equivalence classes of Azumaya algebras, with the addition given by

the tensor product of algebras. Here two Azumaya algebras A, A′ are considered to be equivalent

when M𝑟 (A) (cid:27) M𝑠 (A′) as sheaves of O𝑋-algebras for matrices of size 𝑟 × 𝑟 and 𝑠 × 𝑠 respectively.

As in the case of a field, we define the cohomological Brauer group of a quasi-compact scheme

𝑋 to be the torsion subgroup of the étale cohomology group 𝐻2

ét(𝑋, G𝑚). The cohomology group

ét(𝑋, G𝑚) is torsion for a regular scheme 𝑋, but it may not be torsion in general.
𝐻2
We recall several well-known facts about Brauer groups in the following.

Theorem 2.1.2 (O. Gabber)

The Brauer group of a scheme 𝑋 is equal to the cohomological Brauer group for any scheme with

an ample line bundle.

For example, when 𝑋 is quasi-projective over a field 𝑘, we have the coincidence of two Brauer

groups.

Theorem 2.1.3 (Purity in codimension 1 [44])

11

For a Noetherian, integral, regular scheme 𝑋 with function field 𝐾,

ét(𝑋, G𝑚) =
𝐻2

(cid:217)

𝑥∈𝑋 (1)

ét(O𝑋,𝑥, G𝑚) in 𝐻2
𝐻2

ét(𝐾, G𝑚).

2.2 Structure of 𝑝-primary part of Brauer groups

In this section, we assume all the fields have positive characteristic 𝑝 > 0. We focus on the

𝑝-primary part of the Brauer groups. First, we recall the 𝑝-primary counterpart of the Merkurjev-

Suslin theorem [1982]. The Merkurjev-Suslin theorem states that Br(𝐾) [𝑛] is generated by cyclic

algebras of degree 𝑛 when 𝐾 contains a primitive 𝑛-th root of unity 𝜇𝑛.

Firstly, we recall the Artin-Schreier-Witt theory of cyclic field extensions in positive character-

istic:

Theorem 2.2.1 ([37])

Let 𝑘 be a field of characteristic 𝑝 > 0. Denote by P : 𝑊𝑟 (𝑘) → 𝑊𝑟 (𝑘) the endmorphism of the
length-𝑟 Witt ring that maps (𝑥1, · · · , 𝑥𝑟) ∈ 𝑊𝑟 (𝑘) to (𝑥 𝑝

𝑟 ) − (𝑥1, · · · , 𝑥𝑟). Then there exists

, · · · , 𝑥 𝑝

1

a canonical isomorphism

𝑊𝑟 (𝑘)/P (𝑊𝑟 (𝑘)) (cid:27) 𝐻1

ét(𝑘, Z/𝑝𝑟).

Then we have the following theorem about the 𝑝𝑟-cyclic algebras (symbol algebras).

Proposition 2.2.2

Let 𝐾 be a field of characteristic 𝑝 > 0. For every 𝜔 ∈ Br(𝐾) [ 𝑝𝑟], we can write

∑︁

𝜔 =

[𝑎𝑖, 𝑏𝑖),

𝑖

as a sum of 𝑝𝑟-symbol algebras where 𝑎𝑖 ∈ 𝑊𝑟 (𝐾) and 𝑏𝑖 ∈ 𝐾 ×. The 𝑝𝑟-symbol algebra [𝑎𝑖, 𝑏𝑖) is

defined by

(cid:42)

[𝑎𝑖, 𝑏𝑖) :=

𝑥, 𝑦

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝑥 is a primitive element of the Artin-Schreier-Witt extension defined by

P (𝑥1, . . . , 𝑥𝑟) = 𝑎𝑖 and with a generator 𝜎 of the Galois group such that,

(cid:43)

.

𝑦 𝑝𝑛

= 𝑏𝑖, 𝑦−1𝑥𝑦 = 𝜎(𝑥).

The 𝑝𝑟-symbol algebra has index = period = 𝑝𝑟.

12

Example 2.2.3

Let 𝑎, 𝑏 ∈ 𝐾 and consider the 𝑝-symbol algebra [𝑎, 𝑏). By definition,

[𝑎, 𝑏) := ⟨𝑥, 𝑦 | 𝑥 𝑝 − 𝑥 = 𝑎, 𝑦 𝑝 = 𝑏, 𝑦−1𝑥𝑦 = 𝑥 + 1⟩.

This symbol algebra is the main object of our study, since we can reduce questions related to

𝑝𝑟-torsion Brauer classes to the 𝑝-symbol algebra by Theorem 2.2.7.

Next we relate the 𝑝-primary part of the Brauer group with the de Rham-Witt complex 𝑊𝑟Ω1
𝐾

[22]. We can identify Br(𝐾) [ 𝑝𝑟] with the cokernel of

𝐹 − 𝐼 : 𝑊𝑟Ω1

𝐾 → 𝑊𝑟Ω1

𝐾/𝑑𝑉 𝑟−1(𝐾),

(2.2.1)

where 𝐹 is Frobenius morphism and 𝐼 is the identity morphism.

Lemma 2.2.4 ([22])

Let 𝐾 be a field of characteristic 𝑝 > 0 and 𝑟 ∈ N+.

Br(𝐾) [ 𝑝𝑟] (cid:27) 𝑊𝑟Ω1

𝐾/

(cid:16)

(𝐹 − 𝐼)𝑊𝑟Ω1

𝐾 + 𝑑𝑊𝑟 (𝐾)

(cid:17)

.

(2.2.2)

Proof.

In fact, there exists an exact sequence of étale sheaves over the affine scheme 𝑋 = Spec(𝐾):

0

(cid:47) 𝑊𝑟Ω1

𝑋,log

(cid:47) 𝑊𝑟Ω1
𝑋

𝐹−𝐼 (cid:47)

(cid:47) 𝑊𝑟Ω1

𝑋/𝑑𝑉 𝑟−1O𝑋

(cid:47) 0 ,

which induces the cohomology group sequence

𝑊𝑟Ω1
𝐾

𝐹−𝐼 (cid:47)

(cid:47) 𝑊𝑟Ω1

𝐾/𝑑𝑉 𝑟−1𝐾 𝛿𝑟

(cid:47) 𝐻1

ét(𝐾, 𝑊𝑟Ω1

𝐾,log)

(cid:47) 0 ,

since 𝐻1

ét(𝐾, 𝑊𝑟Ω1

𝐾) = 0 by the quasi-coherence of 𝑊𝑟Ω1
𝐾.

Also, there is another exact sequence of étale sheaves which relates the 𝑝𝑟-torsion part of the

Brauer group with the logarithmic de Rham-Witt complex

0

(cid:47) G𝑚

𝑝𝑟

(cid:47) G𝑚

(cid:47) 𝑊𝑟Ω1

𝑋,log

(cid:47) 0.

13

(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
It induces the long exact cohomology sequence

0

(cid:47) 𝐻1

ét(𝐾, 𝑊𝑟Ω1

𝐾,log)

(cid:47) 𝐻2

ét(𝐾, G𝑚)

𝑝𝑟

(cid:47) 𝐻2

ét(𝐾, G𝑚),

where 𝐻1

ét(𝐾, G𝑚) = 0 by Hilbert’s Theorem 90.

By using the relation 𝑑 = 𝐹𝑟−1𝑑𝑉 𝑟−1 [22, (2.18)], it is easy to see that 𝛿𝑟 induces an isomorphism

between the cokernel of (2.2.1) and Br(𝐾) [ 𝑝𝑟].

□

Now we use the structure of 𝑊𝑟Ω1

𝐾 to describe 𝑝𝑟-torsion part of the Brauer group of 𝐾. We

recall some facts about 𝑊𝑟Ω1

𝐾 [2]. We use the notation [𝑎]𝑟 := (𝑎, 0, · · · , 0) ∈ 𝑊𝑟 (𝐾).

Definition 2.2.5

𝑀 1

𝑟 𝐾 ⊂ 𝑊𝑟Ω1

𝐾 denotes the subgroup generated by the elements [𝑎]𝑟 𝑑 [ 𝑓 ]𝑟 where 𝑎 ∈ 𝐾, 𝑓 ∈ 𝐾 ×.

Lemma 2.2.6 (Lemma 2.4, [2])

Let 𝑀 1

𝑟 𝐾 ⊂ 𝑊𝑟Ω1

𝐾 denote the subgroup generated by multiplicative elements [𝑎]𝑟 𝑑 [ 𝑓 ]𝑟. Then we

have

Moreover,

It follows that

𝑊𝑟Ω1

𝐾 =

𝑟−1
∑︁

𝑖=0

𝑉 𝑖 𝑀 1

𝑟−𝑖𝐾 +

𝑑𝑉 𝑖𝐾.

𝑟−1
∑︁

𝑖=0

𝑑𝑊𝑟 (𝐾) =

𝑟−1
∑︁

𝑖=0

𝑑𝑉 𝑖𝐾 ⊂ 𝑊𝑟Ω1
𝐾 .

Br𝑝𝑟 (𝐾) (cid:27) 𝑊𝑟Ω1
𝐾

(cid:44)

(cid:16)

(𝐹 − 𝐼)𝑊𝑟Ω1

𝐾 + 𝑑𝑊𝑟 (𝐾)

(cid:17) (cid:27)

𝑟−1
∑︁

𝑖=0

(cid:2)𝑉 𝑖 𝑀 1

𝑟−𝑖𝐾(cid:3) .

(2.2.3)

Then we relate the differential forms with symbol algebras by the following map

𝛿𝑟 : 𝑊𝑟Ω1

𝐾/((𝐹 − 𝐼)𝑊𝑟Ω1

𝐾 + 𝑑𝑊𝑟 (𝐾)) −→ Br(𝐾) [ 𝑝𝑟]

where 𝑎 ∈ 𝑊𝑟 (𝐾), 𝑏 ∈ 𝐾 ×, and dlog([𝑏]𝑟) = [𝑏]−1

𝑟 𝑑 [𝑏]𝑟.

𝑎 dlog( [𝑏]𝑟) ↦−→ [𝑎, 𝑏),

14

(cid:47)
(cid:47)
(cid:47)
We denote the composite map 𝑊𝑟Ω1

𝐾 → Br(𝐾) [ 𝑝𝑟] → Br(𝐾) by 𝛿𝑟 as well. We have a

commutative diagram

(cid:47) 𝑊𝑟Ω1
𝐾

𝑉

𝛿𝑟

𝑊𝑟−1Ω1
𝐾

𝛿𝑟 −1

Br(𝐾)

Using the isomorphism (2.2.3), it is easy to give a direct proof of the following theorem.

Theorem 2.2.7 ([25])

For a field 𝐾 of positive characteristic 𝑝 > 0 and 𝑚 ∈ N, we have an exact sequence:

0

(cid:47) Br(𝐾) [ 𝑝𝑚]

𝑉 (cid:47)

(cid:47) Br(𝐾) [ 𝑝𝑚+1]

𝑅1

(cid:47) Br(𝐾) [ 𝑝]

(cid:47) 0,

(2.2.4)

where 𝑅1 : 𝑊𝑚Ω1

𝐾 → Ω1

𝐾 sends [𝑎]𝑚dlog( [𝑏]𝑚) to 𝑎 dlog(𝑏).

The Brauer dimension at 𝑝 for a field of characteristic 𝑝 > 0 is effectively controlled by the

rank of the 𝑝-basis.

Definition 2.2.8 (𝑝-basis and 𝑝-rank)

Let 𝐾 be a field and [𝐾 : 𝐾 𝑝] = 𝑝𝑛, 𝑛 ≥ 0. A 𝑝-basis of 𝐾 is a subset {𝑥𝑖} ⊂ 𝐾 such that the
elements 𝑥𝐸 = (cid:206) 𝑥𝑒𝑖
𝑖 , 0 ≤ 𝑒𝑖 < 𝑝 form a basis of 𝐾 over 𝐾 𝑝, and the 𝑝-rank of 𝐾 is the number of

elements in the subset {𝑥𝑖}. Hence the 𝑝-rank of 𝐾 is 𝑛.

Proposition 2.2.9 ([11, Corollary 3.4])

Let 𝐾 be a field with [𝐾 : 𝐾 𝑝] = 𝑝𝑛. Then Br.dim𝑝 (𝐾) ≤ 𝑛.

Proof. For 𝑟 ∈ N and a 𝑝-basis {𝑎𝑖}𝑛

𝑖=1 of 𝐾, by Theorem 2.2.7 and induction, every 𝑝𝑟-torsion
Brauer class can be written as a sum of 𝑛 symbol algebras [𝑐𝑖, 𝑎𝑖), where 𝑐𝑖 ∈ 𝑊𝑟 (𝐾) for 𝑖 ∈

{1, · · · , 𝑛}. Then the proposition follows from the following lemma.

□

Lemma 2.2.10 ([1, Ch. VII, Lemma 13])

Let 𝐾 be a field of characteristic 𝑝 > 0. If 𝐴, 𝐵 are two symbol algebras of degree 𝑝𝑚 and 𝑝𝑛

respectively, then 𝐴 ⊗ 𝐵 is Brauer equivalent to a symbol algebra of degree no more than 𝑝𝑚+𝑛.

15

(cid:15)
(cid:15)
(cid:47)
(cid:121)
(cid:121)
(cid:47)
(cid:47)
(cid:47)
2.3 Brauer group of a complete discretely valued field

In this section, 𝐾 denotes a complete discretely valued field with valuation ring O𝐾, residue

field 𝐹 and maximal ideal 𝑚𝐾 = (𝜋). The valuation of 𝐾 is denoted by 𝑣𝐾. Recall that a discrete

valuation is a map 𝑣𝐾 : 𝐾 → Z ∪ {∞} that satisfies:

(i) 𝑣𝐾 (𝑎) = ∞ if and only if 𝑎 = 0;

(ii) 𝑣𝐾 (𝑎𝑏) = 𝑣𝐾 (𝑎) + 𝑣𝐾 (𝑏);

(iii) 𝑣𝐾 (𝑎 + 𝑏) ≥ min(𝑣𝐾 (𝑎), 𝑣𝐾 (𝑏)), with equality if 𝑣𝐾 (𝑎) ≠ 𝑣𝐾 (𝑏).

The valuation ring O𝐾 = 𝑣−1

𝐾 (Z≥0) is a complete local ring of Krull dimension 1. For a
complete discretely valued field 𝐾, we can extend the complete valuation 𝑣𝐾 to central simple

division algebras over 𝐾 and consider the residue division algebras. They are summarized in the

following proposition.

Proposition 2.3.1 (Proposition 1.3.1, [5])

Let 𝐷 be a central division 𝐾-algebra.

(i) The function 𝑤 : 𝐷 → Z ∪ {∞} defined by 𝑤(𝑎) = 𝑣𝐾 (det(𝑎)) is a discrete valuation on 𝐷.

(ii) The set 𝐵 (cid:66) {𝑎 ∈ 𝐷 | 𝑤(𝑎) ≥ 0} = {𝑎 | det(𝑎) ∈ O𝐾 } is the unique maximal O𝐾-order in

𝐷.

(iii) 𝐵 is a local domain with maximal ideal 𝐽 (cid:66) {𝑎 | 𝑤(𝑎) > 0}; the residue ring Δ = 𝐵/𝐽 is a

division ring.

(iv) If 𝜋 is an element of 𝐽 such that 𝑤(𝜋) takes the minimal positive value, then 𝐽 = 𝐵𝜋 = 𝜋𝐵

Next we study the unique maximal order 𝐵 in the above proposition. Let 𝐹′ be the center of Δ.

Then we have the integers 𝑑, 𝑒, 𝑒′, 𝑓 , 𝑛 defined as follows:

𝑑 = 𝑤(𝜋), 𝐽 𝑒 = 𝑚𝐾 𝐵, 𝑒′ = [𝐹′ : 𝐹],

𝑓 2 = [Δ : 𝐹′], 𝑛2 = [𝐷 : 𝐾].

(2.3.1)

Here 𝑛 is the degree (index) of 𝐷, and also its degree.

Lemma 2.3.2 ([5, Lemma 1.3.7])

𝑒𝑑 = 𝑛, and 𝑒𝑒′ 𝑓 2 = 𝑛2.

Corollary 2.3.3

16

[𝐷 : 𝐾] = 1 if and only if [Δ : 𝐹] = 1.

Proof. This is immediate from the above lemma.

□

In the latter part, we are interested in the case that the residue field 𝐹 is quasi-algebraically

closed, i.e a 𝐶1 field. Recall that a finite extension of a 𝐶1 field is also 𝐶1. Hence, the central

division algebra Δ over 𝐹′ will be isomorphic to 𝐹′, since Br(𝐹′) = 0. This implies 𝑓 = 1.

Lemma 2.3.4

Suppose that the residue field 𝐹 is 𝐶1 and [𝐹 : 𝐹 𝑝] = 𝑝. Then 𝑒 = 𝑒′ = 𝑛 and 𝑑 = 1.

Proof.

We already have 𝑓 = 1 and so it suffices to show 𝑒′ ≤ 𝑛 by Lemma 2.3.2. We will show that

any field extension 𝐹′ of 𝐹 is simple. In this case, 𝐹′ = 𝐹 [𝛼] and we choose 𝛽 ∈ 𝐵 such that

¯𝛽 = 𝛼 ∈ 𝐹′. Then we have 𝑒′ ≤ [𝐾 (𝛽) : 𝐾] ≤ 𝑛, since ind(𝐷) is 𝑛.

Now we prove that any finite field extension 𝐹′ of 𝐹 is simple. The field extension 𝐹 ⊂ 𝐹′ can

be written as a chain of field extensions 𝐹 ⊂ 𝐸 ⊂ 𝐹′ such that 𝐸 is separable over 𝐹 and 𝐹′ is

purely inseparable over 𝐸. It follows that [𝐸 : 𝐸 𝑝] = 𝑝 by Lemma 2.3.5 below. Then the purely

inseparable extension 𝐹′/𝐸 is simple. Set 𝐹′ = 𝐸 [𝛼1] and 𝛼1 is algebraic over 𝐹. We can also

denote 𝐸 = 𝐹 [𝛼2] by Theorem 2.3.6 below, since 𝐸/𝐹 is finite and separable. Finally, we get

𝐹 ⊂ 𝐹′ = 𝐹 [𝛼1, 𝛼2] is simple by Theorem 2.3.6 again.

□

Lemma 2.3.5 ([9, A.V.135, Corollary 3])

Let 𝑙/𝑘 be a finite or separable field extension of fields of characteristic 𝑝, and let 𝑛 be the 𝑝-rank

of 𝑘. Then the 𝑝-rank of 𝑙 is also 𝑛.

Theorem 2.3.6 ([33, Theorem 5.1])

Let 𝑙 = 𝑘 [𝛼1, · · · , 𝛼𝑟] be a finite extension of 𝑘, and assume that 𝛼2, · · · , 𝛼𝑟 are separable over 𝑘

(but not necessarily 𝛼1). Then there exists a 𝛾 ∈ 𝐸 such that 𝑙 = 𝑘 [𝛾].

17

CHAPTER 3

KATO’S GROUP AND SWAN CONDUCTOR

3.1 Kato’s group

In the 1980s, Kato used differential forms to define groups 𝐻𝑖
ét

(𝑘, (Z)/𝑚( 𝑗)) for a field 𝑘 and

any positive integer 𝑚, especially when 𝑚 is not invertible in 𝑘. These groups generalize many

well-known arithmetic cohomology groups. For example, we have 𝐻1

group classifying cyclic Z/𝑚-extensions of 𝑘 with generators, and 𝐻2

ét(𝑘, Z/𝑚) (cid:27) 𝐻1
ét(𝑘, Z/𝑚), the
ét(𝑘, (Z/𝑚) (1)) (cid:27) Br(𝑘) [𝑚],

the 𝑚-torsion part of the Brauer group of 𝑘.

In fact, there is an explanation for Kato’s groups: Voevodsky’s étale motivic cohomology

groups 𝐻𝑖
ét

(𝑋, 𝐴( 𝑗)) of a scheme 𝑋 over a field 𝑘 are defined for any abelian group 𝐴. They agree

with Kato’s groups when 𝑋 = Spec(𝑘) and 𝐴 = Z/𝑚 for any 𝑚.

It is especially of interest to investigate Kato’s groups of a field 𝑘 when 𝑘 has residual charac-

teristic 𝑝 > 0.

Definition 3.1.1

We say a field 𝑘 has residual characteristic 𝑝 > 0 if it satisfies one of the following conditions:

(i) 𝑘 is of characteristic 𝑝 > 0;

(ii) 𝑘 is a discretely valued field with a residue field of characteristic 𝑝 > 0.

We will describe our approaches to Kato’s groups in both cases.

3.1.1 Case: characteristic 𝑝 > 0

Let us start with the definition of Kato’s groups when 𝑘 is of characteristic 𝑝 > 0 and 𝑚 =

𝑝𝑟, 𝑟 ∈ N. For 𝑗 ≥ 0, let Ω

𝑗
𝑘 := Ω

be the subgroup of Ω

More generally, let 𝑊𝑟Ω

𝑗
𝑘 generated by logarithmic differential

𝑗
𝑘/Z be the group of absolute Kähler differential forms and Ω
𝑑𝑓1
𝑓1

𝑗
𝑘,log
for 𝑓1, . . . , 𝑓 𝑗 ∈ 𝑘 ×.

𝑗
𝑘,log be the analogous group of logarithmic de Rham-Witt differentials [22].

𝑑𝑓 𝑗
𝑓 𝑗

∧ · · · ∧

Then we have the following

ét(𝑘, (Z/𝑝𝑟) ( 𝑗)) (cid:27) 𝐻𝑖− 𝑗
𝐻𝑖

ét

(𝑘, 𝑊𝑟Ω

𝑗
log

).

(3.1.1)

Since the étale 𝑝-cohomological dimension of 𝑘 is at most 1 [17, Proposition 6.1.9],

18

𝐻𝑖
ét

(𝑘, (Z/𝑝𝑟)( 𝑗)) is zero except when 𝑖 is 𝑗 or 𝑗 + 1. When 𝑖 = 𝑗, Bloch, Gabber and Kato [7,

Corollary 2.8] showed that

𝐻 𝑗
ét

(𝑘, (Z/𝑝𝑟)( 𝑗)) (cid:27) 𝐻0

ét(𝑘, 𝑊𝑟Ω

𝑗
log

) (cid:27) 𝑊𝑟Ω

𝑗
𝑘,log

(cid:27) 𝐾 𝑀

𝑗 (𝑘)/𝑝𝑟,

(3.1.2)

where 𝐾 𝑀

𝑗 (𝑘) is the Milnor 𝐾-group. When 𝑖 = 𝑗 + 1, one way to describe these groups is in terms

of Galois cohomology. First, we focus on the case 𝑟 = 1. Let 𝑘 𝑠 be a separable closure of 𝑘. Then

𝐻 𝑗+1
ét

(𝑘, (Z/𝑝) ( 𝑗)) (cid:27) 𝐻1

Gal(𝑘, Ω

𝑗
𝑘 𝑠,log

).

(3.1.3)

To give a more precise description of the case 𝑖 = 𝑗 + 1, we recall the original definition from

Kato [26]. We define a group homomorphism P : Ω

𝑗
𝑘 → Ω

𝑗
𝑘 /𝑑Ω

𝑗−1
𝑘

by

P (𝑎

𝑑𝑏1
𝑏1

∧ · · · ∧

𝑑𝑏 𝑗
𝑏 𝑗

) = (𝑎 𝑝 − 𝑎)

𝑑𝑏1
𝑏1

∧ · · · ∧

𝑑𝑏 𝑗
𝑏 𝑗

.

