Reduced norms of division algebras over complete discrete valuation fields of local-global type

TL;DR

This paper characterizes reduced norms of certain division algebras over complete discrete valuation fields with global residue fields of positive characteristic, extending known results to the $p$-torsion case.

Contribution

It proves that the subgroup of reduced norms equals the kernel of a specific cup product map for division algebras of $p$-power degree under certain conditions, extending previous prime-to-$p$ results.

Findings

Reduced norms form the kernel of a cup product map in the $p$-torsion case.

The result applies to tamely ramified or period-$p$ division algebras.

Extends the theorem of Parimala, Preeti, and Surech to $p$-torsion division algebras.

Abstract

Let $F$ be a complete discrete valuation field whose residue field $k$ is a global field of positive characteristic $p$. Let $D$ be a central division $F$-algebra of $p$-power degree. We prove that the subgroup of $F^*$ consisting of reduced norms of $D$ is exactly the kernel of the cup product map $\lambda\in F^*\mapsto (D)\cup(\lambda)\in H^3(F,\,\mathbb{Q}_{p}/\Z_{p}(2))$, if either $D$ is tamely ramified or of period $p$. This gives a $p$-torsion counterpart of a recent theorem of Parimala, Preeti and Surech, where the same result is proved for division algebras of prime-to-$p$ degree.

Equations

$$F^{\ast }⟶H^3(F,\mathbb{Z}/n(2)):=H^3(F,{\mu }_n^{\otimes 2});\lambda ⟼(D)\cup (\lambda )$$

$${\mathcal{R}}_D:F^{\ast }/\mathrm{Nrd}(D^{\ast })⟶H^3(F,\mathbb{Z}/n(2));\lambda ⟼(D)\cup (\lambda ),$$

$$H^{r+1}(K,\mathbb{Z}/n(r))\times (K^{\ast }/K^{\ast n})⟶H^{r+1}(K,\mathbb{Z}/n(r+1))$$

$$H^1(F,{\mathbf{SL}}_1(D))=F^{\ast }/\mathrm{Nrd}(D^{\ast })⟶H^1(\hat{F},{\mathbf{SL}}_1(D))={\hat{F}}^{\ast }/\mathrm{Nrd}({\hat{D}}^{\ast })$$

$$\begin{array}{rl}{H}_{p^m}^{r+1}(k)& =H^{r+1}(k,\mathbb{Z}/p^m(r)),\\ \text{and}{H}^{r+1}(k)& =\underset{m}{\underrightarrow{\lim }}{H}_{p^m}^{r+1}(k)=\underset{m}{\underrightarrow{\lim }}{H}^{r+1}(k,\mathbb{Z}/p^m(r)).\end{array}$$

$$H_{p^m}^{r+1}(k)\times \left(K_q(k)/p^m\right)⟶H_{p^m}^{r+q+1}(k),$$

$${\mathrm{Res}}_{L/k}:H_{p^m}^{r+1}(k)⟶H_{p^m}^{r+1}(L);\theta ⟼{\theta }_L.$$

$${\mathrm{Cor}}_{L/k}:H_{p^m}^{r+1}(L)⟶H_{p^m}^{r+1}(k).$$

$$\begin{array}{rl}{\mathrm{Cor}}_{L/k}({\theta }_L\cup \mu )& =\theta \cup N_{L/k}(\mu ),\forall \theta \in H_{p^m}^{r+1}(k),\forall \mu \in K_q(L)/p^m,\\ {\mathrm{Cor}}_{L/k}({\theta }^{\prime }\cup {\lambda }_L)& ={\mathrm{Cor}}_{L/k}({\theta }^{\prime })\cup \lambda ,\forall {\theta }^{\prime }\in H_{p^m}^{r+1}(L),\forall \lambda \in K_q(k)/p^m.\end{array}$$

$$H^{r+1}_{p^{m}}(F)_{tr}:=\ker\big{(}\mathrm{Res}\,:\;H^{r+1}_{p^{m}}(F)\longrightarrow H^{r+1}_{p^{m}}(F_{nr})\big{)}\,,$$