Then there is an exact sequence of groups

0

(cid:47) 𝐻0
ét

(𝑘, Ω1

𝑘,log

)

(cid:47) Ω

𝑗
𝑘

P (cid:47)

(cid:47) Ω

𝑗
𝑘 /𝑑Ω

𝑗−1
𝑘

(cid:47) 𝐻1

ét(𝑘, Ω

𝑗
𝑘,log

)

(cid:47) 0.

Therefore, 𝐻1

ét(𝑘, Ω

𝑗
𝑘,log

) is isomorphic to the cokernel of P.

In conclusion, we have the following full description of 𝐻𝑖
ét

(𝑘, (Z/𝑝) ( 𝑗)) for a field 𝑘 of

characteristic 𝑝 > 0:

𝐻𝑖
ét(𝑘, (Z/𝑝)( 𝑗)) (cid:27)

Ω

(cid:16)

Ω

𝑗
𝑘 /

P (Ω

𝑗
𝑘,log
𝑗
𝑘 ) + 𝑑Ω

(cid:17)

𝑗−1
𝑘

0

if 𝑖 = 𝑗

if 𝑖 = 𝑗 + 1

otherwise.






(3.1.4)

Notice that these cohomology groups appear as subgroups or quotient groups of the group of the

absolute Kähler differential forms. Hence, we can express an element in these groups as a sum of

symbols. The symbols can be regarded as either equivalent classes of differential forms or elements

in the Milnor 𝐾-groups by Bloch-Kato-Gabber [7].

3.1.2 Case: characteristic 0

Now let 𝑘 be a field of characteristic 0 and 𝑝 be a prime number. Moreover, assume 𝑘 contains

a primitive 𝑝-th root 𝜁 of unity. This assumption assures that we can use symbols to investigate

19

(cid:47)
(cid:47)
(cid:47)
(cid:47)
Kato’s groups. Recall the norm residue isomorphism theorem (Bloch-Kato conjecture), which is

proved by Voevodsky [45].

Theorem 3.1.2 (Norm residue isomorphism theorem [45])

Let 𝐾 be a field and 𝑝 an integer invertible in 𝐾. Then

𝐻𝑛 (𝐾, (Z/𝑝) (𝑛)) (cid:27) 𝐾 𝑀

𝑛 (𝐾)/𝑝.

The norm residue isomorphism is firstly proved by Bloch and Kato [7] in the case of complete

discretely valued fields. Then Murkerjev and Suslin [32] proved the case 𝑛 = 2. Finally, Voevodsky

[45] used the motivic cohomology to finish the general proof.

Using the primitive 𝑝-th root 𝜁 of the unity , we can identify Z/𝑝 = (Z/𝑝) (1) : 1 ↦→ 𝜁.

Therefore, for any 𝑖 ∈ N, we have

ét(𝑘, (Z/𝑝)(𝑖 − 1)) = 𝐻𝑖
𝐻𝑖

ét(𝑘, (Z/𝑝) (𝑖)) (cid:27) 𝐾 𝑀

𝑖

(𝑘)/𝑝.

(3.1.5)

Then we can describe the elements in Kato’s groups by symbols from Milnor 𝐾-groups again.

3.2 Kato’s Swan conductor

Let 𝐾 be a complete discretely valued field with residue field 𝐹, and 𝐿 be a finite Galois

extension of 𝐾. Classically, the Swan conductor of a character of Gal(𝐿/𝐾) is defined in the case

where the residue field of 𝐿 is separable over 𝐹. Kato [26] provided a natural definition of the Swan

conductor without requiring the residue field extension to be separable. More generally, he defined

Swan conductors for elements in Kato’s groups. The classical Swan conductor measures the wild

ramification of the extension, while Kato’s Swan conductor naturally extends this to measure the

wild ramification of Brauer classes and other elements in higher Kato’s groups.

As Kato’s Swan conductors measure the wild ramification behaviors, we will concentrate on

fields of residual characteristic 𝑝 > 0 and Kato’s groups with coefficient in Z/𝑝.

Notation 3.2.1 ([26])

Let 𝐾 be a field. We define

𝐻𝑞

𝑝 (𝐾) (cid:66) 𝐻𝑞
ét

(𝐾, (Z/𝑝) (𝑞 − 1)).

20

When 𝐾 is of characteristic 𝑝 > 0, we have (Z/𝑝) (𝑞 − 1) ≃ Ω

𝑞−1
log

[−(𝑞 − 1)] in 𝐷 𝑏 (𝐾ét). Then it

follows that

𝐻𝑞

𝑝 (𝐾) = 𝐻1

ét(𝐾, Ω

𝑞−1
𝐾,log

).

(★) In the rest of this chapter, we denote by 𝐾 a henselian discretely valued field with the

valuation 𝑣. Let O𝐾 be the discrete valuation ring of 𝐾

O𝐾 = {𝑥 ∈ 𝐾 | 𝑣(𝑥) ≥ 0}

(3.2.1)

with the maximal ideal 𝑚, and let 𝐹 = O𝐾/𝑚 be the residue field.

Definition 3.2.2 (Unit group filtration)

Let 𝑈𝐾 = (O𝐾)× be the group of units in the ring O𝐾. For each 𝑖 ∈ N, consider the subgroup

𝑈𝑖

𝐾 = {𝑥 ∈ 𝑈𝐾 | 𝑣(𝑥 − 1) ≥ 𝑖} for 𝑖 ≥ 1.

(3.2.2)

Then since 𝑈𝐾 ⊃ 𝑈1

𝐾 ⊃ 𝑈1

𝐾 ⊃ · · · , we have defined a decreasing filtration on 𝑈𝐾.

In the bounded derived category 𝐷 𝑏 (𝐾ét), we have an exact triangle

(Z/𝑝)(1)

(cid:47) G𝑚

𝑝

(cid:47) G𝑚

(cid:47) (Z/𝑝) (1) [1] .

(3.2.3)

Given 𝑎 ∈ 𝐾 × = 𝐻0(𝐾, G𝑚), we denote the image of 𝑎 in 𝐻1

ét(𝐾, (Z/𝑝) (1)) by {𝑎}.

Then we have the product maps:

𝐻𝑞

𝑛 (𝐾) × (𝐾 ×)⊕𝑟 → 𝐻𝑞+𝑟

𝑛

(𝐾)

defined by ( 𝜒, 𝑎1, · · · , 𝑎𝑟) ↦→ { 𝜒, 𝑎1, · · · , 𝑎𝑟 } (cid:66) 𝜒 ∪ {𝑎1} ∪ · · · ∪ {𝑎𝑟 }.

Definition 3.2.3 (Kato’s filtration [26, Proposition 6.3])
The increasing filtration {𝑀 𝑝

𝑛 }𝑛≥0 on 𝐻𝑞

𝑝 (𝐾) is defined by:

𝜒 ∈ 𝑀 𝑝

𝑛 ⇐⇒ { 𝜒𝐿, 1 + 𝜋𝑛+1O𝐿 } = 0 in 𝐻𝑞+1

𝑝

(𝐿)

for any henselian discrete valuation field 𝐿 over 𝐾 such that O𝐾 ⊂ O𝐿 and 𝑚 𝐿 = O𝐿𝑚𝐾.

To see this filtration is well-defined, we refer to Kato’s original paper [26, Proposition 1.8,

Lemma 2.2]. We have 𝐻𝑞

𝑝 (𝐾) = (cid:208)
𝑛≥0

𝑀 𝑝

𝑛 . Now we are ready to define Kato’s Swan conductors.

21

(cid:47)
(cid:47)
(cid:47)
Definition 3.2.4 (Kato’s Swan conductor [26, Definition 2.3])
Let 𝜒 ∈ 𝐻𝑞
such that 𝜒 ∈ 𝑀 𝑝

𝑛 , i.e.

𝑝 (𝐾). We define Kato’s Swan conductor sw( 𝜒) ∈ N to be the minimum integer 𝑛 ≥ 0

sw𝐹 ( 𝜒) (cid:66) min{𝑛 ∈ N | 𝜒 ∈ 𝑀 𝑝

𝑛 }.

As we mentioned earlier, the Kato’s Swan conductor measures the wild ramifications of elements

in Kato’s groups. We usually consider the Kato’s group 𝐻𝑞

𝑝 (𝐾) when the residue field 𝐹 of 𝐾 is of

characteristic 𝑝 > 0.

Notice that the above definition of Kato’s filtration is independent of the characteristic of 𝐾.

The following proposition tells that there is no wild ramification if we look at the Kato’s groups

with torsion away from the residual characteristic.

Proposition 3.2.5 ([26, Corollary 2.5])

Let 𝐾 be a henselian discretely valued field with the residue field 𝐹 of characteristic 𝑝 > 0 and
𝑙 ≠ 𝑝 be a prime. Then 𝐻𝑞

𝑙 (𝐾) = 𝑀 𝑙

0 for all 𝑞 ∈ N.

When the torsion of Kato’s group is understood from the context, we will simply denote the

filtration by {𝑀𝑛}.

In the next two sections, we will describe the consecutive quotients of this

filtration, based on the characteristic of 𝐾.

3.3 Equal characteristic case: char(𝐾) = 𝑝 > 0

Recall that 𝐾 is a henselian discretely valued field with the discrete valuation 𝑣 and the residue

field 𝐹 of characteristic 𝑝 > 0. We assume char(𝐾) = 𝑝 > 0 in this section. Then we have

𝐻𝑞

𝑝 (𝐾) = 𝐻𝑞
ét

(𝐾, (Z/𝑝)(𝑞 − 1)) = 𝐻1

ét(𝐾, Ω

𝑞−1
𝐾,log

) (cid:27)

𝑞−1
Ω
𝐾
𝑞−1
𝐾 + 𝑑Ω
(Fr − 𝐼)Ω

,

𝑞−2
𝐾

where Fr is the Frobenius morphism. Kato generalized Brylinski’s filtration [10] on Witt vectors to
define an increasing filtration {𝑀 𝑗 } 𝑗 ≥0 on the 𝑝-primary Kato’s groups. For our purpose, we only
𝑝 (𝐾). For 𝑗 ≥ 0, 𝑀 𝑗 is the subgroup of 𝐻𝑞
consider the 𝑝-torsion one 𝐻𝑞
𝑝 (𝐾) generated by elements

of the form

𝑎

𝑑𝑏1
𝑏1

∧ · · · ∧

𝑑𝑏𝑞−1
𝑏𝑞−1

22

with 𝑎 ∈ 𝐾, 𝑏1, . . . , 𝑏𝑞−1 ∈ 𝐾 ×, and 𝑣(𝑎) ≥ − 𝑗. It is clear that

0 ⊂ 𝑀 0 ⊂ 𝑀 1 ⊂ · · · ,

𝑀 𝑗 = 𝐻𝑞

with (cid:208) 𝑗 ≥0
𝑀 𝑗 = 𝑀 𝑗 for each 𝑗 [26, Theorem 3.2]. Therefore, we will use 𝑀 𝑗 in the following context for

𝑝 (𝐾). Kato proved that the two filtrations {𝑀 𝑗 } and {𝑀 𝑗 } coincide, that is,

convenience.

Let 𝜋 ∈ O𝐾 be a uniformizer for 𝑣. For any 𝑗 > 0, we define two homomorphisms depending

on whether 𝑗 is relatively prime to 𝑝 or 𝑝 | 𝑗. In each case, a simple computation shows that the

homomorphim is well defined up to a choice of a uniformizer. First, consider the case when 𝑗 is

relatively prime to 𝑝. We define

Ω

𝑞−1
𝐹 → 𝑀 𝑗 /𝑀 𝑗−1

by

¯𝑎

𝑑 ¯𝑏1
¯𝑏1

∧ · · · ∧

𝑑 ¯𝑏𝑞−1
¯𝑏𝑞−1

↦→

𝑎
𝜋 𝑗

𝑑𝑏1
𝑏1

∧ · · · ∧

𝑑𝑏𝑞−1
𝑏𝑞−1

(mod 𝑀 𝑗−1),

for 𝑎 ∈ O𝐾 and 𝑏1, . . . , 𝑏𝑞−1 ∈ O×
𝐾.

Now we define the second homomorphism. Let 𝑍 𝑞−1
𝑞−1
𝐹 , Ω

respectively. For 𝑗 > 0 and 𝑝 | 𝑗, define a homomorphism

𝐹 , 𝑍 𝑞−2

𝑞−2
𝐹

𝐹

be the subgroup of closed forms in

Ω

Ω

𝐹 /𝑍 𝑞−1
𝑞−1

𝐹 ⊕ Ω

𝐹 /𝑍 𝑞−2
𝑞−2

𝐹 → 𝑀 𝑗 /𝑀 𝑗−1

as follows: On the first summand, it is defined as

¯𝑎

𝑑 ¯𝑏1
¯𝑏1

∧ · · · ∧

𝑑 ¯𝑏𝑞−1
¯𝑏𝑞−1

↦→

𝑎
𝜋 𝑗

𝑑𝑏1
𝑏1

∧ · · · ∧

𝑑𝑏𝑞−1
𝑏𝑞−1

(mod 𝑀 𝑗−1),

and for the second summand it is defined as

¯𝑎

𝑑 ¯𝑏1
¯𝑏1

∧ · · · ∧

𝑑 ¯𝑏𝑞−2
¯𝑏𝑞−2

↦→

𝑎
𝜋 𝑗

𝑑𝜋
𝜋

∧

𝑑𝑏1
𝑏1

∧ · · · ∧

𝑑𝑏𝑞−2
𝑏𝑞−2

(mod 𝑀 𝑗−1),

where 𝑎 ∈ O𝐾 and 𝑏1, . . . , 𝑏𝑞−1 ∈ O×
𝐾.

The homomorphisms are well defined (although they depend on the choice of uniformizer 𝜋).

We recall Cartier’s theorem in this context. It says that, for a field 𝑘 of characteristic 𝑝 > 0, 𝑞 ∈ N,

23

the subgroups 𝑍 𝑞
the form 𝑎 𝑝 (𝑑𝑏1/𝑏1) ∧ · · · ∧ (𝑑𝑏𝑞/𝑏𝑞) [23, Lemma 1.5.1].

𝑘 of closed forms in Ω

𝑞
𝑘 is generated by the exact forms together with the forms of

To describe the subgroup 𝑀0, we need to describe tame extensions of 𝐾 [43]. We fix a discrete

valuation 𝑣 as above. An extension field of 𝐾 is called tame with respect to 𝑣 if it is a union of

finite extensions of 𝐾 for which the extension of residue fields is separable and the ramification

degree is invertible in the residue field 𝐹. Let 𝐾tame be the maximal tamely ramified extension of

𝐾 (with respect to 𝑣) in a separable closure of 𝐾. Define the tame (or tamely ramified) subgroup of
𝐻𝑞
ét

(𝐾, (Z/𝑝)(𝑞 − 1)) by

𝐻𝑞

tame(𝐾, (Z/𝑝)/𝑝(𝑞 − 1)) = ker

(cid:16)

𝐻𝑞
ét

(𝐾, (Z/𝑝)/𝑝(𝑞 − 1)) → 𝐻𝑞
ét

(𝐾tame, (Z/𝑝)/𝑝(𝑞 − 1))

(cid:17)

.

There is residue homomorphism on the tamely ramified subgroup

𝜕𝑣 : 𝐻𝑞

tame(𝐾, (Z/𝑝) (𝑞 − 1)) → 𝐻𝑞−1

ét

(𝐹, (Z/𝑝) (𝑞 − 2)),

characterized by the property that

𝜕𝑣 (𝑎

𝑑𝜋
𝜋

∧

𝑑𝑏1
𝑏1

∧ · · · ∧

𝑑𝑏𝑞−2
𝑏𝑞−2

) = ¯𝑎

𝑑 ¯𝑏1
¯𝑏1

∧ · · · ∧

𝑑 ¯𝑏𝑞−2
¯𝑏𝑞−2

,

where 𝑎 ∈ O𝐾, 𝑏1, . . . , 𝑏𝑞−2 ∈ O×
group follows from the theorem below. Then we define the unramified subgroup 𝐻𝑞

𝐾. Note that this description of elements of the tamely ramified sub-
nr(𝐾, (Z/𝑝)(𝑞 −

1)) to be the kernel of the residue homomorphism 𝜕𝑣.

Theorem 3.3.1 (Equal characteristic case: char(𝐾) = 𝑝 > 0 [26, 43])

Let 𝐾 be a henselian discretely valued field of characteristic 𝑝 > 0 with the residue field 𝐹 and
𝑞 be a positive integer. Then the 𝑝-torsion Kato’s group 𝐻𝑞

(𝐾, (Z/𝑝) (𝑞 − 1)) has an

𝑝 (𝐾) = 𝐻𝑞
ét

increasing filtration {𝑀 𝑗 } 𝑗 ≥0 described as above, with isomorphisms (depending on the choice of

a uniformizer)

𝑀 𝑗 /𝑀 𝑗−1 (cid:27)

Ω

𝑞−1
𝐹

if 𝑗 > 0 and 𝑝 ∤ 𝑗,

Ω

𝐹 /𝑍 𝑞−1
𝑞−1

𝐹 ⊕ Ω

𝐹 /𝑍 𝑞−2
𝑞−2

𝐹

if 𝑗 > 0 and 𝑝 | 𝑗 .





24

Moreover, 𝑀0 is the tame subgroup and there is a well-defined residue homomorphism on 𝑀0,

yielding an exact sequence

0

(cid:47) 𝐻𝑞

nr(𝐾, (Z/𝑝)(𝑞 − 1))

(cid:47) 𝐻𝑞

tame(𝐾, (Z/𝑝) (𝑞 − 1))

𝜕𝑣

(cid:47) 𝐻𝑞−1
ét

(𝐹, (Z/𝑝) (𝑞 − 2))

(cid:47) 0,

where 𝐻𝑞
nr(𝐾, (Z/𝑝)(𝑞 − 1)) (cid:27) 𝐻𝑞
𝐻𝑞

nr(𝐾, (Z/𝑝)(𝑞 − 1)) is the unramified subgroup with respect to 𝑣. Finally, notice that
(𝐹, (Z/𝑝)(𝑞 − 1)) by the henselian property of 𝐹.

ét

3.4 Mixed characteristic case: char(𝐾) = 0

Recall that 𝐾 is a henselian discretely valued field with the discrete valuation 𝑣 and the residue

field 𝐹 of characteristic 𝑝 > 0. We assume char(𝐾) = 0 in this section. Furthermore, we will

assume that 𝐾 contains a primitive 𝑝-th root 𝜁 of the unity. In general, when 𝐾 does not contain a

primitive 𝑝-th root of the unity, we can also describe the filtration {𝑀 𝑗 } 𝑗 ≥0 and their consecutive

quotients [26, Proposition 4.1].

Let 𝑒 = 𝑣( 𝑝) and 𝑁 = 𝑒 𝑝( 𝑝 − 1)−1. These two numbers are integers. Notice that 𝑣(𝜁 −

1) = 𝑒( 𝑝 − 1)−1 and 𝑝 | 𝑁. Using the primitive 𝑝-th root 𝜁 of the unity, we can identify
Z/𝑝 = (Z/𝑝)(1) : 1 ↦→ 𝜁 and 𝐻𝑞
𝐻𝑞

(𝐾, (Z/𝑝) (𝑞)). Then we can describe the elements in

𝑝 (𝐾) by symbols from Milnor 𝐾-theory.

𝑝 (𝐾) (cid:27) 𝐻𝑞
ét

Theorem 3.4.1 (Bloch-Gabber-Kato Theorem [7])

Let 𝐹 be a field of characteristic 𝑝 > 0. For all integers 𝑛 ≥ 0, the differential symbol

𝐹 : 𝐾 𝑀
𝜙𝑛

𝑛 (𝐹)/𝑝 → 𝐻𝑛 (𝐹, (Z/𝑝) (𝑛)) = Ω𝑛

𝐹,log

is an isomorphism.

Kato uses the unit group filtration on O𝐾 to define a decreasing filtration {𝑀 𝑗 } 𝑗 ≥0 on 𝐻𝑞

𝑝 (𝐾).

For 𝑗 ≥ 0, 𝑀 𝑗 is the subgroup of 𝐻𝑞

𝑝 (𝐾) generated by elements of the form

{𝑎, 𝑏1, · · · , 𝑏𝑞−1}

with 𝑎 ∈ 𝑈 𝑗

𝐾 (Definition 3.2.2), 𝑏1, . . . , 𝑏𝑞 ∈ 𝐾 ×. It is clear that

𝐻𝑞,𝑞−1(𝐾) = 𝑀 0 ⊃ 𝑀 1 ⊃ · · · ⊃ 𝑀 𝑒 𝑝( 𝑝−1) −1 ⊃ 𝑀 [𝑒 𝑝( 𝑝−1) −1+1] = 0.

25

(cid:47)
(cid:47)
(cid:47)
(cid:47)
Notice that 𝑀 𝑛 = 0 for 𝑛 > 𝑒 𝑝( 𝑝 − 1)−1 by the henselian property of 𝐾. More precisely, when

𝑛 > 𝑒 𝑝( 𝑝 − 1)−1, 1 + 𝜋𝑛O𝐾 ⊂ (1 + 𝜋𝑛−𝑒O𝐾) 𝑝 by

(1 + 𝜋𝑛−𝑒𝑥) 𝑝 = 1 + 𝑝𝜋𝑛−𝑒𝑥 +

𝑝−1
∑︁

𝑖=2

𝑐𝑖𝑥𝑖 + 𝜋 𝑝(𝑛−𝑒)𝑥 𝑝

with 𝑣(𝑐𝑖) > 𝑣( 𝑝𝜋𝑛−𝑒) = 𝑛 and 𝑝(𝑛 − 𝑒) > 𝑛. Kato proved that the two filtrations {𝑀 𝑗 } and

{𝑀 𝑁− 𝑗 } coincide, that is, 𝑀 𝑗 = 𝑀 𝑁− 𝑗 for each 𝑗 [26, Proposition 4.1]. Therefore, we will use 𝑀 𝑗

in the following context for convenience.

Let 𝜋 ∈ O𝐾 be a uniformizer for 𝑣. For any 𝑗 > 0, we define three homomorphisms depending

on whether 𝑝 ∤ 𝑗, 𝑝 |

𝑗 < 𝑁 and 𝑗 = 𝑁.

In each case, a simple computation shows that the

homomorphim is well defined up to a choice of a uniformizer. First, consider the case when 𝑗 is

relatively prime to 𝑝. We define

Ω

𝑞−1
𝐹 → 𝑀 𝑗 /𝑀 𝑗−1

by

𝑑 ¯𝑏𝑞−1
¯𝑏𝑞−1
for 𝑎 ∈ O𝐾 and 𝑏1, . . . , 𝑏𝑞−1 ∈ O×
𝐾.

𝑑 ¯𝑏1
¯𝑏1

∧ · · · ∧

¯𝑎

↦→ {1 + 𝜋𝑁− 𝑗 𝑎, 𝑏1, · · · , 𝑏𝑞−1} (mod 𝑀 𝑗−1),

Now we define the second homomorphism. Let 𝑍 𝑞−1
𝑞−1
𝐹 , Ω

respectively. For 𝑗 > 0 and 𝑝 | 𝑗, define a homomorphism

𝐹 , 𝑍 𝑞−2

𝑞−2
𝐹

𝐹

be the subgroup of closed forms in

Ω

Ω

𝐹 /𝑍 𝑞−1
𝑞−1

𝐹 ⊕ Ω

𝐹 /𝑍 𝑞−2
𝑞−2

𝐹 → 𝑀 𝑗 /𝑀 𝑗−1

as follows: On the first summand, it is defined as

¯𝑎

𝑑 ¯𝑏1
¯𝑏1

∧ · · · ∧

𝑑 ¯𝑏𝑞−1
¯𝑏𝑞−1

↦→ {1 + 𝜋𝑁− 𝑗 𝑎, 𝑏1, · · · , 𝑏𝑞−1} (mod 𝑀 𝑗−1),

and for the second summand it is defined as

¯𝑎

𝑑 ¯𝑏1
¯𝑏1
where 𝑎 ∈ O𝐾 and 𝑏1, . . . , 𝑏𝑞−1 ∈ O×
𝐾.