$$\mathrm{Inf}:H_{p^m}^{r+1}(k)⟶H_{p^m}^{r+1}(F)$$

$$h_{\pi }:H^r(k)⟶H^{r+1}(F);w⟼\mathrm{Inf}(w)\cup (\pi ).$$

$$\mathrm{Inf}\oplus h_{\pi }:H_{p^m}^{r+1}(k)\oplus H_{p^m}^r(k)\stackrel{\sim }{\to }{H}_p^{r+1}(F{)}_{tr}.$$

$$0⟶H_{p^m}^{r+1}(k)\stackrel{\mathrm{Inf}}{⟶}{H}_{p^m}^{r+1}(F{)}_{tr}\stackrel{\partial }{⟶}{H}_{p^m}^r(k)⟶0.$$

$$\partial (\mathrm{Inf}({\chi }_0)\cup (\lambda ))=v_F(\lambda ).{\chi }_0\in H^1(k),$$

$$\partial (\mathrm{Inf}({\chi }_0)\cup (\lambda )\cup (\mu ))={\chi }_0\cup (-1{)}^{v(\lambda )v(\mu )}\overline{{\lambda }^{v(\mu )}{\mu }^{-v(\lambda )}}\in H^2(k),$$

$$0⟶\mathrm{Br}(k)[p^m]\stackrel{\mathrm{Inf}}{⟶}{\mathrm{Br}}_{tr}(F)[p^m]\stackrel{\partial }{⟶}{H}^1(k,\mathbb{Z}/p^m)⟶0,$$

$${\mathrm{Br}}_{tr}(F):=\ker (\mathrm{Br}(F)⟶\mathrm{Br}(F_{nr}))$$

$$\begin{array}{rl}{\mathrm{sd}}_p(k)& :=\inf \{r\in \mathbb{N}|H_p^{r+1}(k^{\prime })=0,\forall \text{finite separable extension }{k}^{\prime }/k\},\\ {\dim }_p(k)& :=\inf \{r\in \mathbb{N}|[k:k^p]\le p^r\text{ and }{H}_p^{r+1}(k^{\prime })=0,\forall \text{finite extension }{k}^{\prime }/k\}.\end{array}$$

$${\log }_p[k:k^p]\le {\dim }_p(k)\le {\log }_p[k:k^p]+1.$$

$${\dim }_p(k)\le 1⟹[k:k^p]\le p⟹{\dim }_p(k)\le 2⟹{\mathrm{sd}}_p(k)\le 2.$$

$$\begin{array}{rl}{v}_L& =\text{the normalized discrete valuation on }L,\\ U_L& =\{x\in L|v_L(x)=0\},\\ \text{and }{U}_L^n& =\{x\in L|v_L(x-1)\ge n\},\forall n\ge 1.\end{array}$$

$$(E/F,b)=\chi \cup (b)\in H^2(F)=\mathrm{Br}(F)[p^{\infty }].$$

$$\alpha ={\alpha }^{\prime }+(E/F,\pi )\in \mathrm{Br}(F)\text{ with }{\alpha }^{\prime }\in \mathrm{Br}(F)\text{ unramified}.$$

$$\lambda =\theta .(-\pi {)}^s\text{with }s=v_F(\lambda )\in \mathbb{Z}\text{ and }\theta \in U_F.$$

$$s{\alpha }^{\prime }=(E/F,\theta )\text{and}s\alpha =(E/F,(-1{)}^s\lambda )$$

$$\lambda =\theta (-\pi {)}^{rp}\text{and}rp{\alpha }^{\prime }=(E/F,\theta )$$

$$N_{L_0/k}({\xi }_0)=\bar{\theta }\text{and}r{\bar{\alpha }}_{L_0}^{\prime }=(E_0{L}_0/L_0,{\xi }_0).$$

$$\begin{array}{rl}{\partial }_L({\alpha }_L\cup (\xi .(-\pi {)}^r))& =r{\bar{\alpha }}_{L_0}^{\prime }+{\partial }_L((E/F,\pi {)}_L\cup (\xi .(-\pi {)}^r))\\ & =r{\bar{\alpha }}_{L_0}^{\prime }-(E_0{L}_0/L_0,\bar{\xi })\\ & =(E_0{L}_0/L_0,{\xi }_0)-(E_0{L}_0/L_0,\bar{\xi }).\end{array}$$

$$\begin{array}{ccccccccc}0& \to & U_L^1& \to & U_L& \to & L_0^{\ast }& \to & 0\\ & & & & & & & & \\ N_{L/F}\downarrow & & \downarrow N_{L/F}& & \downarrow N_{L_0/k}& & & & \\ 0& \to & U_F^1& \to & U_F& \to & k^{\ast }& \to & 0\end{array}$$