𝑑 ¯𝑏𝑞−2
¯𝑏𝑞−2

∧ · · · ∧

↦→ {1 + 𝜋𝑁− 𝑗 𝑎, 𝑏1, · · · , 𝑏𝑞−2, 𝜋} (mod 𝑀 𝑗−1),

Finally, we define the third homomorphism. For 𝑗 = 𝑁, define a homomorphism

𝑞 (𝐹)/𝑝 ⊕ 𝐾 𝑀
𝐾 𝑀

𝑞−1(𝐹)/𝑝 → 𝑀𝑁 /𝑀𝑁−1

26

as follows: On the first summand, it is defined as

{ ¯𝑎1, · · · , ¯𝑎𝑞} ↦→ {𝑎1, · · · , 𝑎𝑞},

and for the second summand it is defined as

{ ¯𝑎1, · · · , ¯𝑎𝑞−1} ↦→ {𝑎1, · · · , 𝑎𝑞−1, 𝜋}.

The homomorphisms are well defined (although they depend on the choice of uniformizer 𝜋).

There is residue homomorphism on the tamely ramified subgroup

𝜕𝑣 : 𝐻𝑞

tame(𝐾, (Z/𝑝)( 𝑝 − 1)) (cid:27) 𝐻𝑞

tame(𝐾, (Z/𝑝) ( 𝑝)) → 𝐻𝑞−1

𝑝

(𝐹),

characterized by the property that

𝜕𝑣 ({1 + 𝜋𝑁 𝑎, 𝑏1, · · · , 𝑏𝑞−2, 𝜋}) = ¯𝑎

𝑑 ¯𝑏1
¯𝑏1

∧ · · · ∧

𝑑 ¯𝑏𝑞−2
¯𝑏𝑞−2

,

where 𝑎 ∈ O𝐾, 𝑏1, . . . , 𝑏𝑞−2 ∈ O×
𝐾.

Theorem 3.4.2 (char(𝐾) = 0, mixed characteristic case)

Let 𝐾 be a henselian discretely valued field of characteristic 0 with the valuation 𝑣 and the residue

field 𝐹 of characteristic 𝑝 > 0. Assume that 𝐾 contains a primitive 𝑝-th root 𝜁 of 1. Let
𝑁 = 𝑣( 𝑝) 𝑝( 𝑝 − 1)−1. Then 𝐻𝑞
filtration {𝑀 𝑗 }𝑁

𝑝 (𝐾) = 𝐻𝑞 (𝐾, (Z/𝑝) (𝑞 − 1)) (cid:27) 𝐻𝑞 (𝐾, (Z/𝑝) (𝑞)) has an increasing

𝑗=0 as above, with isomorphisms (depending on the choice of a uniformizer)

𝑀 𝑗 /𝑀 𝑗−1 (cid:27)

if 0 < 𝑗 < 𝑒 𝑝( 𝑝 − 1)−1 and 𝑝 | 𝑗,

𝑞−1
𝐹

Ω

Ω
𝐹 /𝑍 𝑞−1
𝑞−1
𝐹 ⊕ Ω
𝑞 (𝐹)/𝑝 ⊕ 𝐾 𝑀
𝐾 𝑀






if 𝑝 ∤ 𝑗,

𝐹

𝐹 /𝑍 𝑞−2
𝑞−2
𝑞−1(𝐹)/𝑝 if 𝑗 = 𝑁 .

Moreover, 𝑀0 is the tame subgroup and there is a well-defined residue homomorphism on 𝑀0,

yielding an exact sequence

0

(cid:47) 𝐻𝑞

nr(𝐾, (Z/𝑝)(𝑞 − 1))

(cid:47) 𝐻𝑞

tame(𝐾, (Z/𝑝) (𝑞 − 1))

𝜕𝑣

(cid:47) 𝐻𝑞−1
ét

(𝐹, (Z/𝑝) (𝑞 − 2))

(cid:47) 0,

where 𝐻𝑞
1)) (cid:27) 𝐻𝑞
ét

nr(𝐾, (Z/𝑝)(𝑞−1)) is the unramified subgroup with respect to 𝑣. Finally, 𝐻𝑞
(𝐹, (Z/𝑝)(𝑞 − 1)) by the henselian property of 𝐾.

nr(𝐾, (Z/𝑝)(𝑞−

27

(cid:47)
(cid:47)
(cid:47)
(cid:47)
3.5 Symbol length problem of groups 𝐾 𝑀

2 (𝐹)/𝑝 and Ω1
In this section, let 𝐹 be a field of characteristic 𝑝 > 0. we will investigate the symbol length

𝐹/𝑍 1
𝐹

problems groups 𝐾 𝑀

3.5.1 Symbol length of 𝐾 𝑀

𝐹/𝑍 1
𝐹.

2 (𝐹)/𝑝 and Ω1
2 (𝐹)/𝑝

Definition 3.5.1 (Symbol length in 𝐾 𝑀

2 (𝐹)/𝑝)

Let 𝑘 be a field. Let 𝛼 ∈ 𝐾 𝑀

the minimal integer 𝑚 such that 𝛼 = {𝑎1, 𝑏1} + · · · + {𝑎𝑚, 𝑏𝑚} in 𝐾 𝑀

2 (𝐹)/𝑝. The symbol length len(𝛼) of 𝛼 in 𝐾 𝑀
2 (𝐹)/𝑝.

2 (𝐹)/𝑝 is defined to be

Then we define the symbol length of 𝐾 𝑀

2 (𝐹)/𝑝 by

len(𝐾 𝑀

2 (𝐹)/𝑝) (cid:66) sup
𝛼

{len(𝛼)}.

Recall that the 𝑝-rank of 𝐹 is defined to be the integer log𝑝 ( [𝐹 : 𝐹 𝑝]). We collect the known

results when the 𝑝-rank of 𝐹 is no more than 3.

Lemma 3.5.2 ([35, Lemma 1.3])

Let 𝐹 be field of characteristic 𝑝 > 0 and [𝐹 : 𝐹 𝑝] = 𝑝. Then 𝐾 𝑀

2 (𝐹)/𝑝 = 0.

Theorem 3.5.3 ([6, Theorem 3.4])

Let 𝐹 be a field of characteristic 𝑝 > 0 and [𝐹 : 𝐹 𝑝] = 𝑝𝑛, 2 ≤ 𝑛 ≤ 3. Assume that 𝐹 does not

admit any finite extension of degree prime to 𝑝. Then




Notice that the assumption that 𝐹 does not admit any finite extension of degree prime to 𝑝 can

2 (𝐹)/𝑝) ≤

1, 𝑛 = 2;

3, 𝑛 = 3.

len(𝐾 𝑀

be weakened to 𝐹 = 𝐹 𝑝−1 (cid:66) {𝑥 𝑝−1 | 𝑥 ∈ 𝐹}. In fact, the key lemma in the proof of Theorem 3.5.3

is the following.

Lemma 3.5.4 ([25, Section 1, Lemma 3], [14, Lemma 3.2])

Let 𝐹 be a field of characteristic 𝑝 > 0 and 𝐸 a purely inseparable extension of degree 𝑝 of 𝑘.

Assume 𝐹 = 𝐹 𝑝−1. Let 𝑔 : 𝐸 → 𝐹 be a 𝐹-linear map. Then there exists a non-zero element 𝑐 ∈ 𝐸

such that 𝑔(𝑐𝑖) = 0 for 𝑖 = 1, · · · , 𝑝 − 1.

28

When 𝑝 = 2, the condition 𝐹 = 𝐹 𝑝−1 is naturally satisfied. Then we have the following

corollary.

Corollary 3.5.5

Let 𝐹 be a field of characteristic 𝑝 = 2 and [𝐹 : 𝐹 𝑝] = 𝑝𝑛, 2 ≤ 𝑛 ≤ 3. Then

len(𝐾 𝑀

2 (𝐹)/𝑝) =

1, 𝑛 = 2;

3, 𝑛 = 3.





3.5.2 Symbol length of Ω1

𝐹/𝑍 1
𝐹

Definition 3.5.6 (Symbol length in Ω1

𝐹/𝑍 1
𝐹)

Let 𝐹 be a field of characteristic 𝑝 > 0. Let 𝛼 ∈ Ω1

𝐹/𝑍 1

𝐹. The symbol length len(𝛼) of 𝛼 in Ω1

𝐹/𝑍 1
𝐹

is defined to be the minimal integer 𝑚 such that 𝛼 = 𝑎1𝑑𝑏1 + · · · + 𝑎𝑚𝑑𝑏𝑚 in Ω1

𝐹/𝑍 1
𝐹.

Then we define the symbol length of Ω1

𝐹/𝑍 1

𝐹 by

len(Ω1

𝐹/𝑍 1

𝐹) := sup
𝛼

{len(𝛼)}.

The symbol length of Ω1

𝐹/𝑍 1

i.e. [𝐹 : 𝐹 𝑝] = 𝑝, we have that Ω1

𝐹 is clearly controlled by the 𝑝-rank of 𝐹. If the 𝑝-rank of 𝐹 is 1,
𝐹 = 0, since there is no nontrivial 2-form over 𝐹 and every

𝐹/𝑍 1

1-form over 𝐹 is closed. Meanwhile, if the 𝑝-rank of 𝐹 is 𝑛, the symbol length of Ω1

𝐹/𝑍 1

𝐹 is no

more than 𝑛.

Following the observation for the case [𝐹 : 𝐹 𝑝] = 𝑝, we make the following conjecture.

Conjecture 3.5.7

Let 𝐹 be a field of characteristic 𝑝 > 0 and [𝐹 : 𝐹 𝑝] = 𝑝𝑛 for 𝑛 ∈ N>0. Assume that 𝐹 does not
admit any finite extension of degree prime to 𝑝. Then len(Ω1

𝐹) ≤ 𝑛 − 1.

𝐹/𝑍 1

The following proposition gives us the hint to make the conjecture.

Proposition 3.5.8

Let 𝐹 be a field of characteristic 𝑝 > 0 and [𝐹 : 𝐹 𝑝] = 𝑝𝑛, 𝑛 ∈ N>0. Suppose 𝛼 ∈ Ω1
there exists a degree 𝑝𝑛−1 inseparable field extension 𝐸/𝐹 such that [𝛼𝐸 ] = 0 in Ω1
𝐸 /𝑍 1
𝐸 .

𝐹/𝑍 1

𝐹. Then

29

𝑛
(cid:205)
𝑖=1

Proof. Since [𝐹 : 𝐹 𝑝] = 𝑝𝑛, there exists a 𝑝-basis of 𝐹 given by {𝑥1, · · · , 𝑥𝑛} for some 𝑥𝑖 ∈ 𝐹.
− 𝑥𝑛−1). It
We have 𝛼 =

𝑓𝑖𝑑𝑥𝑖 for some 𝑓𝑖 ∈ 𝐹. Let 𝐸 = 𝐹 [𝑡1, · · · , 𝑡𝑛−1]/(𝑡 𝑝

− 𝑥1, · · · , 𝑡 𝑝
𝑛−1
𝑗 𝑥 𝑗+1
𝑔 𝑝
𝑛
𝑗 + 1
𝐸 by Cartier’s isomorphism. Hence, [𝛼𝐸 ] = 0 in Ω1

𝑛. For 𝑗 ≠ 𝑝 − 1, 𝑔 𝑝

𝑛𝑑𝑥𝑛 = 𝑑 (

𝑝−1
(cid:205)
𝑗=0

𝑗 𝑥 𝑗
𝑔 𝑝

𝑗 𝑥 𝑗

𝐸 /𝑍 1
𝐸 .

1

) ∈ 𝑍 1

𝐸 . When

□

follows that 𝛼𝐸 = 𝑓𝑛𝑑𝑥𝑛, where 𝑓𝑛 =
𝑗 = 𝑝 − 1, 𝑔 𝑝

𝑥 𝑝dlog(𝑥) ∈ 𝑍 1

𝑝−1

Besides the case [𝐹 : 𝐹 𝑝] = 𝑝, we give evidence for Conjecture 3.5.7 in the case 𝑝 = 2 and

𝑛 = 2.

Lemma 3.5.9

Let 𝐹 be a field of characteristic 𝑝 = 2 and [𝐹 : 𝐹 𝑝] = 𝑝2. Then len(Ω1

𝐹/𝑍 1

𝐹) = 1.

Proof.

Since [𝐹 : 𝐹 𝑝] = 𝑝2, there exist 𝑠, 𝑡 ∈ 𝐹 such that the set {𝑠𝑖𝑡 𝑗 }(𝑖, 𝑗) is a basis for 𝐹 as an 𝐹 𝑝-vector

space.

Let 𝛼 ∈ Ω1

𝐹/𝑍 1

𝐹. Then for some 𝑓 , 𝑔 ∈ 𝐹, we get the following equalities modulo 𝑍 1
𝐹:

𝛼 = 𝑓 dlog(𝑠) + 𝑔dlog(𝑡)

(cid:16) ∑︁

=

𝑓 𝑝
𝑖 𝑗 𝑠𝑖𝑡 𝑗 dlog(𝑠)

(cid:17)

+

(cid:16) ∑︁

𝑔 𝑝
𝑖 𝑗 𝑠𝑖𝑡 𝑗 dlog(𝑡)

(cid:17)

0≤𝑖, 𝑗 ≤ 𝑝−1

(cid:16)

=

𝑓 2
01

𝑡dlog(𝑠) + 𝑓 2
11

𝑠𝑡dlog(𝑠)

(cid:17)

0≤𝑖, 𝑗 ≤ 𝑝−1
(cid:16)

+

𝑔2
10

𝑠dlog(𝑡) + 𝑔2
11

𝑠𝑡dlog(𝑡)

(cid:17)

= 𝑓 2
01

𝑡dlog(𝑠) + 𝑔2
10

𝑠dlog(𝑡) +

(cid:16)

(cid:17)

11 − 𝑔2
𝑓 2
11

𝑠𝑡dlog(𝑠)

Now, suppose that 𝛼 = 𝑎𝑑𝑏 ∈ Ω1

𝐹/𝑍 1

𝐹. Then we have that

(cid:16) ∑︁

𝑎𝑑𝑏 =

𝑖 𝑗 𝑠𝑖𝑡 𝑗 (cid:17)
𝑎 𝑝

𝑑

(cid:16) ∑︁

𝑖 𝑗 𝑠𝑖𝑡 𝑗 (cid:17)
𝑏 𝑝

(0≤𝑖, 𝑗 ≤ 𝑝−1

0≤𝑖, 𝑗 ≤ 𝑝−1

(cid:16)

(cid:16)

=

=

𝑎2
01

𝑡 + 𝑎2
10

𝑠 + 𝑎2
11

𝑠𝑡

(cid:16)

(cid:17)

𝑑

𝑏2
01

(cid:17)

𝑠𝑡

𝑠 + 𝑏2
11

𝑎2
11

𝑏2
10

𝑠2 + 𝑎2
10

𝑏2
11

𝑠2(cid:17)

𝑡dlog(𝑠) +

𝑡 + 𝑏2
10
(cid:16)

𝑎2
11

𝑏2
01

𝑡2 + 𝑎2
01

𝑏2
11

𝑡2(cid:17)

𝑠dlog(𝑡) +

(cid:16)

𝑎2
10

01 − 𝑎2
𝑏2
01

𝑏2
10

(cid:17)

𝑠𝑡dlog(𝑡).

30

Hence, it suffices to solve the following system of equations in the variables 𝑎𝑖 𝑗 , 𝑏𝑖 𝑗 for 0 ≤ 𝑖, 𝑗 ≤ 1:

01 =𝑎2
𝑓 2
11

𝑏2
10

𝑠2 + 𝑎2
10

𝑏2
11

𝑠2

10 =𝑎2
𝑔2
11

𝑏2
01

𝑡2 + 𝑎2
01

𝑏2
11

𝑡2

11 − 𝑔2
𝑓 2

11 =𝑎2
10

01 − 𝑎2
𝑏2
01

𝑏2
10

.

Since 𝐹 is of characteristic 2, it follows that

𝑓01 =𝑎11𝑏10𝑠 + 𝑎10𝑏11𝑠

𝑔10 =𝑎11𝑏01𝑡 + 𝑎01𝑏11𝑡

𝑓11 + 𝑔11 =𝑎10𝑏01 + 𝑎01𝑏10.

Now, to solve this system of equations, we can write down a solution explicitly when 𝑓01 ≠ 0.

Let 𝑎11 = 0 and 𝑏11 = 1. Then we have that 𝑎10 =
𝑠( 𝑓11 + 𝑔11)
𝑓01

𝑓01
𝑠 and 𝑎01 =
. The other case follows similarly.

follows that 𝑏01 =

𝑔10
𝑡

. Next, we take 𝑏10 = 0. It

Finally, we finish the proof in the case 𝑝 = 2. More precisely, we have that

𝛼 = (

𝑓 2
01
𝑠

𝑔2
10
𝑡

+

)𝑑 (

𝑠2( 𝑓 2

11 + 𝑔2
11)
𝑓 2
01

𝑡 + 𝑠𝑡) in Ω1

𝐹 .
𝐹/𝑍 1

Hence the symbol length is 1.

(3.5.1)

□

For ( 𝑝, 𝑛) ≠ (2, 2), we can also formulate the system of equations in a similar way. But the

number of equations and variables increase exponentially as 𝑝 and 𝑛 increase. Hence, we will need

a more nuanced approach in the general case.

We will provide a different approach to the symbol length problem of the group Ω1

𝐹/𝑍 1

𝐹 using

the foliation and Galois theory of purely inseparable extensions in the appendix.

31

CHAPTER 4

PERIOD-INDEX PROBLEMS OF HENSELIAN DISCRETELY VALUED FIELDS

In this chapter, we show that it is sufficient to prove the Conjecture 1.2.1 for wildly ramified 𝑝-Brauer

classes. Through out this chapter, let 𝐾 be a henselian discretely valued field with the valuation 𝑣,

valuation ring O𝐾 and residue field 𝐹 of characteristic 𝑝 > 0. Suppose that [𝐹 : 𝐹 𝑝] = 𝑝𝑛, 𝑛 ∈ N.

4.1 Reduction to the 𝑝-torsion part of Brauer group

Proposition 4.1.1 ([46, Proposition 2.1], [31, Proposition 6.1])

Suppose that a field 𝐾 and all its finite extensions 𝐿, have the property that for all central simple

𝐴/𝐿 of period 𝑝 satisfies ind( 𝐴) ≤ 𝑝𝑚. Then, any 𝐴/𝐾 of period 𝑝𝑛 satisfies ind( 𝐴) ≤ 𝑝𝑚𝑛.

Proposition 4.1.2 ([47, Proposition 5.3])

Suppose that 𝐾 is a henselian discretely valued field with the residue field 𝐹 of characteristic 𝑝 > 0

and [𝐹 : 𝐹 𝑝] = 𝑝𝑛. Let 𝐿 be a finite extension of 𝐾. Then 𝐿 is also a henselian discretely valued

field with the residue field 𝐸 and [𝐸 : 𝐸 𝑝] = 𝑝𝑛.

Proof. We reduce to either case of a finite separable extension case or a purely inseparable simple

extension case. When 𝐿/𝐾 is finite separable, the statement follows from Lemma 2.3.5 and [39,

Remark 09E8]. When 𝐸/𝐹 is a purely inseparable simple extension, the statement follows from

[39, Lemma 04GH] and lemmas 4.1.3, 4.1.4 below.

□

Lemma 4.1.3 ([35, Lemma 3.1])

Let 𝐵 be a regular local ring with field of fractions 𝐾, residue field 𝜅 and maximal ideal 𝑚. Let 𝑛

be a natural number and 𝑢 ∈ 𝐵 a unit such that [𝜅(𝑢 1

𝑛 ) : 𝜅] = 𝑛. Then 𝐵[𝑢 1

𝑛 ] is a regular local

ring with residue field 𝜅( ¯𝑢 1

𝑛 ).

Lemma 4.1.4 ([35, Lemma 3.2])

Let 𝐵 be a regular local ring with field of fractions 𝐾, residue field 𝜅 and maximal ideal 𝑚. Let

𝜋 ∈ 𝑚 be a regular prime and 𝑛 a natural number. Then 𝐵[𝜋 1

𝑛 ] is a regular local ring with residue

field 𝜅.

32

Combining these two propositions above, it suffices to verify Conjecture 1.2.1 for 𝑝-torsion

Brauer classes. Moreover, we can assume that the residue field 𝐹 does not admit any finite extension

of degree prime to 𝑝 by the lemma below.

Lemma 4.1.5 ([28])

Let 𝐾 be a field and 𝛼 ∈ Br(𝐾) a class annihilated by 𝑛. If 𝐿/𝐾 is a finite field extension of degree

𝑑 and 𝑛 is relatively prime to 𝑑, then per(𝛼) = per(𝛼| 𝐿) and ind(𝛼) = ind(𝛼| 𝐿).

Next we want to show that the tamely ramified classes satisfy the conjectured period-index

bounds. The tamely ramified Brauer classes are exactly the elements in 𝑀0 (Definition 3.2.3). By

fixing a uniformizer 𝜋 in 𝐾, we have the following split exact sequence

0

(cid:47) Br(𝐾) [ 𝑝]

(cid:47) Brtame(𝐾) [ 𝑝]

𝜕

(cid:47) 𝐻1

ét(𝐹, Z/𝑝)

(cid:47) 0,

(4.1.1)

Since [𝐹 : 𝐹 𝑝] = 𝑝𝑛, it follows that Br.dim𝑝 (𝐹) ≤ 𝑛 [11, Corollary 3.4]. So this takes care of one

of the two components form the split sequence above. The elements arising from the 𝐻1 term are

split by degree-𝑝 extensions and so the conjectural bound follows in this case. So we get:

Lemma 4.1.6 (Tamely ramified 𝑝-torsion Brauer classes)

Let 𝐾 be a henselian discretely valued field with the residue field 𝐹 of characteristic 𝑝 > 0. Assume

that [𝐹 : 𝐹 𝑝] = 𝑝𝑛, 𝑛 ∈ N. Let 𝛼 ∈ Br(𝐾) [ 𝑝] and sw(𝛼) = 0. Then ind(𝛼) | per(𝛼)𝑛+1.

Notice that this lemma works for both equal characteristic case and mixed characteristic case.

4.2 Kato’s results in the 𝑝-rank 1 case

In this section, we will recall Kato’s results and the proof in the 𝑝-rank 1 case.

Proposition 4.2.1 ([24, Section 4, Lemma 5])

Let 𝐾 be a complete field with a discrete valuation 𝑣 and residue field 𝐹. Suppose that char(𝐹) =
𝑝 > 0 and [𝐹 : 𝐹 𝑝] = 𝑝. Suppose that 𝜔 ∈ Br(𝐾) [ 𝑝] and 𝜔 ∉ Br(𝐾tame/𝐾) [ 𝑝]. Then the division

algebra 𝐷 which represents 𝜔 is a degree 𝑝 division algebra whose residue algebra is a purely

inseparable field extension of degree 𝑝 over 𝐹.