Full text

Key words: Reduced norms, Rost invariant, division algebras over valuation fields, Galois cohomology

MSC classification: 11E72, 17A35, 11R52, 16K50

1 Introduction

Let $F$ be a field and let $D$ be a central division algebra of degree $n$ over $F$. First assume $n$ is not divisible by the characteristic of $F$. Using Kummer theory we may consider the Brauer class $(D)$ of $D$ as an element of the Galois cohomomology group $H^{2}(F,\,\mathbb{Z}/n(1)):=H^{2}(F,\,\mu_{n})$; and the isomorphism $F^{*}/F^{*n}\cong H^{1}(F,\,\mu_{n})$ gives rise to a natural map $F^{*}\to H^{1}(F,\,\mu_{n}):\;\lambda\mapsto(\lambda)$. The cup product map

[TABLE]

vanishes on the group $\mathrm{Nrd}(D^{*})$ of (nonzero) reduced norms of $D$, by a norm principle for reduced norms ([GS17, Prop. 2.6.8]) and the projection formula in Galois cohomology. So we have an induced homomorphism

[TABLE]

which we call the Rost invariant of $D$. (This is the Rost invariant of the semisimple simply connected group $\mathbf{SL}_{1}(D)$. See e.g. [Mer03].)

By well known theorems of Merkurjev and Suslin ([MS82, Theorem 12.2], [Sus85, Theorems 24.4 and 24.8]), if $n=\deg(D)$ is square-free or the cohomological $\ell$-dimension of $F$ is $\leq 2$ for all prime divisors $\ell$ of $n$, then the Rost invariant $\mathcal{R}_{D}$ is injective. A recent work of Parimala, Preeti and Suresh [PPS18] proves this Rost injectivity in a remarkable new case: If $F$ is a one-variable function field over a (non-archimedean) local field $K$ and if $n=\deg(D)$ is coprime to the residue characteristic of $K$, then $\mathcal{R}_{D}$ is injective. Their proof relies heavily on patching techniques developed by Harbater, Hartmann and Krashen ([HHK09], [HH10], [HHK15a], [HHK15b], etc.), and it has been an important intermediate step to analyze the situation over the completion of $F$ at a divisorial valuation $v$. The residue field of such a (discrete) valuation is a global field of positive characteristic $p$ or a local field with residue characteristic $p$. For division algebras of prime-to-$p$ degree, the Rost injectivity over the completion $F_{v}$ was proved in [PPS18, Thm. 4.12].

We note that if $F$ is a 2-local field, i.e., a complete discrete valuation field whose residue field $k$ is a (non-archimedean) local field (with finite residue field) in the usual sense, then the Rost invariant $\mathcal{R}_{D}$ is injective for all division algebras $D$ over $F$ (including the case where $\deg(D)$ is divisible by the residue characteristic of $k$). This was a consequence of Kato’s work on two-dimensional local class field theory ([Kat80, p.657, $\S$3.1, Thm. 3 (2)]). So, it is natural to ask whether the same is true when the residue field $k$ is global.

In this paper we prove the following:

Theorem 1.1**.**

Let $F$ be a complete discrete valuation field whose residue field $k$ is a global field of characteristic $p>0$. Let $D$ be a central division algebra of $p$-power degree over $F$.

Then the Rost invariant $\mathcal{R}_{D}$ is injective in each of the following cases:

$D$* is tamely ramified, i.e., $D$ splits over the maximal unramified extension of $F$;* 2. 2.

the period of $D$ is $p$.

After some reviews of preliminary facts in $\S$2, we will prove the theorem in $\S$3 (see (3)) admitting some technical results whose proofs occupy the whole $\S$4.