Moreover, suppose that char(𝐾) = 𝑝 > 0. Let 𝜋 be a uniformizer of 𝐾. In this case, 𝐷 = [𝑎, 𝑏)

33

(cid:47)
(cid:47)
(cid:47)
(cid:47)
where 𝑎 ∈ 𝐾, 𝑏 ∈ 𝐾 ×, and it must have one of the following two forms:

(𝑖) [

, 𝑒𝜋), where 𝑓 ∈ O𝐾,

¯𝑓 ∉ 𝐹 𝑝, 𝑚 > 0, 𝑣(𝑒) = 0.

(𝑖𝑖) [

, 𝑔), where 𝑔 ∈ O𝐾, ¯𝑔 ∉ 𝐹 𝑝, 𝑣(𝑐) = 0 and 𝑛 is prime to 𝑝.

𝑓
𝜋 𝑝𝑚
𝑐
𝜋𝑛

In both case, 𝐷 is decomposed by a totally ramified field extension of degree 𝑝 and a field

extension of degree 𝑝 whose residue field is a purely inseparable extension.

We notice that Kato’s proof can be generalized to the henselian case easily. Hence, we put the

generalized result below with proof:

Theorem 4.2.2

Let 𝐾 be a henselian field of characteristic 𝑝 > 0 with a discrete valuation 𝑣 and residue field 𝐹.
Suppose that [𝐹 : 𝐹 𝑝] = 𝑝. Suppose that 𝜔 ∈ Br(𝐾) [ 𝑝] and 𝜔 ∉ Br(𝐾tame/𝐾) [ 𝑝]. Then the

division algebra 𝐷 which represents 𝜔 is a degree 𝑝 divison algebra with inseparable residue field

extension.

Moreover, let 𝜋 be a uniformizer of 𝐾. Then 𝐷 = [𝑎, 𝑏) for some 𝑎 ∈ 𝐾, 𝑏 ∈ 𝐾 × and it has one

of the following two forms:

(𝑖) [

, 𝑒𝜋), where 𝑓 ∈ O𝐾,

¯𝑓 ∉ 𝐹 𝑝, 𝑚 > 0, 𝑣(𝑒) = 0.

(𝑖𝑖) [

, 𝑔), where 𝑔 ∈ O𝐾, ¯𝑔 ∉ 𝐹 𝑝, 𝑣(𝑐) = 0 and 𝑛 is prime to 𝑝.

𝑓
𝜋 𝑝𝑚
𝑐
𝜋𝑛

The following lemma plays the fundamental role in the proof.

It explains how the Swan

conductor of the class 𝑎dlog(1 + 𝑏) is affected by the valuations of 𝑎, 𝑏.

Lemma 4.2.3 (Kato [26])

Let 𝑎, 𝑏 ∈ 𝐾, 𝑖, 𝑗 ∈ Z, and assume that 𝑣𝐾 (𝑎) ≥ −𝑖, 𝑣𝐾 (𝑏) ≥ 𝑗 > 0. Then we have

𝑎dlog(1 + 𝑏) ∈ 𝑀𝑖− 𝑗 .

More precisely, if 𝑎 ≠ 0, we have

𝑎dlog(1 + 𝑏) + 𝑎𝑏dlog(𝑎) ∈ 𝑀𝑖−2 𝑗 .

34

(4.2.1)

(4.2.2)

Proof.

𝑎dlog(1 + 𝑏) =

𝑎
1 + 𝑏

≡ −(1 + 𝑏)𝑑 (

) mod 𝑑 (𝐾)

𝑑 (1 + 𝑏)
𝑎
1 + 𝑏
) mod 𝑑 (𝐾)
𝑏
1 + 𝑏

𝑎
1 + 𝑏

1
1 + 𝑏

) − (

𝑑𝑎 mod 𝑀𝑖−2 𝑗

≡ −𝑏𝑑 (

= −(𝑎𝑏)𝑑 (

≡ −

𝑏
1 + 𝑏

)𝑑𝑎

≡ −𝑏𝑑𝑎 mod 𝑀𝑖−2 𝑗

= −(𝑎𝑏)dlog(𝑎) mod 𝑀𝑖−2 𝑗 .

□

Proof of Theorem 4.2.2.

The proof follows from the following two steps by induction on 𝑖:

(i) Hypotheses:

𝜔 ∈ Br(𝐾) [ 𝑝],
𝑓
𝜋 𝑝𝑚 dlog(𝜋)] mod 𝑀𝑖,

𝜔 ≡ [

𝑓 ∈ O𝐾,

¯𝑓 ∉ 𝐹 𝑝, 𝑝𝑚 > 𝑖 ≥ 0,

𝜋 is a uniformizer of 𝐾.

Conclusion:

There exist 𝑓 ′ and 𝜋′ such that

𝜔 ≡ [

𝑓 ′
𝜋 𝑝𝑚 dlog(𝜋′)] mod 𝑀𝑖−1
𝑣( 𝑓 ′ − 𝑓 ) ≥ 𝑝𝑚 − 𝑖, 𝜋′/𝜋 ∈ 𝑈 ( 𝑝𝑚−𝑖)

𝐾

.

35

(ii) Hypotheses:

Conclusion:

𝜔 ≡ [

𝜔 ∈ Br(𝐾) [ 𝑝],
𝑐
𝜋𝑛 dlog(𝑔)] mod 𝑀𝑖,
𝑔 ∈ O𝐾, ¯𝑔 ∉ 𝐹 𝑝, 𝑣(𝑐) = 0, 𝑛 ≥ 𝑖 ≥ 0

𝜋 is a uniformizer of 𝐾.

There exist 𝑐′ and 𝑔′ such that

𝜔 ≡ [

𝑐′
𝜋𝑛 dlog(𝑔′)] mod 𝑀𝑖−1
𝑣(𝑐′ − 𝑐) > 𝑛 − 𝑖, 𝑔′/𝑔 ∈ 𝑈 (𝑛−𝑖)

𝐾

.

We prove both of these simultaneously in two cases: 𝑖 > 0 and 𝑖 = 0.
(1) 𝒊 > 0 : For (i), if 𝑝 | 𝑖, the conclusion is clear since 𝑀𝑖/𝑀𝑖−1 (cid:27) 𝐹/𝐹 𝑝 by fixing the uniformizer
𝜋. If 𝑝 ∤ 𝑖, for any 𝑔 ∈ O𝐾, by Lemma 4.2.3,

𝑓
𝜋 𝑝𝑚 dlog(1 + ℎ𝜋 𝑝𝑚−𝑖) ≡ −
= −

dlog(

𝑓
𝜋 𝑝𝑚 )

𝑓 ℎ𝜋 𝑝𝑚−𝑖
𝜋 𝑝𝑚
𝑓 ℎ
𝜋𝑖 dlog( 𝑓 ) mod 𝑀𝑝𝑚−2( 𝑝𝑚−𝑖).

Since [𝑘 : 𝑘 𝑝] = 𝑝, we can find ℎ ∈ O𝐾 such that 𝜔 − [

𝑓
𝜋 𝑝𝑚 dlog(𝜋(1 + ℎ𝜋 𝑝𝑚−𝑖))] ∈ 𝑀𝑖−1. Hence

the conclusion follows.

Then let us look at (ii). If 𝑝 ∤ 𝑖, the conclusion follows since the 𝑝-rank of the residue field 𝐹

is 1 and 𝑀𝑖/𝑀𝑖−1 (cid:27) Ω1

𝐹 by fixing the uniformizer 𝜋. If 𝑝 | 𝑖, then for any 𝑒 ∈ O𝐾, by Lemma 4.2.3

we have

𝑐
𝜋𝑛 dlog(1 + 𝑒𝜋𝑛−𝑖) ≡ −
= −

𝑐𝑒𝜋𝑛−𝑖

𝜋𝑛 dlog(
𝑐𝑒
𝜋𝑖 dlog(𝑐)
(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)
(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)
(cid:125)
(cid:123)(cid:122)
(cid:124)
𝑆𝑤(∗)<𝑖

−

𝑐
𝜋𝑛 )
𝑛𝑐𝑒
𝜋𝑖 dlog(𝜋)

= −

𝑛𝑐𝑒
𝜋𝑖 dlog(𝜋) mod 𝑀𝑖−1.

36

The (∗) term has a Swan conductor smaller than 𝑖, since its residue corresponds to a term in
𝑐
𝜋𝑛 dlog(𝑔(1 + 𝑒𝜋𝑛−𝑖)) ∈ 𝑀𝑖−1. Hence the

𝐹 (cid:27) 0. Now we can find 𝑒 ∈ O𝐾 such that 𝜔 −

𝐹/𝑍 1

Ω1

conclusion follows in this case.

(2) 𝒊 = 0: First fix the uniformizer 𝜋. By Theorem 3.3.1, we have that

𝑀0 (cid:27) Br(𝐹) [ 𝑝] ⊕ 𝐹/P (𝐹).

For both hypotheses, the proof proceeds in two steps: First, we modify the condition in each

hypothesis so that the resulting symbol algebra is congruent to 𝜔 modulo Br(𝐹) [ 𝑝]. In the second

step, we finish the proof. We give details for (𝑖) as an example.

Since 𝜔 − [

𝑓
𝜋 𝑝𝑚 dlog(𝜋)] ∈ 𝑀0, then there exists 𝑓1 ∈ O𝐾 such that 𝜔 − [

𝑓 + 𝑓1𝜋 𝑝𝑚
𝜋 𝑝𝑚

dlog(𝜋)] ∈

Br(𝐹) [ 𝑝]. Set 𝑓 ′ = 𝑓 + 𝑓1𝜋 𝑝𝑚. For any ℎ ∈ O𝐾, by Lemma 4.2.3,

𝑓 ′
𝜋 𝑝𝑚 dlog(1 + ℎ𝜋 𝑝𝑚) ≡ − 𝑓 ′ℎdlog(

𝑓 ′
𝜋 𝑝𝑚 )
= − 𝑓 ′ℎdlog( 𝑓 ′) in Br(𝐹) [ 𝑝].

We can find ℎ ∈ O𝐾 such that 𝜔 − [

𝑓 + 𝑓1𝜋 𝑝𝑚
𝜋 𝑝𝑚

conclusion follows.

dlog(𝜋(1 + ℎ𝜋 𝑝𝑚))] ∈ Br(𝐹) [ 𝑝]. Therefore the

□

The following theorem was first proved in [12]. We give a different proof using ideas here.

Theorem 4.2.4 ([12, Theorem 2.3])

Let 𝐾 be a henselian discretely valued field of characteristic 𝑝 > 0 with the residue field 𝐹. Suppose

that 𝐹 is a local field. Then Br.dim(𝐾) = 1.

Proof. Since 𝐹 is a local field of characteristic 𝑝 > 0, we have that 𝑘 (cid:27) F𝑞 ((𝑠)), 𝑞 = 𝑝𝑛. By

Theorem 4.2.2, a wildly ramified Brauer class in Br(𝐾) [ 𝑝] is represented by a symbol algebra of

degree 𝑝. So it suffices to show that a tamely ramified Brauer class in Br(𝐾) has symbol length

1. Let 𝜔 ∈ Br(𝐾tame/𝐾) [ 𝑝]. Then 𝜔 = [𝑎, 𝜋) + [𝑏, 𝑐) where 𝑎 defines an unramified degree 𝑝

Artin-Schreier extension of 𝐾 and [𝑏, 𝑐) ∈ Br(𝐹) [ 𝑝]. By [37, Corollary 3, Page 194], [𝑏, 𝑐) is

split by the degree 𝑝 Artin-Schreier extension defined by 𝑎. Hence, 𝛼 = [𝑎, 𝑒) for some 𝑒 ∈ 𝐾 ×.

37

Notice that a finite extension of a local field is still a local field. Hence, combining with Theorem

4.1.1, we get the desired conclusion.

4.3 Equal characteristic case

□

In this section, we will prove the period-index result for 𝑝-torsion part of the Brauer group of a

henselian discretely valued field of characteristic 𝑝 > 0.

Let 𝐾 be a henselian discretely valued field of characteristic 𝑝 > 0 with the valuation 𝑣,

valuation ring O𝐾 and residue field 𝐹 with [𝐹 : 𝐹 𝑝] = 𝑝𝑛, 𝑛 ∈ N. Given a 𝑝-torsion Brauer class

𝛼 ∈ Br(𝐾) [ 𝑝], there are three cases: (i) sw(𝛼) = 0, (ii) 𝑝 ∤ sw(𝛼) > 0 and (iii) 𝑝 | sw(𝛼) > 0.

Recall Conjecture 3.5.7 mentioned in the previous chapter. We should point out that only Case (iii)

is relevant to this conjecture. Now, Case (i) is already discussed in Lemma 4.1.6.

4.3.1 Case (ii): 𝑝 ∤ sw(𝛼) > 0

We will prove the following theorem in this subsection.

Theorem 4.3.1

Let 𝐹 be a field of characteristic 𝑝 > 0 and [𝐹 : 𝐹 𝑝] = 𝑝𝑛, 𝑛 ∈ N>0. Let 𝐾 be a henselian
discretely valued field of characteristic 𝑝 > 0 with the residue field 𝐹. Suppose that 𝛼 ∈ Br(𝐾) [ 𝑝]

and 𝑝 ∤ sw(𝛼) > 0. Then ind(𝛼) | per(𝛼)𝑛.

Proof.

Let { ¯𝑥1, · · · , ¯𝑥𝑛} be a 𝑝-basis of 𝐹. Let {𝑥1, · · · , 𝑥𝑛} be the lifting of the 𝑝-basis in 𝐾 and
𝑎1
𝜋 be a uniformizer of 𝐾. Since 𝑝 ∤ sw(𝛼) = 𝑘 > 0, we have that 𝛼 ≡
𝜋𝑘 dlog(𝑥1) + · · · +
𝑎𝑛
𝜋𝑘 dlog(𝑥𝑛) mod 𝑀𝑘−1, where either 𝑎𝑖 = 0 or 𝑣(𝑎𝑖) = 0, 𝑎𝑖 ≠ 0 for at least one 𝑖. The proof is
based on the following downward induction on 𝑗:

38

Hypotheses:

Conclusion:

𝛼 ≡ [

𝛼 ∈ Br(𝐾) [ 𝑝], 0 ≤ 𝑗 < 𝑘,
𝑎𝑛
𝜋𝑘 dlog(𝑥𝑛)] mod 𝑀 𝑗 ,
either ¯𝑎𝑖 = 0 or 𝑣(𝑎𝑖) = 0 for all 𝑖 ∈ {1, · · · , 𝑛},

𝑎1
𝜋𝑘 dlog(𝑥1) + · · · +

¯𝑎𝑖 ≠ 0 for at least one 𝑖,

{ ¯𝑥1, · · · , ¯𝑥𝑛}is a 𝑝-basis of 𝐹, and 𝜋 is a uniformizer of 𝐾.

𝑖 }𝑖 and 𝜋′ for 𝑖 ∈ {1, · · · , 𝑛} such that

𝑖}𝑖, {𝑥′

𝛼 ≡ [

There exist {𝑎′
𝑎′
𝜋′𝑘 dlog(𝑥′
1
𝑖 = 0 or 𝑣(𝑎′

either ¯𝑎′

1) + · · · +

𝑎′
𝑛
𝜋′𝑘 dlog(𝑥′
𝑖) = 0 for all 𝑖 ∈ {1, · · · , 𝑛},

𝑛)] mod 𝑀 𝑗−1,

¯𝑎′
𝑖 ≠ 0 for at least one 𝑖,

{ ¯𝑥′
1

, · · · , ¯𝑥′

𝑛}is a 𝑝-basis of 𝐹, and 𝜋′ is a uniformizer of 𝐾.

If 𝑝 ∤ 𝑗, by fixing the uniformizer 𝜋, we have 𝑀 𝑗 /𝑀 𝑗−1 (cid:27) Ω1

𝐹. Since { ¯𝑥1, · · · , ¯𝑥𝑛} is a 𝑝-basis of

𝐹, the conclusion easily follows.

If 𝑝 | 𝑗 > 0, by fixing the uniformizer 𝜋, we have 𝑀 𝑗 /𝑀 𝑗−1 (cid:27) Ω1

𝐹 ⊕ 𝐹/𝐹 𝑝. Denote the
projections from 𝑀 𝑗 /𝑀 𝑗−1 to two direct components by 𝑃1, 𝑃2 respectively. WLOG, we assume

𝐹/𝑍 1

that ¯𝑎1 ≠ 0. For any 𝑐 ∈ O𝐾, by Lemma 4.2.3,

𝑎1
𝜋𝑘 dlog(1 + 𝑐𝜋𝑘− 𝑗 ) ≡ −
= −

𝑎1𝑐𝜋𝑘− 𝑗
𝜋𝑘

dlog(

𝑎1𝑐
𝜋 𝑗 dlog(𝑎1) +

𝑎1
𝜋𝑘 )
𝑘𝑎1𝑐
𝜋 𝑗 dlog(𝜋) mod 𝑀𝑘−2(𝑘− 𝑗).

(4.3.1)

(4.3.2)

𝑎1
𝑎𝑛
𝜋𝑘 dlog(𝑥1) + · · · +
𝜋𝑘 dlog(𝑥𝑛)] ∈ 𝑀 𝑗 , we can choose 𝑐 such that 𝑘𝑎1𝑐 = 𝑃2(𝛼 −
𝑎𝑛
𝜋𝑘 dlog(𝑥𝑛)]). Then {𝑥1(1 + 𝑐𝜋𝑘− 𝑗 ), 𝑥2, · · · , 𝑥𝑛} gives a different lifting of the
𝑎1
𝜋𝑘 dlog(𝑥1(1 +

Since 𝛼 − [
𝑎1
𝜋𝑘 dlog(𝑥1) + · · · +
[
𝑝-basis { ¯𝑥1, ¯𝑥2, · · · , ¯𝑥𝑛}. We use this new lifting to match the class from (cid:0)𝑃1(𝛼 − [
𝑐𝜋𝑘− 𝑗 )) + · · · +
𝐹/𝑍 1

𝐹 ⊕ 𝐹/𝐹 𝑝. The conclusion follows.

𝑎𝑛
𝜋𝑘 dlog(𝑥𝑛)]), 0(cid:1) ∈ Ω1

39

If 𝑗 = 0, the proof is similar to the case ( 𝑝 | 𝑗 > 0), since we treat the elements from Ω1

𝐹 and

Ω1

𝐹/𝑍 1

𝐹 in the same way.

□

4.3.2

(iii) 𝑝 | sw(𝛼) > 0

Proposition 4.3.2

Let 𝐹 be a field of characteristic 𝑝 > 0 and [𝐹 : 𝐹 𝑝] = 𝑝𝑛, 𝑛 ∈ N>0. Let 𝐾 be a henselian
discretely valued field of characteristic 𝑝 > 0 with the residue field 𝐹. Assume that Conjecture

3.5.7 is true and 𝐹 does not admit any finite extension of degree prime to 𝑝. Let 𝛼 ∈ Br(𝐾) [ 𝑝] and

𝑝 | sw(𝛼) > 0. Then ind(𝛼) | per(𝛼)𝑛.

Proof.

𝑝 | sw(𝛼) = 𝑘 > 0, we have that 𝛼 ≡ [

The Conjecture 3.5.7 implies that the symbol length of the group Ω1

𝐹 is no more than 𝑛−1. Since
𝑏
𝑎1
𝜋𝑘 dlog(𝜋)] mod 𝑀𝑘−1.
𝜋𝑘 dlog(𝑥1) + · · · +
For 𝑖 ∈ {1, · · · , 𝑛 − 1}, either ¯𝑎𝑖 = 0 or 𝑣(𝑎𝑖) = 0, while either 𝑏 = 0 or 𝑣(𝑏) = 0, ¯𝑏 ∉ 𝐹 𝑝. Not all
¯𝑎𝑖, 𝑏 are zero and 𝜋 is a uniformizer. 𝑣(𝑥𝑖) = 0, { ¯𝑥1, · · · , ¯𝑥𝑛−1} is a 𝐹 𝑝-linearly independent set.

𝐹/𝑍 1
𝑎𝑛
𝜋𝑘 dlog(𝑥𝑛−1) +

We shall discuss in cases 𝑣(𝑏) = 0, ¯𝑏 ∉ 𝐹 𝑝 and 𝑏 = 0 separately.

4.3.2.1

𝑣(𝑏) = 0, ¯𝑏 ∉ 𝐹 𝑝

The proof is based on the following induction on 𝑗:

Hypotheses:

𝛼 ∈ Br(𝐾) [ 𝑝], 0 ≤ 𝑗 < 𝑘,

𝛼 ≡ [

𝑎1
𝜋𝑘 dlog(𝑥1) + · · · +

𝑎𝑛−1
𝜋𝑘 dlog(𝑥𝑛−1) +

𝑏
𝜋𝑘 dlog(𝜋)] mod 𝑀 𝑗 ,

either ¯𝑎𝑖 = 0 or 𝑣(𝑎𝑖) = 0 for all 𝑖 ∈ {1, · · · , 𝑛},

𝑣(𝑏) = 0, ¯𝑏 ∉ 𝐹 𝑝, and 𝜋 is a uniformizer of 𝐾,

𝑣(𝑥𝑖) = 0, { ¯𝑥1, · · · , ¯𝑥𝑛−1} is a 𝐹 𝑝-linearly independent set.

40

Conclusion:

𝑖 }𝑖, 𝑏 and 𝜋′ for 𝑖 ∈ {1, · · · , 𝑛 − 1} such that

𝑖}𝑖, {𝑥′

𝛼 ≡ [

There exist {𝑎′
𝑎′
𝜋′𝑘 dlog(𝑥′
1
𝑖 = 0 or 𝑣(𝑎′

either ¯𝑎′

1) + · · · +

𝑎′
𝑛−1
𝜋′𝑘 dlog(𝑥′
𝑖) = 0 for all 𝑖 ∈ {1, · · · , 𝑛},

𝑛−1) +

𝑏′
𝜋′𝑘 dlog(𝜋′)] mod 𝑀 𝑗−1,

𝑣(𝑏′) = 0, ¯𝑏′ ∉ 𝐹 𝑝, and 𝜋′ is a prime element of 𝐾,

𝑣(𝑥′

𝑖) = 0, { ¯𝑥′
1

, · · · , ¯𝑥′

𝑛−1} is a 𝐹 𝑝-linearly independent set.

𝑎1
𝜋𝑘 dlog(𝑥1) + · · · +

Let 𝛼′ = 𝛼 − [

If 𝑝 |
𝑗 > 0, by fixing the
𝐹 ⊕ 𝐹/𝐹 𝑝. Since { ¯𝑥1, · · · , ¯𝑥𝑛−1} is a 𝐹 𝑝-linearly
uniformizer 𝜋, we have 𝑀 𝑗 /𝑀 𝑗−1 (cid:27) Ω1
independent set and [𝐹 : 𝐹 𝑝] = 𝑝𝑛, we can choose 𝑥𝑛 ∈ O𝐾 such that { ¯𝑥1, · · · , ¯𝑥𝑛} is a 𝑝-basis of
𝐹. Denote the projections from 𝑀 𝑗 /𝑀 𝑗−1 to two direct components by 𝑃1, 𝑃2 respectively. Then

𝑎𝑛−1
𝜋𝑘 dlog(𝑥𝑛−1) +
𝐹/𝑍 1

𝑏
𝜋𝑘 dlog(𝜋)].