A few remarks about the theorem are worth mentioning at this moment.

First, in the theorem the field $F$ can have characteristic 0 or $p$. In fact, for any field $K$ of characteristic $p$ and any $p$-power $n$, the groups $H^{r+1}(K,\,\mathbb{Z}/n(r)),\,r\in\mathbb{N}$ can be defined in a suitable way so that they have almost all the properties the Galois cohomology groups $H^{r+1}(K,\,\mu_{n}^{\otimes r})$ have in the case $p\nmid n$ (or at least, all the properties we need in this paper remain valid as in the case $p\nmid n$). In $\S$2 we will recall some most useful facts in this respect and we refer the reader to [Kat80, $\S$3.2] and [GMS03, Appendix A] for more details. In particular, $H^{2}(K,\,\mathbb{Z}/n(1))$ can be identified with the $n$-torsion subgroup $\mathrm{Br}(K)[n]$ of the Brauer group of $K$ and “cup product” maps

[TABLE]

can be defined and have similar properties. So the Rost invariant map $\mathcal{R}_{D}$ in Theorem 1.1 can be defined in the same way as before.

Secondly, for the field $F$ in Theorem 1.1, if $D$ has period $p$ and is not tamely ramified, then by [Kat79, p.337, $\S$3, Lemma 5], the degree of $D$ must be $p$. Therefore, the result in this case follows directly form the Merkurjev–Suslin theorem [MS82, Thm. 12.2] or its $p$-primary counterpart [Gil00, p.94, Thm. 6]. So we need only to consider the first case of the theorem.

Still another remark is that in the theorem we may have only assumed $F$ henselian and excellent instead of the completeness assumption. In fact, letting $\hat{F}$ denote the completion of $F$ and $\hat{D}=D\otimes_{F}\hat{F}$, the natural map

[TABLE]

is injective by Greenberg’s theorem [Gre66]. Therefore, the injectivity of $\mathcal{R}_{\hat{D}}$ implies that of $\mathcal{R}_{D}$.

We expect that the theorem remain true without any restriction on $D$ and that the same result can be shown for function fields of curves over local fields, thereby extending [PPS18, Thm. 1.1] to the case of $p$-power degree algebras.

2 Kato–Milne cohomology and Brauer groups in characteristic $p$

Fix a prime number $p$ and a field $k$ of characteristic $p$.

We need to use the Kato–Milne cohomology groups $H^{r+1}(k,\,\mathbb{Z}/p^{m}(r)),\,r\in\mathbb{N},\,m\in\mathbb{N}^{*}$, which can be viewed as the $p$-primary counterpart of the Galois cohomology groups $H^{r+1}(k,\,\mu_{n}^{\otimes r}),\,r\in\mathbb{N}$ with $n$ coprime to $p$. It does not seem to be absolutely necessary to give the precise definition oof the Kato–Milne cohomology here. The interested readers are referred to [Kat80] and [Mil76], or for another approach, [GMS03, Appendix A]. However, we do need to know some basic facts about the groups $H^{r+1}(k,\,\mathbb{Z}/p^{m}(r))$ which we shall now recall.

**(2.1) ** Let $k$ be a field of characteristic $p$. For integers $r\geq 0$ and $m\geq 1$, the group $H^{r+1}(k,\,\mathbb{Z}/p^{m}(r))$ can be defined as in [Kat80, $\S$3.2] or [GMS03, Appendix A].. The following statements hold:

$H^{1}(k,\,\mathbb{Z}/p^{m}(0))$ coincides with the usual Galois cohomology group $H^{1}(k,\,\mathbb{Z}/p^{m})$, with the Galois action on $\mathbb{Z}/p^{m}$ being trivial. 2. 2.

$H^{2}(k,\,\mathbb{Z}/p^{m}(1))$ can be identified with the $p^{m}$-torsion subgroup $\mathrm{Br}(k)[p^{m}]$ of the Brauer group of $k$. 3. 3.

We write

[TABLE]

For every $m\in\mathbb{N}^{*}$, the natural map $H^{r+1}_{p^{m}}(k)\to H^{r+1}(k)$ is injective and its image is the subgroup of $p^{m}$-torsion elements in $H^{r+1}(k)$. 4. 4.