𝑃1(𝛼′) = 𝑓1dlog( ¯𝑥1) + · · · + 𝑓𝑛dlog( ¯𝑥𝑛). For any 𝑐 ∈ O𝐾, by Lemma 4.2.3,

𝑏
𝜋𝑘 dlog(1 + 𝑐𝜋𝑘− 𝑗 ) ≡ −
= −

𝑏
𝜋𝑘 )

𝑏𝑐
𝜋 𝑗 dlog(
𝑏𝑐
𝜋 𝑗 dlog(𝑏) mod 𝑀𝑘−2(𝑘− 𝑗).

(4.3.3)

(4.3.4)

We can choose 𝑐 ∈ O𝐾 such that −𝑏𝑐dlog( ¯𝑏) coincides with 𝑓𝑛 on dlog( ¯𝑥𝑛) part. Let 𝑔 ∈ O𝐾 be a
𝑎1
lifting of 𝑃2(𝛼′) ∈ 𝑘/𝑘 𝑝. Then 𝑃1
𝜋𝑘 dlog(𝑥1) + · · · +
𝑐𝜋𝑘−𝑙)](cid:1) is supported away from dlog( ¯𝑥𝑛) and 𝑃2(•) = 0. Hence the conlusion follows.

𝑎𝑛−1
𝜋𝑘 dlog(𝑥𝑛−1) +

𝑏 + 𝑔𝜋𝑘−𝑙
𝜋𝑘

dlog(𝜋(1 +

(cid:0)𝛼 − [

If 𝑗 = 0 or 𝑝 ∤ 𝑗 > 0, the proof is similar to the case ( 𝑝 | 𝑗 > 0), since we treat the elements

from Ω1

𝐹, Ω1

𝐹/𝑍 1

𝐹, Br(𝐹) [ 𝑝] in the same way. More precisely, we are using their liftings in Ω1
𝐹.

4.3.2.2

𝑏 = 0

In this case, the proof can be reduced to either the case ( 𝑝 ∤ sw(𝛼 > 0) or the above case.

Hence we finish the proof.

4.4 Mixed characteristic case

□

In this section, we will prove the period-index result for 𝑝-torsion part of the Brauer group of a

henselian discretely valued field of characteristic 0 with residual characteristic 𝑝 > 0.

41

Let 𝐾 be a henselian discretely valued field of characteristic 0 with the valuation 𝑣, valuation

ring O𝐾 and residue field 𝐹 of characteristic 𝑝 > 0. Let [𝐹 : 𝐹 𝑝] = 𝑝𝑛, 𝑛 ∈ N. In the mixed

characteristic case, we are not always able to express a Brauer class as a sum of symbol algebras,

since 𝐾 may not contain a primitive 𝑝-th root of unity. However, when addressing period-index

problems, we can always reduce to the case where 𝐾 contains a primitive 𝑝-th root of unity (as

noted in Lemma 4.1.5).

In our setting, if the field 𝐾 does not contain a primitive 𝑝-th root of unity, we can adjoin a

primitive 𝑝-th root of unity 𝜁 to the field 𝐾. The field extension 𝐾 (𝜁)/𝐾 is of degree 𝑝 − 1. Hence

it suffices to consider the period-index problem over 𝐾 (𝜁).

In the rest of this section, we assume that 𝐾 contains a primitive 𝑝-th root 𝜁 of the unity. Notice

that 𝑣(𝜁 − 1) =

𝑝𝑣( 𝑝)
𝑝−1 . Given a 𝑝-torsion Brauer class 𝛼 ∈ Br(𝐾) [ 𝑝], there are
four cases: (i) sw(𝛼) = 0, (ii) 𝑝 ∤ sw(𝛼) > 0, (iii) 𝑝 | sw(𝛼), 0 < sw(𝛼) < 𝑁 and (iv) sw(𝛼) = 𝑁.

𝑣( 𝑝)
𝑝−1 and 𝑝 | 𝑁 :=

We mentioned Conjecture 3.5.7 in the previous chapter. It is important to note that the case (iii)

and (iv) are relevant to Conjecture 3.5.7. Additionally, the case (iv) requires the bound of symbol

length of the group 𝐾2(𝐹)/𝑝𝐾2(𝐹).

The case (i) has been addressed in Lemma 4.1.6. We will discuss the other three cases separately

in the subsequent subsections.

4.4.1

(ii) 𝑝 ∤ sw(𝛼) > 0

We will prove the following theorem in this subsection.

Theorem 4.4.1

Let 𝐹 be a field of characteristic 𝑝 > 0 and [𝐹 : 𝐹 𝑝] = 𝑝𝑛, 𝑛 ∈ N>0. Let 𝐾 be a henselian
discretely valued field of characteristic 0 with the residue field 𝐹. Suppose that 𝛼 ∈ Br(𝐾) [ 𝑝] and

𝑝 ∤ sw(𝛼) > 0. Then ind(𝛼) | per(𝛼)𝑛.

Proof. Let { ¯𝑥1, · · · , ¯𝑥𝑛} be a 𝑝-basis of 𝐹. Let {𝑥1, · · · , 𝑥𝑛} be a lifting of the 𝑝-basis and 𝜋
be a uniformizer. Since 𝑝 ∤ sw(𝛼) = 𝑘 > 0, we have that 𝛼 ≡ [{1 + 𝜋𝑁−𝑘 𝑎1, 𝑥1} + · · · + {1 +
𝜋𝑁−𝑘 𝑎𝑛, 𝑥𝑛}] mod 𝑀𝑘−1, where either 𝑎𝑖 = 0 or 𝑣(𝑎𝑖) = 0, 𝑎𝑖 ≠ 0 for at least one 𝑖. The proof is
based on the following induction on 𝑗:

42

Hypotheses:

Conclusion:

𝛼 ∈ Br(𝐾) [ 𝑝], 0 ≤ 𝑗 < 𝑘,

𝛼 ≡ [{1 + 𝜋𝑁−𝑘 𝑎1, 𝑥1} + · · · + {1 + 𝜋𝑁−𝑘 𝑎𝑛, 𝑥𝑛}] mod 𝑀 𝑗 ,

either ¯𝑎𝑖 = 0 or 𝑣(𝑎𝑖) = 0 for all 𝑖 ∈ {1, · · · , 𝑛},

¯𝑎𝑖 ≠ 0 for at least one 𝑖,

{ ¯𝑥1, · · · , ¯𝑥𝑛}is a 𝑝-basis of 𝐹, and 𝜋 is a uniformizer of 𝐾.

There exist {𝑎′

𝑖}𝑖, {𝑥′

𝑖 }𝑖 and 𝜋′ for 𝑖 ∈ {1, · · · , 𝑛} such that

𝛼 ≡ [{1 + 𝜋′𝑁−𝑘 𝑎′
1

either ¯𝑎′

𝑖 = 0 or 𝑣(𝑎′

, 𝑥′

1} + · · · + {1 + 𝜋′𝑁−𝑘 𝑎′
𝑖) = 0 for all 𝑖 ∈ {1, · · · , 𝑛},

𝑛, 𝑥′

𝑛}] mod 𝑀 𝑗−1,

¯𝑎′
𝑖 ≠ 0 for at least one 𝑖,

{ ¯𝑥′
1

, · · · , ¯𝑥′

𝑛}is a 𝑝-basis of 𝐹, and 𝜋′ is a uniformizer of 𝐾.

If 𝑝 ∤ 𝑗, by fixing the uniformizer 𝜋, we have 𝑀 𝑗 /𝑀 𝑗−1 (cid:27) Ω1

𝐹. Since { ¯𝑥1, · · · , ¯𝑥𝑛} is a 𝑝-basis of

𝐹, the conclusion easily follows.

If 𝑝 | 𝑗 > 0, by fixing the uniformizer 𝜋, we have 𝑀 𝑗 /𝑀 𝑗−1 (cid:27) Ω1

𝐹 ⊕ 𝐹/𝐹 𝑝. Denote the
projections from 𝑀 𝑗 /𝑀 𝑗−1 to two direct components by 𝑃1, 𝑃2 respectively. WLOG, we assume

𝐹/𝑍 1

that ¯𝑎1 ≠ 0. For any 𝑐 ∈ O𝐾, by Lemma 4.2.3,

{1 + 𝜋𝑁−𝑘 𝑎1, 1 + 𝑐𝜋𝑘− 𝑗 } ≡ −{1 + 𝜋𝑁− 𝑗 𝑎1𝑐, −𝜋𝑁−𝑘 𝑎1}

= −{1 + 𝜋𝑁− 𝑗 𝑎1𝑐, −𝑎1} − {1 + 𝜋𝑁− 𝑗 𝑎1𝑐, 𝜋𝑁−𝑘 } mod 𝑀𝑘−2(𝑘− 𝑗)

= −{1 + 𝜋𝑁− 𝑗 𝑎1𝑐, 𝑎1} + {1 − (𝑁 − 𝑘)𝜋𝑁− 𝑗 𝑎1𝑐, 𝜋} mod 𝑀 𝑗−1

Since 𝛼− [{1+𝜋𝑁−𝑘 𝑎1, 𝑥1}+· · ·+{1+𝜋𝑁−𝑘 𝑎𝑛, 𝑥𝑛}] ∈ 𝑀 𝑗 , we can choose 𝑐 such that −(𝑁 − 𝑘)𝑎1𝑐 =
𝑃2(𝛼 − [{1 + 𝜋𝑁−𝑘 𝑎1, 𝑥1} + · · · + {1 + 𝜋𝑁−𝑘 𝑎𝑛, 𝑥𝑛}]). Then {𝑥1(1 + 𝑐𝜋𝑘− 𝑗 ), 𝑥2, · · · , 𝑥𝑛} gives a
different lifting of the 𝑝-basis { ¯𝑥1, ¯𝑥2, · · · , ¯𝑥𝑛}. We use this new lifting to match the class from

43

(cid:0)𝑃1(𝛼 − [{1 + 𝜋𝑁−𝑘 𝑎1, 𝑥1(1 + 𝑐𝜋𝑘− 𝑗 )} + · · · + {1 + 𝜋𝑁−𝑘 𝑎𝑛, 𝑥𝑛}]), 0(cid:1) ∈ Ω1

𝐹/𝑍 1

𝐹 ⊕ 𝐹/𝐹 𝑝. The

conclusion follows.

If 𝑗 = 0, the proof is similar to the case ( 𝑝 | 𝑗 > 0), since we treat the elements from Ω1

𝐹 and

Ω1

𝐹/𝑍 1

𝐹 in the same way.

□

4.4.2

(iii) 𝑝 | sw(𝛼), 0 < sw(𝛼) < 𝑁

We will prove the following theorem in this subsection.

Theorem 4.4.2

Let 𝐹 be a field of characteristic 𝑝 > 0 and [𝐹 : 𝐹 𝑝] = 𝑝𝑛, 𝑛 ∈ N>0. Let 𝐾 be a henselian
discretely valued field of characteristic 0 with the residue field 𝐹. Assume that Conjecture 3.5.7 is

true and 𝐹 does not admit any finite extension of degree prime to 𝑝. Suppose that 𝛼 ∈ Br(𝐾) [ 𝑝]

and 𝑝 | sw(𝛼), 0 < sw(𝛼) < 𝑁. Then ind(𝛼) | per(𝛼)𝑛.

Proof.

𝐹 is no more than 𝑛 − 1.
The Conjecture 3.5.7 suggests that the symbol length of the group Ω1
Since 𝑝 | sw(𝛼) = 𝑘 > 0, we have that 𝛼 ≡ [{1 + 𝜋𝑁−𝑘 𝑎1, 𝑥1} + · · · + {1 + 𝜋𝑁−𝑘 𝑎𝑛−1, 𝑥𝑛−1} + {1 +
𝜋𝑁−𝑘 𝑏, 𝜋}] mod 𝑀𝑘−1. For 𝑖 ∈ {1, · · · , 𝑛 − 1}, either 𝑎𝑖 = 0 or 𝑣(𝑎𝑖) = 0, while either 𝑏 = 0 or
𝑣(𝑏) = 0, ¯𝑏 ∉ 𝐹 𝑝. Not all 𝑎𝑖, 𝑏 are zero and 𝜋 is a uniformizer. 𝑣(𝑥𝑖) = 0, { ¯𝑥1, · · · , ¯𝑥𝑛−1} is a
𝐹 𝑝-linearly independent set.

𝐹/𝑍 1

We shall discuss in cases 𝑣(𝑏) = 0, ¯𝑏 ∉ 𝐹 𝑝 and 𝑏 = 0 separately.

4.4.2.1

𝑣(𝑏) = 0, ¯𝑏 ∉ 𝐹 𝑝

The proof is based on the following induction on 𝑗:

44

Hypotheses:

𝛼 ∈ Br(𝐾) [ 𝑝], 0 ≤ 𝑗 < 𝑘,

𝛼 ≡ [{1 + 𝜋𝑁−𝑘 𝑎1, 𝑥1} + · · · + {1 + 𝜋𝑁−𝑘 𝑎𝑛−1, 𝑥𝑛−1} + {1 + 𝜋𝑁−𝑘 𝑏, 𝜋}] mod 𝑀 𝑗 ,

either ¯𝑎𝑖 = 0 or 𝑣(𝑎𝑖) = 0 for all 𝑖 ∈ {1, · · · , 𝑛},

𝑣(𝑏) = 0, ¯𝑏 ∉ 𝐹 𝑝, and 𝜋 is a prime element of 𝐾,

𝑣(𝑥𝑖) = 0, { ¯𝑥1, · · · , ¯𝑥𝑛−1} is a 𝐹 𝑝-linearly independent set.

Conclusion:

There exist {𝑎′

𝑖}𝑖, {𝑥′

𝑖 }𝑖, 𝑏 and 𝜋′ for 𝑖 ∈ {1, · · · , 𝑛 − 1} such that

𝛼 ≡ [{1 + 𝜋′𝑁−𝑘 𝑎′
1

either ¯𝑎′

𝑖 = 0 or 𝑣(𝑎′

, 𝑥′

1} + · · · + {1 + 𝜋′𝑁−𝑘 𝑎′
𝑖) = 0 for all 𝑖 ∈ {1, · · · , 𝑛},

𝑛−1

, 𝑥′

𝑛−1} + {1 + 𝜋′𝑁−𝑘 𝑏′, 𝜋′}] mod 𝑀 𝑗−1,

𝑣(𝑏′) = 0, ¯𝑏′ ∉ 𝐹 𝑝, and 𝜋′ is a prime element of 𝐾,

𝑣(𝑥′

𝑖) = 0, { ¯𝑥′
1

, · · · , ¯𝑥′

𝑛−1} is a 𝐹 𝑝-linearly independent set.

Let 𝛼′ = 𝛼 − [{1 + 𝜋𝑁−𝑘 𝑎1, 𝑥1} + · · · + {1 + 𝜋𝑁−𝑘 𝑎𝑛−1, 𝑥𝑛−1} + {1 + 𝜋𝑁−𝑘 𝑏, 𝜋}]. If 𝑝 | 𝑗 > 0, by fixing
𝐹 ⊕ 𝐹/𝐹 𝑝. Since { ¯𝑥1, · · · , ¯𝑥𝑛−1} is a 𝐹 𝑝-linearly
the uniformizer 𝜋, we have 𝑀 𝑗 /𝑀 𝑗−1 (cid:27) Ω1
independent set and [𝐹 : 𝐹 𝑝] = 𝑝𝑛, we can choose 𝑥𝑛 ∈ O𝐾 such that { ¯𝑥1, · · · , ¯𝑥𝑛} is a 𝑝-basis of
𝐹. Denote the projections from 𝑀 𝑗 /𝑀 𝑗−1 to two direct components by 𝑃1, 𝑃2 respectively. Then

𝐹/𝑍 1

𝑃1(𝛼′) = 𝑓1dlog( ¯𝑥1) + · · · + 𝑓𝑛dlog( ¯𝑥𝑛). For any 𝑐 ∈ O𝐾, by Lemma 4.2.3,

{1 + 𝜋𝑁−𝑘 𝑏, 1 + 𝑐𝜋𝑘− 𝑗 } ≡ −{1 + 𝜋𝑁− 𝑗 𝑏𝑐, −𝜋𝑁−𝑘 𝑏}

= −{1 + 𝜋𝑁− 𝑗 𝑏𝑐, −𝑏} mod 𝑀𝑘−2(𝑘− 𝑗)

= {1 − 𝜋𝑁− 𝑗 𝑏𝑐, 𝑏} mod 𝑀 𝑗−1

(4.4.1)

(4.4.2)

(4.4.3)

We can choose 𝑐 ∈ O𝐾 such that −𝑏𝑐dlog( ¯𝑏) coincides with 𝑓𝑛 on dlog( ¯𝑥𝑛) part. Let 𝑔 ∈ O𝐾 be
(cid:0)𝛼 − [{1 + 𝜋𝑁−𝑘 𝑎1, 𝑥1} + · · · + {1 + 𝜋𝑁−𝑘 𝑎𝑛−1, 𝑥𝑛−1} + {1 +
a lifting of 𝑃2(𝛼′) ∈ 𝐹/𝐹 𝑝. Then 𝑃1
𝜋𝑁−𝑘 (𝑏 + 𝑔𝜋𝑘− 𝑗 ), 𝜋(1 + 𝑐𝜋𝑘− 𝑗 )}](cid:1) is supported away from dlog( ¯𝑥𝑛) and 𝑃2(•) = 0. Hence the

conclusion follows.

45

If 𝑗 = 0 or 𝑝 ∤ 𝑗 > 0, the proof is similar to the case ( 𝑝 | 𝑗 > 0), since we treat the elements

from Ω1

𝐹, Ω1

𝐹/𝑍 1

𝐹, Br(𝐹) [ 𝑝] in the same way. More precisely, we are using their liftings in Ω1
𝐹.

4.4.2.2

𝑏 = 0

In this case, the proof can be reduced to either the case ( 𝑝 ∤ sw(𝛼 > 0) or the above case.

Hence we finish the proof.

4.4.3

(iv) sw(𝛼) = 𝑁

We will prove the following theorem in this subsection.

Theorem 4.4.3

□

Let 𝐹 be a field of characteristic 𝑝 > 0 and [𝐹 : 𝐹 𝑝] = 𝑝𝑛, 𝑛 ∈ N>0. Let 𝐾 be a henselian
discretely valued field of characteristic 0 with the residue field 𝐹. Assume that Conjecture 3.5.7 is

true and 𝐹 does not admit any finite extension of degree prime to 𝑝. Suppose that 𝛼 ∈ Br(𝐾) [ 𝑝]

and sw(𝛼) = 𝑁. Then ind(𝛼) | per(𝛼)𝑛.

Proof.

The proof is similar to the case (iii). Fixing a uniformizer 𝜋, it follows that 𝑀𝑁 /𝑀𝑁−1 (cid:27)

𝐾2(𝐹)/𝑝𝐾2(𝐹) ⊕ 𝐾1(𝐹)/𝑝𝐾1(𝐹). Hence, at the starting point, we need both the symbol length

bounds of 𝐾2(𝐹)/𝑝𝐾2(𝐹) and Ω1

𝐹/𝑍 1
𝐹.

□

4.5 Symbol length problems of higher Kato’s groups

In this section, we generalize the ideas from previous sections to investigate the symbol length

problems of higher Kato’s groups. Let 𝐾 be a henselian discretely valued field of residual charac-

teristic 𝑝 > 0. We prove that any wildly ramified element in 𝐻3
ét

(𝐾, (Z/𝑝) (2)) is split by a purely

inseparable extension of degree 𝑝.

Theorem 4.5.1

Let 𝐾 be a henselian discretely valued field of characteristic 𝑝 > 0 with the residue field 𝐹.

Suppose that [𝐹 : 𝐹 𝑝] = 𝑝2. Let 𝛼 ∈ 𝐻3
ét
𝐾 and 𝑐 ∈ 𝐾 ×.

𝑑𝑐
𝑐 for some 𝜔 ∈ Ω1

that 𝛼 = 𝜔 ∧

(𝐾, (Z/𝑝) (2)) such that 𝑝 ∤ sw(𝛼) > 0. Then we have

46

Proof. First, notice that Ω2

𝐹/𝑍 2

𝐹 = 0, since [𝐹 : 𝐹 𝑝] = 𝑝2, which forces there is no non-trivial

3-form over 𝐹. By Theorem 3.3.1, we have that

𝑀 𝑗 /𝑀 𝑗−1 (cid:27)

Ω2
𝐹

if 𝑗 > 0 and 𝑝 ∤ 𝑗,

Ω1

𝐹/𝑍 1
𝐹

if 𝑗 > 0 and 𝑝 | 𝑗 .





Let { ¯𝑥1, ¯𝑥2} be a 𝑝-basis of 𝐹. Let {𝑥1, 𝑥2} be the lifting of the 𝑝-basis in 𝐾 and let 𝜋 be a

uniformizer of 𝐾. Then {𝑑𝑥1 ∧ 𝑑𝑥2} gives a basis of Ω2

𝐹. Since 𝑝 ∤ sw(𝛼) = 𝑘 > 0, we have
𝑎
𝜋𝑘 dlog(𝑥1) ∧ dlog(𝑥2) mod 𝑀𝑘−1, where 𝑣(𝑎) = 0. The proof is based on the following

that 𝛼 ≡

induction on 𝑗:

Hypotheses:

Conclusion:

ét(𝐾, (Z/𝑝)(2)), 0 ≤ 𝑗 < 𝑘,
𝛼 ∈ 𝐻3
𝑎
𝑏
𝜋𝑘 dlog(𝑥2) ∧ dlog(𝜋)] mod 𝑀 𝑗 ,
𝜋𝑘 dlog(𝑥1) ∧ dlog(𝑥2) +

𝛼 ≡ [

¯𝑎 ≠ 0, 𝑣(𝑏) > 0,

{ ¯𝑥1, ¯𝑥2}is a 𝑝-basis of 𝐹, and 𝜋 is a uniformizer of 𝐾.

There exist 𝑎′, 𝑏′ and 𝑥′

𝛼 ≡ [

𝑎′
𝜋𝑘 dlog(𝑥1) ∧ dlog(𝑥′

2 such that
𝑏′
𝜋𝑘 dlog(𝑥′

2) +

2) ∧ dlog(𝜋)] mod 𝑀 𝑗−1,

¯𝑎′ ≠ 0, 𝑣(𝑏′) > 0,

{ ¯𝑥1, ¯𝑥′

2}is a 𝑝-basis of 𝐹, and 𝜋 is a uniformizer of 𝐾.

Let 𝛼′ = 𝛼 − [

𝑎
𝜋𝑘 dlog(𝑥1) ∧ dlog(𝑥2) +
uniformizer 𝜋, we have 𝑀 𝑗 /𝑀 𝑗−1 (cid:27) Ω1

𝑏
𝜋𝑘 dlog(𝑥2) ∧ dlog(𝜋)].
𝐹. Since { ¯𝑥1, ¯𝑥2} is a 𝑝-basis for 𝐹, we have [𝛼′] ≡

𝑗 > 0, by fixing the

If 𝑝 |

𝐹/𝑍 1

𝑓1dlog( ¯𝑥1) + 𝑓2dlog( ¯𝑥2) in 𝑀 𝑗 /𝑀 𝑗−1, where 𝑓1, 𝑓2 ∈ 𝐹. For any 𝑐 ∈ O𝐾,

𝑎
𝜋𝑘 dlog(𝑥1) ∧ dlog(1 + 𝑐𝜋𝑘− 𝑗 ) ≡ −

𝑎𝑐
𝜋 𝑗 dlog(𝑥1) ∧ dlog(
𝑘𝑎𝑐
𝜋 𝑗 dlog(𝑥1) ∧ dlog(𝜋) mod 𝑀𝑘−2(𝑘− 𝑗).