For every $q\in\mathbb{N}$, let $K_{q}$ be the functor that associates to each field its $q$-th Milnor $K$-group. Then there are natural pairings

[TABLE]

which we call cup products and denote by $(\alpha,\,\lambda)\mapsto\alpha\cup\lambda$, by analogy to the prime-to-$p$ case. (In this paper, for any abelian group $A$ and any integer $n$, we write $A/n=A\otimes_{\mathbb{Z}}(\mathbb{Z}/n\mathbb{Z})$.) 5. 5.

If $L/k$ is a field extension, there are restriction maps

[TABLE]

If $L/k$ is a finite extension, we have corestriction maps

[TABLE]

As in the prime-to-$p$ case, $\mathrm{Cor}_{L/k}\circ\mathrm{Res}_{L/k}$ is the multiplication by $[L:k]$. Moreover, the usual projection formulas hold:

[TABLE]

Here for Milnor $K$-groups we denote by $N_{L/k}:K_{q}(L)\to K_{q}(k)$ the norm homomorphism.

**(2.2) ** Now let $F$ be a henselian excellent discrete valuation field with residue field $k$ (of characteristic $p$). ($F$ may have characteristic [math] or $p$.) For all $r\in\mathbb{N}$ and $m\in\mathbb{N}^{*}$ we define

[TABLE]

where $F_{nr}$ denotes the maximal unramified extension of $F$. There is a natural inflation map (cf. [Kat80, p.659, $\S$3.2, Definition 2])

[TABLE]

and the choice of a uniformizer $\pi\in F$ defines a homomorphism (cf. [Kat80, p.659, $\S$3.2, Lemma 3])

[TABLE]

(By convention, $H^{r}_{p^{m}}=0$ if $r=0$.) The images of $\mathrm{Inf}$ and $h_{\pi}$ are both contained in $H^{r+1}_{p}(F)_{tr}$. It is proved in [Kat82, p.219, Thm. 3] that the above two maps induces an isomorphism

[TABLE]

We can thus define a residue map $\partial:H^{r+1}_{p}(F)_{tr}\to H^{r}_{p}(k)$, which fits into the split exact sequence

[TABLE]

In this paper we are mostly interested in the residue maps defined on $H^{2}$ and $H^{3}$. We have the following formulae:

[TABLE]

and

[TABLE]

for all $\lambda,\,\mu\in F^{*}$ and $\chi_{0}\in H^{1}(k)$.

In terms of Brauer groups, the case $r=1$ of (2.2.3) can be interpreted as the exact sequence

[TABLE]

where

[TABLE]

is called the tame or tamely ramified part of $\mathrm{Br}(F)$.

**(2.3) ** For a field $k$ of characteristic $p$, there are two useful variants of the cohomological $p$-dimension $\mathrm{cd}_{p}(k)$: the separable $p$-dimension $\mathrm{sd}_{p}(k)$ ([Gil00, p.62]) and Kato’s $p$-dimension $\dim_{p}(k)$ ([Kat82, p.220]). They are defined as follows:

[TABLE]

It is easy to see that $\mathrm{sd}_{p}(k)\leq\dim_{p}(k)$. Moreover, it can be shown that

[TABLE]

Therefore, we have the following implications:

[TABLE]

Notice also that the condition $\mathrm{sd}_{p}(k)\leq 1$ is equivalent to saying that $\mathrm{Br}(L)$ has no $p$-torsion for all algebraic extensions $L/k$ (cf. [Ser94, Chap. II, $\S$3.1]).

If $k$ is a global function field of characteristic $p$, then $[k:k^{p}]=p$ and $\dim_{p}(k)=\mathrm{sd}_{p}(k)=2$.

For fields of characteristic different from $p$, both the separable $p$-dimension and Kato’s $p$-dimension are defined to be the same as the cohomological $p$-dimension.

3 Proof of main theorem

We now proceed to the proof of Theorem 1.1. We basically follow the same strategy as Parimala, Preeti and Suresh’s paper [PPS18]. Our main contribution is the extension of some key lemmas in [PPS18] to the characteristic $p$ case. At some points the proofs of our generalized versions involve some subtleties and have to be treated with special care.