𝑎
𝜋𝑘 )

=

(4.5.1)

(4.5.2)

47

We can choose 𝑐 ∈ O𝐾 such that 𝑘𝑎𝑐 coincides with 𝑓1 and let 𝑔 ∈ O𝐾 be a lifting of 𝑓2. Then

𝛼 − [

𝑎
𝜋𝑘 dlog(𝑥1) ∧ dlog(𝑥2(1 + 𝑐𝜋𝑘− 𝑗 )) +
2 = 𝑥2(1 + 𝑐𝜋𝑘− 𝑗 ). It follows that 𝛼 − [
𝑥′
𝑀 𝑗−1. Hence the conclusion follows.

𝑏 + 𝑔𝜋𝑘− 𝑗
𝜋𝑘

dlog(𝑥2) ∧ dlog(𝜋)] ∈ 𝑀 𝑗−1. Moreover, denote

𝑎
𝜋𝑘 dlog(𝑥1) ∧ dlog(𝑥′

2) +

𝑏 + 𝑔𝜋𝑘− 𝑗
𝜋𝑘

dlog(𝑥′

2) ∧ dlog(𝜋)] ∈

If 𝑝 | 𝑗 > 0, by fixing the uniformizer 𝜋, we have 𝑀 𝑗 /𝑀 𝑗−1 (cid:27) Ω2

𝐹. Then the conclusion follows

easily by using 𝑥1, 𝑥2 as the liftings of 𝑝-basis.

If 𝑗 = 0, the proof is similar to the case ( 𝑝 |

𝐻3
ét

(𝐹, (Z/𝑝)(2)). It is a combination of these two previous arguments.

Theorem 4.5.2

𝑗 > 0), since 𝑀0 (cid:27) 𝐻2

ét(𝐹, (Z/𝑝) (1)) ⊕
□

Let 𝐾 be a henselian discretely valued field of characteristic 𝑝 > 0 with the residue field 𝐹. Suppose

that [𝐹 : 𝐹 𝑝] = 𝑝2. Let 𝛼 ∈ 𝐻3
ét

(𝐾, (Z/𝑝) (2)) such that 𝑝 | sw(𝛼) > 0. Then Conjecture 3.5.7

implies that 𝛼 = 𝜔 ∧

𝑑𝑐
𝑐 for some 𝜔 ∈ Ω1

𝐾 and 𝑐 ∈ 𝐾 ×.

Proof. First, notice that Ω2

𝐹/𝑍 2

𝐹 = 0, since [𝐹 : 𝐹 𝑝] = 𝑝2, which forces there is no non-trivial

3-form over 𝐹. By Theorem 3.3.1, we have that

𝑀 𝑗 /𝑀 𝑗−1 (cid:27)

Ω2
𝐹

if 𝑗 > 0 and 𝑝 ∤ 𝑗,

Ω1

𝐹/𝑍 1
𝐹

if 𝑗 > 0 and 𝑝 | 𝑗 .





Let { ¯𝑥1, ¯𝑥2} be a 𝑝-basis of 𝐹. Let {𝑥1, 𝑥2} be the lifting of the 𝑝-basis in 𝐾 and let 𝜋 be a

3.5.7 implies that 𝛼 ≡

uniformizer of 𝐾. Then {𝑑𝑥1 ∧ 𝑑𝑥2} gives a basis of Ω2

𝐹. Since 𝑝 | sw(𝛼) = 𝑘 > 0, Conjecture
𝑎
𝜋𝑘 dlog(𝑏) ∧ dlog(𝜋) mod 𝑀𝑘−1, where 𝑣(𝑎) = 𝑣(𝑏) = 0. Notice that {𝑎, 𝑏}
𝐹/𝑍 1
𝐹. The proof is based on the following

defines a 𝑝-basis of 𝐹, since ¯𝑎dlog( ¯𝑏) is non-trivial in Ω1

induction on 𝑗:

48

Hypotheses:

𝛼 ≡ [

ét(𝐾, (Z/𝑝)(2)), 0 ≤ 𝑗 < 𝑘,
𝛼 ∈ 𝐻3
𝑐
𝜋𝑘 dlog(𝑎) ∧ dlog(𝑏) +
𝑣(𝑐) > 0, ¯𝑎dlog( ¯𝑏) is nonzero in Ω1

𝐹,
𝐹/𝑍 1

𝑎
𝜋𝑘 dlog(𝑏) ∧ dlog(𝜋)] mod 𝑀 𝑗 ,

𝜋 is a uniformizer of 𝐾.

Conclusion:

𝛼 ≡ [

There exist 𝑎′, 𝑏′ and 𝑐′ such that
𝑎′
𝜋𝑘 dlog(𝑏′) ∧ dlog(𝜋)] mod 𝑀 𝑗−1,

𝑐′
𝜋𝑘 dlog(𝑎′) ∧ dlog(𝑏′) +
𝑣(𝑐′) > 0, ¯𝑎′dlog( ¯𝑏′) is nonzero in Ω1

𝐹,
𝐹/𝑍 1

𝜋 is a uniformizer of 𝐾.

Let 𝛼′ = 𝛼 − [

𝑐
𝜋𝑙 dlog(𝑎) ∧ dlog(𝑏) +
uniformizer 𝜋, we have 𝑀 𝑗 /𝑀 𝑗−1 (cid:27) Ω1

𝑎
𝜋𝑘 dlog(𝑏) ∧ dlog(𝜋)].
𝐹/𝑍 1

𝐹. Since { ¯𝑎, ¯𝑏} is a 𝑝-basis for 𝐹, we have [𝛼′] ≡

If 𝑝 |

𝑗 > 0, by fixing the

𝑓1dlog( ¯𝑎1) + 𝑓2dlog( ¯𝑏2) in 𝑀 𝑗 /𝑀 𝑗−1, where 𝑓1, 𝑓2 ∈ 𝐹. For any 𝑒 ∈ O𝐾,

𝑎
𝜋𝑘 dlog((1 + 𝑒𝜋𝑘− 𝑗 )) ∧ dlog(𝜋) ≡ −
= −

𝑎
𝜋𝑘 ) ∧ dlog(𝜋)

𝑎𝑒
𝜋 𝑗 dlog(
𝑎𝑒
𝜋 𝑗 dlog(𝑎) ∧ dlog(𝜋) mod 𝑀𝑘−2(𝑘− 𝑗).

(4.5.3)

(4.5.4)

We can choose 𝑒 ∈ O𝐾 such that −𝑎𝑒 coincides with 𝑓1 and let 𝑔 ∈ O𝐾 be a lifting of 𝑓2. Then
dlog(𝑏(1 + 𝑒𝜋𝑘− 𝑗 )) ∧ dlog(𝜋)] ∈ 𝑀 𝑗−1. Moreover, denote

𝑐
𝜋𝑙 dlog(𝑎) ∧ dlog(𝑏) +

𝑎 + 𝑔𝜋𝑘− 𝑗
𝜋𝑘

𝛼 − [

𝑎′ = 𝑎 + 𝑔𝜋𝑘− 𝑗 and 𝑏′ = 𝑏(1 + 𝑒𝜋𝑘− 𝑗 ). It follows that 𝛼 − [

dlog(𝜋)] ∈ 𝑀 𝑗−1. Hence, the conclusion follows.

𝑐
𝜋𝑙 dlog(𝑎′) ∧ dlog(𝑏′) +

𝑎′
𝜋𝑘 dlog(𝑏′) ∧

If 𝑝 | 𝑗 > 0, by fixing the uniformizer 𝜋, we have 𝑀 𝑗 /𝑀 𝑗−1 (cid:27) Ω2

𝐹. Then the conclusion follows

easily by using 𝑥1, 𝑥2 as the liftings of 𝑝-basis.

If 𝑗 = 0, the proof is similar to the case ( 𝑝 |

𝐻3
ét

(𝐹, (Z/𝑝)(2)). It is a combination of these two previous arguments.

49

𝑗 > 0), since 𝑀0 (cid:27) 𝐻2

ét(𝐹, (Z/𝑝) (1)) ⊕
□

CHAPTER 5

LOGARITHMIC DE RHAM COHOMOLOGY WITH SUPPORT

In this chapter, we will continue on our discussion of 𝑝-torsion part of Brauer groups of 𝐶𝑚 fields.

As we noted in the introduction„ there is one common class of 𝐶𝑚 fields: the function fields of

dimension 𝑚 − 𝑛 algebraic varieties over 𝐶𝑛 fields for 0 ≤ 𝑛 ≤ 𝑚. When studying the Brauer group

of such a function field, an important concept is its behavior in codimension 1, since the Brauer

group of a Noetherian, integral, regular scheme has purity in codimension 1 (Theorem 2.1.3). We

restate this theorem here for clarity.

Theorem 5.0.1 (Purity in codimension 1 [44])

For a Noetherian, integral, regular scheme 𝑋 with function field 𝐾,

ét(𝑋, G𝑚) =
𝐻2

(cid:217)

𝑥∈𝑋 (1)

ét(O𝑋,𝑥, G𝑚) in 𝐻2
𝐻2

ét(𝐾, G𝑚).

Recall that there is an injection from the Brauer group of the scheme 𝑋 into the Brauer group

of its function field 𝐾:

Br(𝑋) ↩→ Br(𝐾).

Therefore, to understand the Brauer group of the function field 𝐾, it is essential to understand the

cokernel of this homomorphism.

Artin and Mumford [4] provides significant insight into this area. When 𝑋 is a surface, they

showed that there exists an exact sequence that relates the Brauer group of 𝑋, the Brauer group of

the function field 𝐾, and the ramification behavior of Brauer classes at both codimension 1 points

and closed points (points of codimenison 2).

Theorem 5.0.2 ([41])

Let 𝑆 be a smooth projective surface over an algebraically closed field 𝑘 with char(𝑘) = 𝑝 ≥ 0 and

𝑙 be a prime number different from 𝑝. Suppose that 𝐻1

ét(𝑆, Q/Z) = 0. There is a canonical exact

sequence

0

(cid:47) Br𝑙 (𝑆)

(cid:47) Br𝑙 (𝑘 (𝑆))

(cid:47) (cid:201)
curves 𝐶

ét(𝑘 (𝐶), Z/𝑙)
𝐻1

(cid:47) (cid:201)

closed points

𝜇−1
𝑙

(cid:47) 𝜇−1
𝑙

(cid:47) 0,

50

(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
where 𝜇𝑙 denotes the 𝑙-th roots of unity.

In fact, this sequence could be derived from the Bloch-Ogus spectral sequence.

Theorem 5.0.3

Assume that 𝑋 is smooth over a perfect field 𝑘 of characteristic 𝑝 ≥ 0 and 𝑙 be a prime number

different from 𝑝. Then there is a spectral sequence

𝐸 𝑝𝑞
1 =

(cid:202)

𝑥∈𝑋 ( 𝑝)

𝐻𝑞−𝑝 (𝑥, 𝜇⊗𝑛−𝑝

𝑙

) ⇒ 𝐻 𝑝+𝑞 (𝑋, 𝜇⊗𝑛

𝑙

)

(5.0.1)

Here 𝑋 ( 𝑝) are the points of codimension 𝑝 in 𝑋.

We should notice that the above discussion are restricted to the case when the torsion is prime

to the base characteristic. We will give a systematic investigation of the case when the torsion is

equal to the base characteristic.

5.1 Bloch-Ogus spectral sequence in positive characteristic

For the logarithmic de Rham cohomology, we also have the Bloch-Ogus spectral sequence.

Theorem 5.1.1 ([8, 15])

Let 𝑋 be an equidimensional scheme over F𝑝. Then we have the coniveau spectral sequence

𝐸 𝑠,𝑡
1 =

(cid:202)

𝐻 𝑠+𝑡
𝑥

𝑥∈𝑋 (𝑠)

(𝑋, Ω𝑖

𝑋,log) ⇒ 𝐸 𝑠+𝑡 = 𝐻 𝑠+𝑡 (𝑋, Ω𝑖

𝑋,log)

converging to the logarithmic de Rham cohomology, where

𝐻𝑚

𝑥 (𝑋, Ω𝑖

𝑋,log) (cid:66) lim
−−→
𝑥∈𝑈

¯{𝑥}∩𝑈 (𝑈, Ω𝑖
𝐻𝑚

𝑋,log) = 𝐻𝑚

𝑥 (𝑋𝑥, Ω𝑖

𝑋,log)

and 𝑈 runs through open neighborhoods of 𝑥 in 𝑋 (the last equality follows from the excision).

The complex of 𝐸 •,𝑞

1 -terms

0

(cid:47) (cid:201)
𝑥∈𝑋 (0)

𝐻𝑞

𝑥 (𝑋, Ω𝑖

𝑋,log

)

(cid:47) (cid:201)
𝑥∈𝑋 (1)

𝐻𝑞+1
𝑥

(𝑋, Ω𝑖

𝑋,log

)

(cid:47) · · ·

(cid:47) (cid:201)
𝑥∈𝑋 (𝑠)

𝐻𝑞+𝑠
𝑥

(𝑋, Ω𝑖

𝑋,log

)

(cid:47) · · ·

51

(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
is usually called the Bloch-Ogus complex and denoted by 𝐵𝑞,𝑖 (𝑋)•. It is a cohomological analogue

of the Brown-Gersten-Quillen complex in algebraic 𝐾-theory.

We can also describe the 𝐸2 page when considering 𝑘-schemes. Let 𝑘 be a field of characteristic

𝑝 > 0. For every integer 𝑚, we have the cohomology functor on 𝑘-schemes:

𝑋 ↦→ 𝐻𝑚−𝑖

ét

(𝑋, Ω𝑖

𝑋,log) = 𝐻𝑚

ét (𝑋, Z/𝑝(𝑖))

for a 𝑘-scheme 𝑋. For shorthand, we would write 𝐻𝑚

𝑝 (𝑋, 𝑖) in stead of the precise notation

𝐻𝑚−𝑖
ét

(𝑋, Ω𝑖

𝑋,log

). The Zariski sheaf associated to the presheaf 𝑈 ↦→ 𝐻𝑚

𝑝 (𝑈, 𝑖) is denoted by ℋ𝑏

𝑝 (𝑖).

For a smooth connected 𝑘-variety 𝑋, we define the unramified cohomology group

𝐻𝑚

nr(𝑋, Z/𝑝(𝑖)) (cid:66) 𝐻0

Zar(𝑋, ℋ𝑚

𝑝 (𝑖)).

Then we have the following theorem which collects some well-known results.

Theorem 5.1.2

Let 𝑋 be a smooth connected 𝑘-variety.

1. We have the Bloch-Ogus spectral sequence

𝐸 𝑠,𝑡
2 = 𝐻 𝑠

Zar(𝑋, ℋ𝑡

𝑝 (𝑖)) ⇒ 𝐸 𝑠+𝑡 = 𝐻 𝑠+𝑡

ét (𝑋, Z/𝑝(𝑖))

with

and

𝐸 𝑠,𝑡
2 = 0 if 𝑏 ∉ {𝑖, 𝑖 + 1}, or if 𝑎 > 𝑏 = 𝑖

𝐸𝑖,𝑖
2 = 𝐻𝑖

Zar(𝑋, ℋ𝑖

𝑝 (𝑖)) (cid:27) 𝐶𝐻𝑖 (𝑋)/𝑝.

2. There are natural isomorphisms

𝐻𝑖 (𝑋, (Z/𝑝)(𝑖)) (cid:27) 𝐻0

Zar(𝑋, ℋ𝑖
𝐻2𝑖+ 𝑗 (𝑋, (Z/𝑝) (𝑖)) (cid:27) 𝐻 𝑗+𝑖−1

Zar

𝑝 (𝑖)) = 𝐻𝑖

nr(𝑋, (Z/𝑝) (𝑖)),

(𝑋, ℋ𝑖+1

𝑝

(𝑖)) for 𝑗 ≥ 1.

and

𝐻2

nr(𝑋, (Z/𝑝) (1)) = 𝐻0

Zar(𝑋, ℋ2

𝑝 (1)) (cid:27) Br(𝑋) [ 𝑝].

52

(5.1.1)

(5.1.2)

(5.1.3)

(5.1.4)

(5.1.5)

(5.1.6)

3. For smooth proper connected 𝑘-varieties, the group 𝐻𝑚

nr(𝑋, Z/𝑝(𝑖)) is a 𝑘-birational invari-

ant.

We want to finish up this section with the Gersten-type theorem which plays the most important

role in the rest of this chapter. As an analogue of Gersten conjecture in algebraic 𝐾-theory, it is

natural to expect that, if 𝑋 is the spectrum of a regular local ring over F𝑝, the Bloch-Ogus complex

is acyclic in positive degree. In fact, we have the following:

Theorem 5.1.3 (Gersten-type theorem for Bloch-Ogus complex [38])

Let 𝑋 be the spectrum of an equidimensional regular local ring over F𝑝. Then we have

𝐻𝑛 (𝐵𝑞,𝑖 (𝑋)•) =

𝐻𝑞 (𝑋, Ω𝑖

𝑋,log

)

(𝑛 = 0)

0

(𝑛 > 0).





It is proved in the case where 𝑋 is a localization of a smooth scheme over a perfect field by

Gros-Suwa [18].

This Gersten-type theorem provides us with the fundation to analyse the ramification behavior

of a 𝑝-torsion Brauer class affine locally. We will mainly use the following version of Theorem

5.1.3.

Theorem 5.1.4 (Gersten-type theorem [38])

Let 𝑋 be the spectrum of a 2-equidimensional regular local ring over F𝑝 with the unique closed

point 𝑃 and quotient field 𝐾. Then we have an exact sequence

0

(cid:47) 𝐻1

ét(𝑋, Ω1

𝑋,log)

(cid:47) 𝐻1

ét(𝐾, Ω1

𝐾,log)

𝛿1

(cid:47) (cid:201)
𝑥∈𝑋 1

𝐻2

𝑥 (𝑋, Ω1

𝑋,log)

𝛿2

(cid:47) 𝐻3

𝑃 (𝑋, Ω1

𝑋,log)

(cid:47) 0.

5.2 Logarithmic de Rham cohomology of affine schemes with support

The goal of this section is to identify the morphisms 𝛿1, 𝛿2 and cohomology groups appeared

in Theorem 5.1.4.

5.2.1 The morphism 𝛿1

Let 𝑥 ∈ 𝑋 (1). By the étale excision theorem [33], we have the following lemma.

53

(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
Lemma 5.2.1

Proof.

𝐻2

𝑥 (𝑋, Ω1

𝑋,log) = 𝐻2

𝑥 (𝑋𝑥, Ω1

•,log) (cid:27) 𝐻2

𝑥 (O ℎ

𝑋,𝑥, Ω1

•,log).

The first isomorphism follows from the étale excision theorem. The second isomorphism follows

from [34, Corollary 1.28].

Furthermore, we have the following commutative diagram:

0

0

(cid:47) 𝐻1(O ℎ

𝑋,𝑥, Ω1

•,log)

𝐻1(𝐾 ℎ, Ω1

•,log)

𝛿1

𝐻2

𝑥 (O ℎ

𝑋,𝑥, Ω1

•,log)

∼

∼

(cid:47) Br(O ℎ

𝑋,𝑥) [ 𝑝]

(cid:47) Br(𝐾 ℎ) [ 𝑝]

𝛿′
1

(cid:47) Br(𝐾 ℎ) [ 𝑝]

(cid:46)

Br(O ℎ

𝑋,𝑥) [ 𝑝]

□

0

(cid:47) 0

The first exact row comes from the long exact sequence in local cohomology in étale topology and

the second exact row is the canonical exact sequence.

It follows that

𝐻2

𝑥 (𝑋, Ω1

𝑋,log) (cid:27) Br(𝐾 ℎ) [ 𝑝]

(cid:46)

Br(O ℎ

𝑋,𝑥) [ 𝑝] ,

(5.2.1)

and we can identify the morphism 𝛿1 with 𝛿′

1, i.e. 𝛿1 = 𝛿′

1

.

5.2.2 The Morphism 𝛿2

Let 𝑦 ∈ 𝑋 (1) and 𝑌 (cid:66) {𝑦} be the closure of 𝑦 in 𝑋.

Lemma 5.2.2 ([38])

Let 𝑋, 𝑍 be regular schemes over F𝑝 and let 𝑖 : 𝑍 ↩→ 𝑋 be a regular closed immersion of
codimension 𝑟. Then we have 𝐻 𝑗

𝑍 (𝑋, O𝑋) = 0, 𝐻 𝑗

𝑋/𝑑𝑉 𝑚−1Ω𝑖−1

𝑋) = 0, 𝐻 𝑗

𝑍 (𝑋, 𝑊𝑚Ω𝑖

𝑍 (𝑋, 𝑊𝑚Ω1

𝑋 ) =

0 for 𝑗 ≠ 𝑟.

Corollary 5.2.3
Let the notation be as above. Then we have 𝐻 𝑗

𝑍 (𝑋, 𝑊𝑚Ω𝑖

𝑋,log

) = 0 for 𝑗 ≠ 𝑟, 𝑟 + 1.

54

(cid:47)
(cid:47)
(cid:47)
(cid:15)
(cid:15)
(cid:47)
(cid:47)
(cid:15)
(cid:15)
(cid:47)
(cid:47)
(cid:15)
(cid:15)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
Also we have the following exact diagram:

0

0

0

0

0

0

𝐻2

𝑃 (𝑋, Ω1

•,log)

(cid:47) 𝐻1

𝑌 (𝑋, Ω1

•,log)

𝐻1

𝑦 (𝑋 − {𝑃}, Ω1

•,log)

(cid:47) 𝐻1

𝑌 (𝑋, Ω1
•)

𝐹−𝐼

𝐻1

𝑦 (𝑋 − {𝑃}, Ω1
•)

𝐹−𝐼

(cid:47) 𝐻1

𝑌 (𝑋, Ω1

•/𝑑O)

𝐻1

𝑦 (𝑋 − {𝑃}, Ω1

•/𝑑O)

(cid:47) 𝐻2

𝑌 (𝑋, Ω1

•,log)

𝐻2

𝑦 (𝑋 − {𝑃}, Ω1

•,log)

𝛿𝑦
1

𝛿𝑦
1

𝛿𝑦
1

𝛿𝑦
2

𝐻2

𝑃 (𝑋, Ω1

•,log)

𝐻2

𝑃 (𝑋, Ω1
•)

𝐹−𝐼

𝐻2

𝑃 (𝑋, Ω1

•/𝑑O)

𝐻3

𝑃 (𝑋, Ω1

•,log)

0

0

0

(cid:47) 0

(cid:47) 0

(cid:47) 0

(cid:47) 0

(5.2.2)

Notice that we have 𝐻 𝑗

𝑦 (𝑋, •) = 𝐻 𝑗

𝑦 (𝑋 − {𝑃}, •) by excision. In the above diagram, we are using

cohomology groups instead of cohomology sheaves, since 𝑋 is a strictly henselian local scheme.

In order to compute 𝛿2 and 𝐻3

𝑃 (𝑋, Ω1

𝑋,log), recall the following facts about the (étale) local

cohomology.

Lemma 5.2.4 ([39, Lemma 0G74])

Let (𝑋, O𝑋) be a ringed space. Let 𝑍 ⊂ 𝑋 be a closed subset. Let 𝐾 be an object of 𝐷 (O𝑋) and

denote 𝐾ab its image in 𝐷 (Z𝑋). Then there is a canonical map 𝑅Γ𝑍 (𝑋, 𝐾) → 𝑅Γ𝑍 (𝑋, 𝐾ab) in

𝐷 (Ab).

Proposition 5.2.5 ([39, Lemma 0A46])

Let 𝑆 be a scheme. Let 𝑍 ⊂ 𝑆 be a closed subscheme. Let F be a quasi-coherent O𝑆-module and

denote F 𝑎 the associated quasi-coherent sheaf on the small étale site of 𝑆. Then

𝑍 (𝑆Zar, F ) = 𝐻𝑞
𝐻𝑞

𝑍 (𝑆, F 𝑎).

Proposition 5.2.6

For any étale morphism 𝑓 : 𝑋 → 𝑌 , 𝑓 ∗Ω1

𝑌 → Ω1

𝑋 is an isomorphism of O𝑋-modules.

55

(cid:15)
(cid:15)
(cid:15)
(cid:15)
(cid:15)
(cid:15)
(cid:47)
(cid:47)
(cid:47)
(cid:15)
(cid:15)
(cid:47)
(cid:47)
(cid:15)
(cid:15)
(cid:15)
(cid:15)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:15)
(cid:15)
(cid:47)
(cid:47)
(cid:15)
(cid:15)
(cid:15)
(cid:15)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:15)
(cid:15)
(cid:47)
(cid:47)
(cid:15)
(cid:15)
(cid:15)
(cid:15)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:15)
(cid:15)
(cid:47)
(cid:47)
(cid:15)
(cid:15)
(cid:15)
(cid:15)
(cid:47)
By Proposition 5.2.6, we get (Ω1

𝑆 on the small étale site of 𝑆. It is also known that
(O𝑆)𝑎 = O𝑆 (or 𝐺 𝑎), where 𝐺 𝑎 is the additive group. Then by Proposition 5.2.5, the étale local

𝑆)𝑎 = Ω1

cohomology groups of 𝑋 agree with the Zariski local cohomology groups.

Now it suffices to calculate the étale local cohomology groups of Ω1

𝑋/𝑑O𝑋. In fact, we have

the following exact sequences on the small étale site of 𝑋

0

0

(cid:47) O𝑋

𝐹

(cid:47) O𝑋

(cid:47) 𝑑O𝑋

(cid:47) 𝑑O𝑋

(cid:47) Ω1
𝑋

(cid:47) Ω1

𝑋/𝑑O𝑋

(cid:47) 0,

(cid:47) 0.

(5.2.3)

These sequences follow from the Cartier isomorphism [38, Corollary 2.5], since 𝑋 is affine

regular and 𝐹-finite. Passing to the cohomology sequences, we have the exact sequences

0

0

(cid:47) 𝐻1

𝑌 (𝑋, O𝑋)/𝐻1

𝑌 (𝑋, O𝑋) 𝑝

(cid:47) 𝐻2

𝑃 (𝑋, O𝑋)/𝐻2

𝑃 (𝑋, O𝑋) 𝑝

𝑑

𝑑

(cid:47) 𝐻1

𝑌 (𝑋, Ω1
𝑋)

(cid:47) 𝐻2

𝑃 (𝑋, Ω1
𝑋)

(cid:47) 𝐻1

𝑌 (𝑋, Ω1

𝑋/𝑑O𝑋)

(cid:47) 𝐻2

𝑃 (𝑋, Ω1

𝑋/𝑑O𝑋)

(cid:47) 0.

(cid:47) 0.

(5.2.4)

Using (5.2.4), it suffices to calculate the local cohomology groups of Ω1

𝑋 and O𝑋. They are

computed by the Cech complex in the below.

Lemma 5.2.7 ([39, Lemma 0A6R])

Let 𝐴 be a noetherian ring and let 𝐼 = ( 𝑓1, · · · , 𝑓𝑟) ⊂ 𝐴 be an ideal. Set 𝑍 = 𝑉 (𝐼) ⊂ Spec( 𝐴).

Then

𝑅Γ𝑍 ( 𝐴) ≃ ( 𝐴

(cid:47) (cid:206)𝑖0

𝐴 𝑓𝑖

0

(cid:47) · · ·

(cid:47) 𝐴 𝑓1··· 𝑓𝑟 )

in 𝐷 ( 𝐴). If 𝑀 is an 𝐴-module, then we have

𝑅Γ𝑍 (𝑀) ≃ ( 𝑀

(cid:47) (cid:206)𝑖0

𝑀 𝑓𝑖

0

(cid:47) · · ·

(cid:47) 𝑀 𝑓1··· 𝑓𝑟 )

in 𝐷 ( 𝐴).

Recall that 𝑋 = Spec 𝑘 [[𝜋, 𝑡]], where 𝑘 is an algebraically closed field. Let 𝑅 = 𝑘 [[𝜋, 𝑡]].

Then 𝜋 and 𝑡 are regular primes of 𝑅. Denote by 𝑉 (𝜋), 𝑉 (𝑡) the closures of codimension 1 points

(𝜋) and (𝑡) respectively.

56

(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
Then we have the following

𝑉 (𝑡) (𝑋, Ω1
𝐻1

𝑋) (cid:27) Ω1

𝑅[ 1
𝑡 ]

(cid:46)

Ω1

𝑅 ,

𝐻1

(𝑡) (𝑋, Ω1

𝑋) (cid:27) 𝐻1

(𝑡) (𝐷 (𝜋), Ω1
(cid:46)

𝐻2

𝑃 (𝑋, Ω1

𝑋) (cid:27) Ω1

(Ω1

𝑅[ 1
𝜋 ]

+ Ω1

𝑅[ 1
𝑡 ]

)

𝐷 (𝜋)) (cid:27) Ω1

𝑅[ 1

𝜋𝑡 ]

(cid:46)

Ω1

𝑅[ 1
𝜋 ]

,

𝑉 (𝑡) (𝑋, O𝑋) (cid:27) 𝑅[
𝐻1

] /𝑅 .

𝜋𝑡 ]

𝑅[ 1
1
𝑡

𝐻1

(𝑡) (𝑋, O1

𝑋) (cid:27) 𝐻1

𝐻2

𝑃 (𝑋, O1

𝑋) (cid:27) 𝑅[

𝐷 (𝜋)) (cid:27) 𝑅[

1
𝜋𝑡

]

(cid:30)

𝑅[

1
𝜋

] ,

(𝑡) (𝐷 (𝜋), O1
(cid:30) (cid:16)

1
𝜋𝑡

]

𝑅[

1
𝜋

] + 𝑅[

(cid:17)

]

.

1
𝑡

Combining with the exact sequence (5.2.4), it follows that

𝑉 (𝑡) (𝑋, Ω1
𝐻2

𝑋,log) (cid:27) Ω1

𝑅[ 1
𝑡 ]

(cid:30) (cid:16)

Ω1

𝑅 + (𝐹 − 𝐼)Ω1

𝑅[ 1
𝑡 ]

+ 𝑑 (𝑅[

(cid:17)

])

1
𝑡

(cid:27) Br(𝑅[

𝐻2

(𝑡) (𝑋, Ω1

𝑋,log) = Ω1

𝑅[ 1

𝜋𝑡 ]

(cid:27) Br(𝑅[

𝐻3

𝑃 (𝑋, Ω1

𝑋,log) = Ω1

𝑅[ 1

𝜋𝑡 ]

𝑅[ 1
𝜋 ]
(cid:30)

]) [ 𝑝]

1
𝜋𝑡
(cid:30) (cid:16)

Ω1

𝑅[ 1
𝜋 ]
(cid:30) (cid:16)

1
]) [ 𝑝] /Br(𝑅) [ 𝑝] ,
𝑡
(cid:30) (cid:16)

Ω1

+ (𝐹 − 𝐼)Ω1

+ 𝑑 (𝑅[

(cid:17)

])

1
𝜋𝑡

𝑅[ 1

𝜋𝑡 ]

Br(𝑅[

1
𝜋

]) [ 𝑝] ,

+ Ω1

𝑅[ 1
𝑡 ]

+ (𝐹 − 𝐼)Ω1

𝑅[ 1

𝜋𝑡 ]

+ 𝑑 (𝑅[

(cid:17)

])

1
𝜋𝑡

(cid:27) Br(𝑅[

1
𝜋𝑡

]) [ 𝑝]

Br(𝑅[

1
𝜋

]) [ 𝑝] + Br(𝑅[

]) [ 𝑝]

(cid:17)

.

1
𝑡

Notice that we used the fact that the localization of a unique factorization domain (UFD) at a

multiplicatively closed subset is also a UFD and the Picard group of a UFD is zero. Furthermore,

from the last row of (5.2.2), we get the following identification

Br(𝑅[ 1

𝑡 ]) [ 𝑝] /Br(𝑅) [ 𝑝]

(cid:47) Br(𝑅[ 1

𝜋𝑡 ]) [ 𝑝]

(cid:46)

Br(𝑅[ 1

𝜋 ]) [ 𝑝]

(5.2.5)

𝛿 𝑦
2

(cid:47) Br(𝑅[ 1

𝜋𝑡 ]) [ 𝑝]

(cid:46) (cid:16)

Br(𝑅[ 1

𝜋 ]) [ 𝑝] + Br(𝑅[ 1

𝑡 ]) [ 𝑝]

(cid:17)

(cid:47) 0.

5.3 Period-index problems of semi-global fields in positive characteristic

In this section, we are going to prove the following theorem as an application of the logarithmic

de Rham cohomology with support.

57

(cid:47)
(cid:47)
(cid:47)
Theorem 5.3.1

Let 𝑋 be a smooth projective curve over 𝑘 ((𝑡)) where 𝑘 is an algebraically closed fields of

characteristic 𝑝 > 0. Suppose that there is a model X over 𝑘 [[𝑡]] with good reduction. Suppose

that 𝜔 ∈ Br(𝑋) [ 𝑝] satisfies swX (𝜔) < 𝑝. Then per(𝜔) = ind(𝜔).

Here the geometric Swan conductor swX is defined based on the smooth model X of 𝑋. It is

not clear if the definition is independent of the choice of the smooth model. We will review the

notations and define the geometric Swan conductor (Definition 5.3.4) in the next subsection.

5.3.1 Notations

Let 𝑋 be an algebraic curve over 𝐾 = 𝑘 ((𝑡)), where 𝑘 = ¯𝑘 is an algebraically closed field

of characteristic 𝑝 > 0. Denote by 𝐹 = 𝐾 (𝑋) the function field of 𝑋. Let O𝐾 = 𝑘 [[𝑡]] be the

complete discrete valuation ring with the field of fraction 𝐾 and 𝑡 the uniformizer. Denote by 𝑇 the

unique closed point of Spec(O𝐾).

Definition 5.3.2

An integral model X of 𝑋 is a 2-dimensional regular O𝐾-scheme such that

(i) 𝑝 : X → Spec(O𝐾) is flat and proper;

(ii) There is an isomorphism of 𝐾-schemes 𝑋 ≃ X𝐾;

(iii) The reduced scheme (𝑌 = X × 𝑇)red is a 1-dimensional (proper) schemes over 𝑇 whose

irreducible components are all regular and has normal crossings (i.e. X𝑇 only has ordinary

double points as singularities).

The existence of an integral model follows from the resolution of singularities of excellent

2-dimensional schemes ([29]), and the embedded resolution of the special fiber ([30]). If 𝑋 admits

a smooth integral model X over O𝐾, we say that 𝑋 has good reduction over O𝐾. In this case, the

special fiber X𝑇 will have a single irreducible component that is a proper smooth curve over 𝑇.

𝑋

X

𝑌

Spec(𝐾)

(cid:47) Spec(O𝐾)

𝑇 = Spec(𝑘)

For a closed point 𝑃 of X, let OX,𝑃 denote the local ring at 𝑃, ˆOX,𝑃 the completion of the regular

58

(cid:47)
(cid:47)
(cid:15)
(cid:15)
(cid:15)
(cid:15)
(cid:111)
(cid:111)
(cid:15)
(cid:15)
(cid:47)
(cid:111)
(cid:111)
local ring OX,𝑃 at its maximal ideal and 𝐹𝑃 the field of fractions of ˆOX,𝑃. For an open subset 𝑈 of

an irreducible component of 𝑌 , let 𝑅𝑈 be the ring consisting of elements in 𝐹 which are regular on

𝑈. Then O𝐾 ⊂ 𝑅𝑈. Let ˆ𝑅𝑈 be the (𝑡)-adic completion of 𝑅𝑈 and 𝐹𝑈 the field of fractions of ˆ𝑅𝑈.

Now suppose that the algebraic curve 𝑋 has good reduction over O𝐾. We have the following

exact sequence by purity of the Brauer groups in codimension 1:

0

(cid:47) Br(𝑋)

(cid:47) Br(𝐾 (𝑋))

⊕𝑖 𝑥 (cid:47)

(cid:47) (cid:201)
𝑥∈𝑋0

Br(Quot( ˆO𝑋,𝑥)).

For 𝜔 ∈ Br(𝑋), we have that 𝜔 ∈ Br(O𝑋,𝑥) for all 𝑥 ∈ 𝑋0.

Lemma 5.3.3

Let 𝑓 : 𝑋 → 𝑌 be a morphism of schemes. Let 𝑦 ∈ 𝑌 and 𝑞 : 𝑋′ = 𝑋 ×𝑌 SpecO𝑌 ,𝑦 → 𝑋 be the

projection morphism. Then O𝑋 ′,𝑞−1 (𝑥) (cid:27) O𝑋,𝑥 for any 𝑥 ∈ 𝑋𝑦.

Recall that every effective irreducible divisor 𝐷 ⊂ X is either 𝑌 (𝐷 is vertical), or the closure

of a closed point 𝑥 ∈ 𝑋0 of the generic fiber (𝐷 is horizontal). Using this lemma, it follows that

𝜔 ∈ Br(OX,𝑥) for all 𝑥 ∈ 𝑋0 ⊂ X (1). Hence we have that 𝜔 is ramified only along the vertical

divisor 𝑌 . Hence we define Kato’s Swan conductor of 𝜔 ∈ Br(𝑋) in the following way.

Definition 5.3.4 (Swan conductor for Brauer groups of curves)

Let 𝑘 be an algebraically closed field of characteristic 𝑝 > 0 and let 𝑋 be an algebraic curve over

𝑘 ((𝑡)). Suppose 𝑋 has good reduction with the associated model X → Spec 𝑘 [[𝑡]]. Denote by 𝑣𝑌

the valuation associated to the divisor 𝑌 and 𝐹 the function field 𝑘 (𝑋).

Then we define the X-Swan conductor of 𝜔 ∈ Br(𝑋) [ 𝑝] by

swX (𝜔) = sw𝐹,𝑣𝑌 (𝜔).

5.3.2 Sketch of the proof

In order to prove Theorem 5.3.1, we will use the patching method from [19], which reduces the

global period-index problem to two types of local period-index problems. We continue to use the

notations from last section.

59

(cid:47)
(cid:47)
Let 𝜂 be a generic point of an irreducible component of 𝑌 and 𝐹𝜂 the completion of 𝐹 at the

discrete valuation given by 𝜂. Let 𝐷 be a central simple algebra over 𝐹. By [20, 5.8], there exists

an irreducible open set 𝑈𝜂 of 𝑌 containing 𝜂 such that ind(𝐷 ⊗𝐹 𝐹𝑈𝜂 ) = ind(𝐷 ⊗𝐹 𝐹𝜂).

Theorem 5.3.5 (Patching, [19, Theorem 5.1], [35, Page 228])

Let 𝐷 be a central simple algebra over 𝐹 of period 𝑝. Let 𝑆0 be a finite set of closed points of X

containing all the points of intersection of the components of 𝑌 and the support of the ramification

divisor of 𝐷. Let 𝑆 be a finite set of closed points of X containing 𝑆0 and 𝑌 \ (∪ 𝑈𝜂), where 𝜂

varies over generic points of 𝑌 . Then

ind(𝐷) = lcm (cid:8)ind(𝐷 ⊗ 𝐹𝜁 )(cid:9) ,

where 𝜁 runs over 𝑆 and irreducible components of 𝑌 \ 𝑆.

We apply this theorem in our situation. First, suppose 𝜁 = 𝑈 for some irreducible component

𝑈 of 𝑌 \ 𝑆. Let 𝜂 be the generic point of 𝑈. Then 𝑈 ⊂ 𝑈𝜂. Since 𝐹𝑈𝜂 ⊂ 𝐹𝑈, ind(𝐷 ⊗𝐹 𝐹𝑈) |

ind(𝐷 ⊗𝐹 𝐹𝑈𝜂 ) = ind(𝐷 ⊗𝐹 𝐹𝜂). Since the residue field of the generic point of 𝑈 is a function field

of the curve over an algebraically closed field, by Theorem 4.2.2, we have ind(𝐷 ⊗ 𝐹𝜂)| 𝑝. Hence,

ind(𝐷 ⊗𝐹 𝐹𝑈) | 𝑝.

Next suppose 𝜁 = 𝑃 ∈ 𝑆, where 𝑃 is a closed point of X. By the Cohen structure theorem for

an equi-characteristic field [39, Tag 0C0S], we have

ˆOX,𝑃 (cid:27) 𝑘 [[𝜋, 𝑡]],

where 𝜋, 𝑡 are local uniformizers at 𝑃. Notice that it is actually a 𝑘-algebra isomorphism, since the

residue field 𝑘 is naturally embedded into the complete local ring. In general, the Cohen’s structure

theorem only provides a ring isomorphism instead of a 𝑘-algebra isomorphism.

To analyze the period-index problem for the field 𝐹𝑃 = 𝑘 ((𝜋, 𝑡)), we will apply Theorem 5.1.4

to the 2-dimensional regular local ring 𝑘 [[𝜋, 𝑡]]. Notice that we have the condition swX (𝜔) < 𝑝.

We would relate the X-Swan conductor to the local Kato’s Swan conductor in the next subsection.

60

5.3.3 Local Swan Conductor

We use the notations from Theorem 5.1.4. There is a commutative diagram

Br(𝐹) [ 𝑝] ≃ 𝐻1(𝐹, Ω1

𝐹,log)

𝐻1(OX,𝑃, Ω1

•,log)

(cid:16)

(cid:47) 𝐻2
(𝑡)

OX,𝑃, Ω1

•,log

(cid:17)

,

𝐻1( ˆOX,𝑃, Ω1

•,log)

(cid:47) 𝐻2
(𝑡)

(cid:16) ˆOX,𝑃, Ω1

•,log

(cid:17)

where the horizontal row is part of the long exact sequence in local étale cohomology associated

to the sheaf Ω1

•,log. By (5.2.1), we have the following isomorhisms

𝐻2

(𝑡) (OX,𝑃, Ω1

𝐻2

(𝑡) ( ˆOX,𝑃, Ω1

•,log) (cid:27) Br(Frac((OX,𝑃) ℎ
•,log) (cid:27) Br(Frac(( ˆOX,𝑃) ℎ

(𝑡))) [ 𝑝]/Br((OX,𝑃) ℎ
(𝑡))) [ 𝑝]/Br(( ˆOX,𝑃) ℎ

(𝑡)) [ 𝑝],

(𝑡)) [ 𝑝].

(5.3.1)

(5.3.2)

Notice that, for a prime ideal 𝔭 of a ring 𝑅, we denote by 𝑅ℎ

𝔭 the henselization of the localization

𝑅𝔭 of the ring 𝑅 at the prime ideal 𝔭.

Using the diagram above, we can give a result which relates the X-Swan conductor to the local

cohomology groups as in (5.3.1) and (5.3.2).

Proposition 5.3.6

Let 𝑋 be an algebraic curve over 𝑘 ((𝑡)) with a smooth integral model X → Spec 𝑘 [[𝑡]] and

𝜔 ∈ Br(𝑋) [ 𝑝]. Then

swFrac(( ˆOX,𝑃)ℎ

(𝑡 ) ) (𝜔) = swFrac((OX,𝑃)ℎ

(𝑡 ) ) (𝜔) = swX (𝜔).

Proof of Proposition 5.3.6.

The second equality follows from Definition 5.3.4. For the first one, by Lemma 5.3.7, we can

take 𝐾 = Frac((OX,𝑃) ℎ

(𝑡)). Then it suffices to show that the residue
field extension is separable, since 𝑡 is the uniformizer in both fields. The residue field extension is

(𝑡))) and 𝐿 = Frac(( ˆOX,𝑃) ℎ

given by 𝑘 (𝜋) → 𝑘 ((𝜋)), which is a completion morphism. The separability is given by Lemma

5.3.8.

Lemma 5.3.7 ([26, Lemma 6.2, Page 119])

□

Let 𝐾 ⊂ 𝐿 be two henselian discretely valued fields such that O𝐾 ⊂ O𝐿 and 𝑚 𝐿 = O𝐿𝑚𝐾. Assume

61

(cid:47)
(cid:47)
(cid:41)
(cid:41)
(cid:15)
(cid:15)
(cid:47)
(cid:15)
(cid:15)
(cid:47)
that the residue field of 𝐿 is separable over the residue field of 𝐾. Then, for any 𝜔 ∈ Br(𝐾) [ 𝑝], we

have

Lemma 5.3.8

sw𝐾 (𝜔) = sw𝐿 (𝜔).

Let 𝐹 be a discretely valued field with [𝐹 : 𝐹 𝑝] = 𝑝 and ˆ𝐹 be its completion. Then the completion

morphism 𝐹 → ˆ𝐹 is separable.

Proof.

We prove it by contradiction. Suppose that there exists an algebraic extension 𝐸/𝐹 inside ˆ𝐹 which

is not separable. Then we can decompose 𝐸/𝐹 as a chain of field extensions 𝐸/𝐿/𝐹 where 𝐿 is

separable over 𝐹 and 𝐸/𝐿 is purely inseparable. Moreover, let 𝜋 be a uniformizer of 𝐹. Then

inseparable, there exists 𝑎 ∈ 𝐸 ⊂ ˆ𝐹 such that 𝑎 𝑝 ∈ 𝐿 \ 𝐿 𝑝. Let 𝑏 = 𝑎 𝑝. It follows that 𝑏 =

we have that 𝜋 is still a uniformizer in 𝐿, and it gives a 𝑝-basis of 𝐿/𝐿 𝑝. Since 𝐸/𝐿 is purely
𝑓 𝑝
𝑖 𝜋𝑖
in 𝐿 (also in ˆ𝐹) such that 𝑓𝑖 ≠ 0 for some 𝑖 > 0. However, notice that 𝜋 is a uniformizer of ˆ𝐹.

𝑝−1
(cid:205)
𝑖=0

Therefore, it implies that 𝑏 is not a 𝑝-power in ˆ𝐹, which is a contradiction. Hence the conclusion

follows.

5.3.4 The end of the proof

Theorem 5.3.9

□

Let 𝑅 = 𝑘 [[𝜋, 𝑡]], 𝑋 = Spec(𝑅), 𝐾 = Frac(𝑅) and 𝜔 ∈ Br(𝐾) [ 𝑝] which ramifies only along (𝑡)

with sw𝐾,(𝑡) (𝜔) = 𝑚 < 𝑝. Then 𝜔 = [∗, 𝜋).

Proof. By Theorem 5.1.4, we have the exact sequence

0

(cid:47) 𝐻1(𝑋, Ω1

𝑋,log)

(cid:47) 𝐻1(𝐾, Ω1

𝐾,log)

𝛿1

(cid:47) (cid:201)
𝑥∈𝑋 1

𝐻2

𝑥 (𝑋, Ω1

𝑋,log)

𝛿2

(cid:47) 𝐻3

𝑃 (𝑋, Ω1

𝑋,log)

(cid:47) 0.