**(3.1) ** Let us fix the following notation for the whole section:

For each $r\in\mathbb{N}$ we write $H^{r+1}(\cdot)=\varinjlim_{m}H^{r+1}(\cdot\,,\,\mathbb{Z}/p^{m}(r))$ as in $\S$2.

Here $H^{r+1}(\cdot,\,\mathbb{Z}/p^{m}(r))$ is the Kato–Milne cohomology in characteristic $p$ and is the Galois cohomology $H^{r+1}(\cdot,\,\mu_{p^{m}}^{\otimes r})$ in characteristic different from $p$. 2. 2.

Let $F$ be a complete discrete valuation field with residue field $k$.

We assume that $\mathrm{char}(k)=p>0$. 3. 3.

Let $v=v_{F}$ denote the normalized discrete valuation on $F$ and fix a uniformizer $\pi\in F$. 4. 4.

If $L/F$ is a finite extension, we put

[TABLE] 5. 5.

Let $0\neq\alpha\in\mathrm{Br}(F)[p^{\infty}]$ be a nonzero Brauer class of $p$-power index over $F$.

We assume that $\alpha$ is tamely ramified, i.e., $\alpha\in\mathrm{Br}_{tr}(F)=\ker(\mathrm{Br}(F)\to\mathrm{Br}(F_{nr}))$. 6. 6.

Let $\chi_{0}=\partial(\alpha)\in H^{1}(k)$ be the residue of $\alpha$, which is well defined according to the tameness assumption on $\alpha$ (2.2.3).

The character $\chi_{0}\in H^{1}(k)$ can be determined by a pair $(E_{0}/k,\,\overline{\sigma})$, where $E_{0}/k$ is the cyclic extension and $\overline{\sigma}$ is a generator of the cyclic Galois group $\mathrm{Gal}(E_{0}/k)$. The correspondence between $\chi_{0}$ and $(E_{0}/k,\,\overline{\sigma})$ is established by requiring that the continuous homomorphism $\chi_{0}:\mathrm{Gal}(k_{s}/k)\to\mathbb{Q}_{p}/\mathbb{Z}_{p}$ has kernel $\mathrm{Gal}(k_{s}/E_{0})$, $k_{s}$ denoting a fixed separable closure of $k$, and that $\overline{\sigma}\in\mathrm{Gal}(E_{0}/k)$ is the generator which is mapped to the canonical generator of the cyclic group $\mathrm{Im}(\chi_{0})$. Since the role played by $\overline{\sigma}$ is almost never explicit in our arguments, we will simply write $\chi_{0}=(E_{0}/k)$.

By the canonical lifting $\chi\in H^{1}(F)$ of $\chi_{0}$ we shall mean the image of $\chi_{0}$ under the inflation map $H^{1}(k)\to H^{1}(F)$. Explicitly, $\chi$ is defined by the pair $(E/F,\,\sigma)$ where $E/F$ is the unramified extension with residue field extension $E_{0}/k$ and $\sigma\in\mathrm{Gal}(E/F)$ is the generator corresponding to $\overline{\sigma}$ via the natural isomorphism $\mathrm{Gal}(E/F)\xrightarrow{\sim}\mathrm{Gal}(E_{0}/k)$. Just as for $\chi_{0}$, we will write $\chi=(E/F)$ for short. For any element $b\in F^{*}$, we write

[TABLE] 7. 7.

As in [PPS18, Lemma 4.1], we may write

[TABLE]

Here we call a Brauer class $\beta\in\mathrm{Br}(F)[p^{\infty}]$ unramified if $\beta\in\mathrm{Br}_{tr}(F)$ and if $\partial(\beta)=0$. Equivalently, an unramified element of $\mathrm{Br}(F)[p^{\infty}]$ is an element in the image of the inflation map $\mathrm{Br}(k)[p^{\infty}]\to\mathrm{Br}_{tr}(F)[p^{\infty}]$ (2.2.6).

We denote by $\bar{\alpha}^{\prime}\in\mathrm{Br}(k)[p^{\infty}]$ the element that is mapped to $\alpha^{\prime}\in\mathrm{Br}(F)$ under the inflation map. 8. 8.

Let $\lambda\in F^{*}$ be such that $\alpha\cup(\lambda)=0\in H^{3}(F)$. We write

[TABLE]

Computing the residue of $\alpha\cup(\lambda)$ we see that

[TABLE]

(cf. [PPS18, Lemma 4.7]).

To prove our main theorem, we need to show $\lambda$ is a reduced norm for $\alpha$ when $k$ is a global function field.

**(3.2) ** With notation as above, by an inductive pair for $(\alpha,\,\lambda)$ we mean a pair $(L,\,\mu)$ consisting of a separable field extension $L/F$ of degree $p$ and an element $\mu\in L^{*}$ such that $N_{L/F}(\mu)=\lambda$, the index $\mathrm{ind}(\alpha_{L})$ of $\alpha_{L}$ is strictly smaller than $\mathrm{ind}(\alpha)$, and $\alpha_{L}\cup(\mu)=0\in H^{3}(L)$.

By the norm principle for reduced norms, we may use induction on the index of $\alpha$ to conclude that $\lambda$ is a reduced norm of $\alpha$, as soon as an inductive pair exists.

Lemma 3.3**.**

With notation and hypotheses as above, suppose that $s=v_{F}(\lambda)$ is coprime to $p$. Let $L=F(\sqrt[p]{-\lambda})$ and $\mu=-\sqrt[p]{-\lambda}$.

Then $\mathrm{ind}(\alpha_{L})<\mathrm{ind}(\alpha)$ and $\partial(\alpha_{L}\cup(\mu))=0$. Therefore, if $\mathrm{sd}_{p}(k)\leq 2$ (e.g., $k$ is a global function field), then $(L,\,\mu)$ is an inductive pair for $(\alpha,\,\lambda)$.

Proof.

The proof can be done in the same manner as in the prime-to-$p$ case, for which the reader is referred to [PPS18, Lemma 4.7]. ∎

Thanks to the above lemma, we may assume $p\,|\,s$. In the rest of this section, we write $s=rp$, so that

[TABLE]

according to (3.1.2) and (3.1.3).

Lemma 3.4**.**

With notation and hypotheses as above (in particular $s=v_{F}(\lambda)=rp$), suppose that $L_{0}/k$ is a separable field extension of degree $p$ and that $\xi_{0}\in L_{0}^{*}$ satisfies

[TABLE]

Let $L/F$ be the unramified extension with residue field $L_{0}/k$.

Then there exists an element $\xi\in U_{L}$ such that

$\bar{\xi}=\xi_{0}$* in $L_{0}$;* 2. 2.

$N_{L/F}(\xi)=\theta$; 3. 3.

$\partial(\alpha_{L}\cup(\xi.(-\pi)^{r}))=0$. (Thus, if $\mathrm{sd}_{p}(k)\leq 2$, we have $\alpha_{L}\cup(\xi.(-\pi)^{r})=0$.)

Proof.

A direct calculation (using (2.2.5)) shows

[TABLE]

So condition (3) is implied by condition (1). We need only to find a unit $\xi\in L$ satisfying (1) and (2).

Since $L/F$ is an unramified extension, we have a commutative diagram with exact rows

[TABLE]

and $N_{L/F}(U^{1}_{L})=U^{1}_{F}$ by [Ser79, $\S$V.2, Prop. 3]. In particular, the natural map $U_{F}\to k^{*};\,x\mapsto\bar{x}$ induces an isomorphism

[TABLE]

and the induced map

[TABLE]

is surjective. By assumption we have $\bar{\theta}\in N_{L_{0}/k}(L_{0}^{*})$. So the injectivity of $w$ implies that $\theta=N_{L/F}(\xi_{1})$ for some $\xi_{1}\in U_{L}$. Using the diagram (3.4.1) we find that $\xi.\bar{\xi}_{1}^{-1}\in L_{0}^{*}$ lies in $\ker(N_{L_{0}/k}:L_{0}^{*}\longrightarrow k^{*})$. Now the surjectivity of $w^{\prime}$ yields an element $z\in\ker(N_{L/F}:U_{L}\longrightarrow U_{F})$ such that $\bar{z}=\xi_{0}.\bar{\xi}_{1}^{-1}$. Taking $\xi=z.\xi_{1}$, we get

[TABLE]

and $\bar{\xi}=\bar{z}\cdot\bar{\xi}_{1}=\xi_{0}$ in $L_{0}$. This completes the proof. ∎

The proof of the lemma below is long. It will be postponed to the next section (see (4)).