Since 𝑋 is affine and regular, 𝐻1(𝑋ét, Ω1

𝑋,log) ≃ Br(𝑅) [ 𝑝] ≃ Br(𝑘) [ 𝑝] = 0. So the last exact

sequence reduces to

0

(cid:47) 𝐻1(𝐾, Ω1

𝐾,log)

𝛿1

(cid:47) (cid:201)
𝑥∈𝑋 1

𝐻2

𝑥 (𝑋, Ω1

𝑋,log)

𝛿2

62

(cid:47) 𝐻3

𝑃 (𝑋, Ω1

𝑋,log)

(cid:47) 0.

(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
(cid:47)
Our goal is to find a symbol algebra that represents 𝜔 in the Brauer group of 𝐾. Since sw𝐾,(𝑡) (𝜔) =

𝑚 < 𝑝, by Theorem 4.2.2, we have that

𝜔 = [

𝑓𝑚
𝑡𝑚

, 𝜋) + [

𝑓𝑚−1
𝑡𝑚−1

, 𝜋) + · · · + [ 𝑓0, 𝑡) in Br(𝐾) [ 𝑝],

where 𝑓𝑖 ∈ 𝜋 · 𝑘 [[𝜋]] for all 𝑖. The choices of 𝑓𝑖 follow from the identification in (5.2.5). Since

that 𝜋 is a regular element, we have a consistent way to lift the elements from 𝑘 ((𝜋)).

Since 𝑓0 ∈ 𝑘 [[𝜋]] and 𝑘 is algebraically closed, by Hensel’s lemma, we have [ 𝑓0, 𝑡) ≃ 0.

Hence, 𝜔 = [

𝑓𝑚
𝑡𝑚 + · · · +

𝑓1
𝑡

, 𝜋).

□

Now we are ready to finish the proof of Theorem 5.3.1 based on Theorem 5.3.5.

Proof of Theorem 5.3.1.

First, suppose 𝜁 = 𝑈 for some irreducible component 𝑈 of 𝑌 \ 𝑆. Let 𝜂 be the generic point of

𝑈. Then 𝑈 ⊂ 𝑈𝜂. Since 𝐹𝑈𝜂 ⊂ 𝐹𝑈, ind(𝐷 ⊗𝐹 𝐹𝑈) | ind(𝐷 ⊗𝐹 𝐹𝑈𝜂 ) = ind(𝐷 ⊗𝐹 𝐹𝜂). Since the

residue field of the generic point of 𝑈 is a function field of the curve over an algebraically closed

field, by Theorem 4.2.2, we have ind(𝐷 ⊗ 𝐹𝜂)| 𝑝. Hence, ind(𝐷 ⊗𝐹 𝐹𝑈) | 𝑝.

Second, suppose 𝜁 = 𝑃 ∈ 𝑆, where 𝑃 is a closed point of X. Combining Proposition 5.3.6 and

Theorem 5.3.9, we have ind(𝐷 ⊗ 𝐹𝜁 )| 𝑝.

Finally, by Theorem 5.3.5, we have that per(𝜔) = ind(𝜔).

□

63

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67

APPENDIX A

SYMBOL LENGTH AND FOLIATION THEORY

Let 𝐹 be a field of characteristic 𝑝 > 0 and [𝐹 : 𝐹 𝑝] = 𝑝𝑛, 𝑛 ∈ N>0. We can approach the symbol

length problem of Ω1

𝐹/𝑍 1

𝐹 using the Galois theory of purely inseparable extensions.

Let 𝐿/𝐾 be a field extension of characteristic 𝑝 > 0. The vector space Der𝐾 (𝐿) of 𝐾-derivations

𝐷 : 𝐿 → 𝐿 is closed under forming commutators [𝐷, 𝐷′] and 𝑝-fold compositions 𝐷 [ 𝑝] in the

associative ring End𝐾 (𝐿). We can view Der𝐾 (𝐿) as a Lie algebra over 𝐾, endowed the map

𝐷 ↦→ 𝐷 [ 𝑝] as an additional structure. This phenomenon only happens in characteristic 𝑝 > 0. We

call them restricted Lie algebras.

A restricted Lie algebra (𝑝-Lie algebra) over 𝐾 is a Lie algebra 𝔤 over 𝐾, together with a map

𝔤 → 𝔤, 𝑥 ↦→ 𝑥 [ 𝑝] called the 𝑝-map, subject to the following three axioms:

(R 1) We have ad𝑥 [ 𝑝] = (ad𝑥) 𝑝 for all vectors 𝑥 ∈ 𝔤.
(R 2) Moreover (𝜆 · 𝑥) [ 𝑝] = 𝜆 𝑝 · 𝑥 [ 𝑝] for all vectors 𝑥 ∈ 𝔤 and scalars 𝜆 ∈ 𝐾.
(R 3) The formula (𝑥 + 𝑦) [ 𝑝] = 𝑥 [ 𝑝] + 𝑦 [ 𝑝] + (cid:205)𝑝−1
𝑟=1

𝑠𝑟 (𝑥, 𝑦) holds for all 𝑥, 𝑦 ∈ 𝔤.

Here the summands 𝑠𝑟 (𝑥, 𝑦) are universal expressions defined by

𝑠𝑟 (𝑡0, 𝑡1) = −

1
𝑟

∑︁

𝑢

(ad𝑡𝑢(1) ◦ · · · ◦ (ad𝑡𝑢( 𝑝−1) (𝑡1),

where ad𝑎 (𝑥) = [𝑎, 𝑥] denotes the adjoint representation, and the index runs over all maps 𝑢 :

{1, · · · , 𝑝 − 1} → {0, 1} taking the value zero exactly 𝑟 times.

Restricted Lie algebras were introduced and studied by Jacobson. It appears in the Galois theory

of purely inseparable extensions. We will recall it below. We say 𝐿/𝐾 has exponent 1 if 𝑥 𝑝 ∈ 𝐾

for all 𝑥 ∈ 𝐿.

Theorem A.0.1 (Jacobson)

Let 𝐿/𝐾 be a finite purely inseparable field extension of exponent one. There is an inclusion-

reversing bijection between 𝐿/𝐾-restricted Lie algebra Der𝐸 (𝐿) ⊂ Der𝐾 (𝐿) and intermediate

field extension 𝐾 ⊂ 𝐸 ⊂ 𝐿.

Recall that there is an isomorphism of 𝐹-vector spaces Hom𝐹 (Ω1

𝐹, 𝐹) (cid:27)𝜙 Der𝐹 𝑝 (𝐹), where

68

𝑓 ∈ Hom𝐹 (Ω1

𝐹, 𝐹) ↦→ 𝜙( 𝑓 )(𝑎) = 𝑓 (𝑑𝑎). Notice that 𝑍 1

𝐹 has dimension ( 𝑝𝑛 + 𝑛 − 1) as a 𝐹 𝑝-vector

space.

Let 𝛼 ∈ Ω1

𝐹 and 𝛽 ∈ 𝑍 1

𝐹, 𝐹) defined by 𝑉 (𝛼 + 𝛽) =
𝐹/𝑍 1
𝐹, 𝐹) | 𝑓 (𝛼 + 𝛽) = 0}. Denote the image of 𝑉 (𝛼 + 𝛽) in Der𝐹 𝑝 (𝐹) also by 𝑉 (𝛼 + 𝛽).
Then the existence of a restricted 𝐹-subspace of 𝑉 (𝛼 + 𝛽) would be equivalent to the symbol length

𝐹. Consider the 𝐹-subspace of Hom𝐹 (Ω1

{ 𝑓 ∈ Hom𝐹 (Ω1

conjecture 3.5.7. Let us investigate the special case ( 𝑝, 𝑛) = (2, 2) using this approach.

𝜕
𝜕
Let 𝜔 ∈ Der𝐹 𝑝 (𝐹) and {
𝜕𝑠
𝜕𝑡
𝜕
𝜕𝑡 for 𝑓 , 𝑔 ∈ 𝐹.

we have that 𝜔 = 𝑓

𝜕
𝜕𝑠

+ 𝑔

,

} be a 𝐹-basis of Der𝐹 𝑝 (𝐹) given by a 𝑝-basis {𝑠, 𝑡} of 𝐹. Then

Since dim𝐹 Der𝐹 𝑝 (𝐹) = 2, we want to find out the conditions on 𝑓 and 𝑔 such that 𝜔[ 𝑝] = 𝑘𝜔

for 𝑘 ∈ 𝐹. If either of 𝑓 or 𝑔 is 0, it is obvious 𝜔[ 𝑝] = 0. Hence we assume that 𝑓 , 𝑔 ≠ 0. Since

𝑝 = 2,

𝜔[2] (𝑠) =𝜔[2] (𝑑𝑠) = 𝜔( 𝑓 )

=𝜔(𝑑𝑓 ) = 𝜔( 𝑓𝑠𝑑𝑠 + 𝑓𝑡 𝑑𝑡)

= 𝑓𝑠 𝑓 + 𝑓𝑡𝑔.

(A.0.1)

(A.0.2)

(A.0.3)

Similarly, we have that 𝜔[2] (𝑡) = 𝑔𝑠 𝑓 + 𝑔𝑡𝑔. Hence, 𝜔[2] = 𝑘𝜔 is equivalent to the existence of a

solution to the following equation

( 𝑓𝑠 𝑓 + 𝑓𝑡𝑔)𝑔 = (𝑔𝑠 𝑓 + 𝑔𝑡𝑔) 𝑓 .

(A.0.4)

Since we are considering the 𝐹-vector space {𝑙 · 𝜔 | 𝑙 ∈ 𝐹}, we can further assume that 𝑔 = 1.

Then the equation reduces to

𝑓𝑠 𝑓 + 𝑓𝑡 = 0.

(A.0.5)

Now we take the expansion of 𝑓 over 𝐹 𝑝. Let 𝑓 = 𝑓 2

00 + 𝑓 2

01

𝑡 + 𝑓 2
10

𝑠 + 𝑓 2
11

𝑠𝑡. Then 𝑓𝑠 = 𝑓 2

10 + 𝑓 2

11

𝑡

and 𝑓𝑡 = 𝑓 2

01 + 𝑓 2

11

𝑠. It follows that

( 𝑓 2

01 + 𝑓 2
00

10 + 𝑓 2
𝑓 2
11

𝑓 2
01

𝑡2) + ( 𝑓 2

11 + 𝑓 4

10 + 𝑓 4
11

𝑡2)𝑠 + ( 𝑓 2
11

00 + 𝑓 2
𝑓 2
10

01)𝑡 = 0.
𝑓 2

(A.0.6)

69

It is equivalent to the following system of equations

𝑓01 + 𝑓00 𝑓10 + 𝑓11 𝑓01𝑡 = 0
10 + 𝑓 2
11
𝑓11 𝑓00 + 𝑓10 𝑓01

𝑓11 + 𝑓 2

= 0

= 0.

𝑡






(A.0.7)

If 𝑓11 = 0, it implies 𝑓10 = 𝑓01 = 0. Hence we get the first kind of solutions 𝑓 = 𝑓 2

00. When 𝑓11 ≠ 0,
we notice that the determinant of the first and the third equations respect to variables 𝑓00 and 𝑓01 is

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝑓10 1 + 𝑓11𝑡

𝑓11

𝑓10

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

= 𝑓11 + 𝑓 2

10 + 𝑓 2
11

𝑡.

(A.0.8)

Therefore, the second equation of (A.0.7) is more independent comparing with others. Moreover,

since 𝑓11 ≠ 0, we can divide both sides of the second equation by 𝑓 2
11

(cid:0)

𝑓10
𝑓11

(cid:1) 2 +

1
𝑓11

+ 𝑡 = 0.

Let 𝑚 =

𝑓10
𝑓11

∈ 𝐹. Then we have 𝑓11 =

𝑛 ∈ 𝐹, and 𝑓 = 𝑚2𝑛2 + 𝑛2𝑡 +

and 𝑓10 =

1
𝑚2 + 𝑡
𝑠𝑡
(𝑚2 + 𝑡)2

.

𝑚2𝑠
(𝑚2 + 𝑡)2

+

(A.0.9)

𝑚
𝑚2 + 𝑡

. Hence 𝑓00 = 𝑚𝑛,

𝑓01 = 𝑛 for

Summarizing the above discussions, we get the following theorem which classifies all the proper

restricted 𝐹-subspaces of Der𝐹 𝑝 (𝐹).

Theorem A.0.2 (Classification of proper restricted subspaces of Der𝐹 𝑝 (𝐹))

Let 𝐹 be a field of characteristic 𝑝 = 2 and [𝐹 : 𝐹 𝑝] = 𝑝2. Let {𝑠, 𝑡} be a 𝑝-basis of 𝐹 and

𝜔 = 𝑓

𝜕
𝜕𝑠

+ 𝑔

𝜕
𝜕𝑡

∈ Der𝐹 𝑝 (𝐹) (cid:27) Hom𝐹 (Ω1

𝐹, 𝐹). Then the 𝐹-subspace of Der𝐹 𝑝 (𝐹) generated by 𝜔

is restricted if and only if [ 𝑓 : 𝑔] ∈ P1

𝐹 takes values in the following cases:

(𝑖) [1 : 0], [0 : 1];

(𝑖𝑖) [𝑘 2 : 1] for 𝑘 ∈ 𝐹×;

(𝑖𝑖𝑖) [𝑚2𝑛2 + 𝑛2𝑡 +

𝑚2𝑠
(𝑚2 + 𝑡)2

+

𝑠𝑡
(𝑚2 + 𝑡)2

: 1] for 𝑚, 𝑛 ∈ 𝐹.

Now we can turn back to the symbol length problem of the group Ω1

𝐹. Keep the assumptions
𝐹. Then 𝛼 = 𝑚2𝑡dlog(𝑠) + 𝑛2𝑠dlog(𝑡) + 𝑙2𝑠𝑡dlog(𝑠), where

𝐹/𝑍 1

in Theorem A.0.2. Let 𝛼 ∈ Ω1

𝐹/𝑍 1

70

𝑓 , 𝑔, ℎ ∈ 𝐹. Let 𝛽 ∈ 𝑍 1

𝐹. Then 𝛽 = 𝑎2dlog(𝑠) + 𝑏2dlog(𝑡) + 𝑑 (𝑐) = 𝑎2dlog(𝑠) + 𝑏2dlog(𝑡) +

𝑑 (𝑐2
01

𝑡 + 𝑐2
10

𝑠 + 𝑐2
11

𝑠𝑡), where 𝑎, 𝑏, 𝑐 = 𝑐2

00 + 𝑐2

01

𝑡 + 𝑐2
10

𝑠 + 𝑐2
11

𝑠𝑡 ∈ 𝐹.

𝛼 + 𝛽 = (

𝑚2𝑡
𝑠

+ 𝑙2𝑡 +

𝑎2
𝑠

+ 𝑐2

10 + 𝑐2
11

𝑡)𝑑𝑠 + (

𝑛2𝑠
𝑡

+ 𝑏2 + 𝑐2

01 + 𝑐2
11

𝑠)𝑑𝑡.

(A.0.10)

Let 𝜔 ∈ 𝑉 (𝛼 + 𝛽), i.e. 𝜔(𝛼 + 𝛽) = 0. Suppose that 𝜔 = 𝑓

𝜕
𝜕𝑠

+ 𝑔

𝜕
𝜕𝑡 . It follows that

𝜔(𝛼 + 𝛽) = (

𝑚2𝑡
𝑠

+ 𝑙2𝑡 +

𝑎2
𝑠

+ 𝑐2

10 + 𝑐2
11

𝑡) 𝑓 + (

𝑛2𝑠
𝑡

+ 𝑏2 + 𝑐2

01 + 𝑐2
11

𝑠)𝑔 = 0

(A.0.11)

Hence

𝑓 =

𝑛2𝑠
𝑡 + 𝑏2 + 𝑐2

01 + 𝑐2

𝑠

𝑚2𝑡

𝑠 + 𝑙2𝑡 + 𝑎2

𝑠 + 𝑐2
10

11
+ 𝑐2
11

.

𝑡

The computation here will get complicated.

From (3.5.1), we find that 𝛼 is split by the purely inseparable extension defined by 𝑦2 =
𝑠2𝑙2
𝑚2

The restricted 𝐹-subspace corresponding to the 1-form 𝑑 (
𝑠2𝑙2
𝑚2𝑡

𝑡 + 𝑠𝑡) = [𝑡𝑑𝑠 + (

𝑠2𝑙2
𝑚2

𝜕
𝜕𝑡 .

𝜕
𝜕𝑠

𝑠
𝑡

+

+

(cid:0)

(cid:1)

𝑠2𝑙2
𝑚2

𝑡+𝑠𝑡.

+ 𝑠)𝑑𝑡] is

71

APPENDIX B

RAMIFICATION OF CENTRAL DIVISION ALGEBRAS (𝑝-RANK 1 CASE)

In this appendix, let 𝐾 be a complete discretely valued field with the valuation 𝑣 and the residue

field 𝐹 of characteristic 𝑝 > 0, where [𝐹 : 𝐹 𝑝] = 𝑝. We want to use the structure of 𝑝-torsion part

of Br(𝐾) to understand the ramification behavior of 𝑝𝑛-torsion part of Br(𝐾). We mainly consider

two cases: (1) Br.dim𝑝 (𝐸) = 0 for all finite extension 𝐸/𝐹, and (2) 𝐹 is a local field.

B.1 Br.dim𝑝 (𝐸) = 0 for all finite extension 𝐸/𝐹

In this case, Br(𝐾) [ 𝑝] has symbol length 1. Hence, every central division algebra of period

𝑝 over 𝐾 is cyclic and has ramification index 𝑝 and a degree 𝑝 residue field extension. Now for

a period 𝑝𝑛 central division algebra 𝐴 over 𝐾, it has degree 𝑝𝑛 by Theorem 4.2.2. Hence we can

assume [ 𝐴 : 𝐾] = 𝑝2𝑛. This gives 𝑒 = 𝑒′ = 𝑝𝑛 (Notation 2.3.1). The residue division algebra of 𝐴

is commutative and hence a field. It has degree 𝑝𝑛 over 𝐾. We want to describe this residue field

using the structure of Br(𝐾) [ 𝑝].

Next we explain how to read the information related to the residue field extension. Consider

𝑝𝑛−1 [ 𝐴], where [ 𝐴] indicates the class of 𝐴 as above in Br(𝐾). This class has period 𝑝. Hence,

there exists a field extension 𝐿1/𝐾 of degree 𝑝 with the degree 𝑝 residue field extension 𝐸1/𝐹.

Then we consider [ 𝐴] 𝐿1, the image of [ 𝐴] in Br(𝐿1). The field 𝐸1 is a complete discretely
valued field with the residue field 𝐸1 which is a degree 𝑝 field extension of 𝐹. Now 𝐸1/𝐹 is

either an Artin-Schreier extension or a purely inseparable extension. By Proposition 4.1.2 and the
assumptions on 𝐹, Br(𝐸1) [ 𝑝] = 0 and [𝐸1 : 𝐸 𝑝
] = 𝑝. The period of [ 𝐴] 𝐿1 is 𝑝𝑛−1. Otherwise,
the index of [ 𝐴] 𝐿1 is less than 𝑝𝑛−1. Then a splitting field of [ 𝐴] 𝐿1 would have the degree over 𝐾
less than 𝑝𝑛 (= 𝑝 · 𝑝𝑛−1), which is a contradiction.

1

Hence, we can repeat the argument above to get a composition of field extensions of degree 𝑝,

𝐾 ⊂ 𝐿1 ⊂ · · · ⊂ 𝐿𝑛, such that the composition of residue field extensions, 𝐹 ⊂ 𝐸1 ⊂ · · · ⊂ 𝐸𝑛,

consists of either Artin-Schreier extension or purely inseparable extension of degree 𝑝. 𝐿𝑛 is a

splitting field of 𝐴 with degree 𝑝𝑛.

This gives the following theorem:

72

Theorem B.1.1

Suppose that the field 𝐹 satisfies [𝐹 : 𝐹 𝑝] = 𝑝 and Br.dim𝑝 (𝐸) = 0 for all finite extensions 𝐸/𝐹.

Let 𝐾 be a complete discretely valued field with the residue field 𝐹. Then the degree of a central

division algebra of period 𝑝𝑛 over 𝐾 is 𝑝𝑛. Moreover, it admits a splitting field of degree 𝑝𝑛 such

that the residue field extension is of degree 𝑝𝑛 which is a composition of either Artin-Schreier

extension of degree 𝑝 or purely inseparable extension of degree 𝑝. The ramification index is also

𝑝𝑛.

In fact, we have an easy way to determine the separable degree and the inseparable degree of

the residue field extension.

Corollary B.1.2

We continue with the same assumptions as in Theorem B.1.1. Let 𝐴 be a central division algebra

over 𝐾 of period 𝑝𝑛. Denote the order of [ 𝐴]tame in Br(𝐾tame) by 𝑝𝑚 (≤ 𝑝𝑛), where 𝐾tame is
the maximal tame extension of 𝑘 ((𝜋)). Then the residue field 𝐵 of 𝐴 has degree 𝑝𝑛 over 𝑘 with

separable degree [𝐵 : 𝐹] 𝑠 = 𝑝𝑛−𝑚 and inseparable degree [𝐵 : 𝐹]𝑖 = 𝑝𝑚.

Proof.

Consider the class 𝑝𝑚 [ 𝐴]. It is split by a tame extension of degree 𝑝𝑛−𝑚 over 𝐾. More precisely,

this tame extension has ramification index 1 and residual degree 𝑝𝑛−𝑚. Hence, the proof reduces

to the case 𝑚 = 𝑛. This case just follows from Theorem B.1.1.

□

B.1.1 𝐹 a local field

In general, if Br.dim𝑝 (𝐸) > 0 for a finite extension 𝐸/𝐹, the situation is more complicated, since

there exist nontrivial division algebras over the residue field 𝐸. However, when 𝐹 is a local field,

we have the following theorem similar to Theorem B.1.1.

Theorem B.1.3

Suppose that 𝐹 is a local field, i.e. 𝐹 (cid:27) F𝑞 ((𝑡)), 𝑞 = 𝑝𝑛. Then the degree of a central division

algebra of period 𝑝𝑛 over 𝐾 is 𝑝𝑛. Moreover, it admits a splitting field of degree 𝑝𝑛 such that the

73

residue field extension is of degree 𝑝𝑛 and it is a composition of either Artin-Schreier extension of

degree 𝑝 or purely inseparable extension of degree 𝑝. The ramification index is also 𝑝𝑛.

Proof.

The proof is similar to the proof of Theorem B.1.1. The local field condition on 𝐹 is used for

Theorem 4.2.4.

It follows that a tamely ramified Brauer class in Br(𝐾) [ 𝑝] is split by a tame

Artin-Schreier extension of degree 𝑝 over 𝐾.

□

Similarly, we have the following corollary.

Corollary B.1.4

Suppose that 𝐹 is a local field, i.e. 𝐹 (cid:27) F𝑞 ((𝑡)), 𝑞 = 𝑝𝑛. Let 𝐴 be a central division algebra over 𝐾

of period 𝑝𝑛. Denote the order of [ 𝐴]tame in Br(𝐾tame) by 𝑝𝑚 (≤ 𝑝𝑛), where 𝐾tame is the maximal
tame extension of 𝐾. Then the residue field 𝐵 of 𝐴 has degree 𝑝𝑛 over 𝐹 with separable degree

[𝐵 : 𝐹] 𝑠 = 𝑝𝑛−𝑚 and inseparable degree [𝐵 : 𝐹]𝑖 = 𝑝𝑚.

Remark B.1.5

In fact, the condition on 𝐹 can be replaced by 𝐹 is 𝑝-quasilocal and almost perfect using [12,

Theorem 2.3]. The key ingredient of the proof is the fact that Br.dim𝑝 (𝐸) = 1 for all fintie extension

𝐸/𝐹.

74