Lemma 3.5**.**

With notation and hypotheses as in $\eqref{3p1}$, suppose that $s=v_{F}(\lambda)=rp$ and that $k$ is a global function field.

Then there exists a separable field extension $L_{0}/k$ of degree $p$ and an element $\xi_{0}\in L_{0}$ such that the following hold:

$N_{L_{0}/k}(\xi_{0})=\bar{\theta}\in k$, and $r\bar{\alpha}^{\prime}_{L_{0}}=(E_{0}L_{0}/L_{0},\,\xi_{0})\in\mathrm{Br}(L_{0})$. 2. 2.

If $\bar{\alpha}^{\prime}_{E_{0}}=0$, then $L_{0}\subseteq E_{0}$ (up to $k$-isomorphism). 3. 3.

If $\bar{\alpha}^{\prime}_{E_{0}}\neq 0$ and $L_{0}\not\subseteq E_{0}$, then $\mathrm{ind}(\bar{\alpha}^{\prime}_{E_{0}L_{0}})<\mathrm{ind}(\bar{\alpha}^{\prime}_{E_{0}})$.

**(3.6) ** Proof of Theorem 1.1. Let us prove our main theorem assuming Lemma 3.5. Let $\alpha$ be the Brauer class of the division algebra $D$ in Theorem 1.1. As we have said in the introduction, it suffices to consider the case where $\alpha$ is tamely ramified, and of course we may assume $\alpha\neq 0$. The previous discussions in this section reduce the problem to the construction of an inductive pair in the case $s=v_{F}(\lambda)=rp$.

Let $L_{0}/k$ and $\xi_{0}\in L_{0}$ be chosen as in Lemma 3.5. Let $L/F$ be the unramified extension with residue field $L_{0}/k$. By Lemma 3.4, there exists $\xi\in U_{L}$ such that $\bar{\xi}=\xi_{0}\in L_{0}$, $N_{L/F}(\xi)=\theta$ and the element $\mu:=\xi.(-\pi)^{r}$ satisfies

[TABLE]

It remains to show $\mathrm{ind}(\alpha_{L})<\mathrm{ind}(\alpha)$.

We distinguish three cases.

Case 1. $\bar{\alpha}^{\prime}_{E_{0}}=0$.

In this case we have $L_{0}\subseteq E_{)}$ by condition (2) in Lemma 3.5. Thus, $L\subseteq E$. Moreover, $\bar{\alpha}^{\prime}_{E_{0}}$ implies $\alpha^{\prime}_{E}=0$. Hence $\alpha^{\prime}=(E/F,\,u)$ for some $u\in E^{*}$ and since $\alpha^{\prime}$ is unramified, we have $u\in U_{E}$. Now $\alpha=\alpha^{\prime}+(E/F,\,\pi)=(E/F,\,u\pi)$. Therefore,

[TABLE]

It follows that

[TABLE]

Case 2. $\bar{\alpha}^{\prime}_{E_{0}}\neq 0$ and $L_{0}\not\subseteq E_{0}$.

In this case we have $E\cap L=F$ and $[EL:L]=[E:F]$, because $[L:F]$ is a prime and $L\not\subseteq E$. Applying the index formula in [JW90, Thm. 5.15 (a)] to the decompositions of $\alpha$ and of $\alpha_{L}=\alpha^{\prime}_{L}+(EL/L,\,\pi)$, we obtain

[TABLE]

the inequality here following from condition (3) of (3.5).

Case 3. $\bar{\alpha}^{\prime}_{E_{0}}\neq 0$ and $L_{0}\subseteq E_{0}$.

Now we have $L\subseteq E$ and $\alpha_{L}=\alpha^{\prime}_{L}+(E/L,\,\pi)$. Using the index formula [JW90, Thm. 5.15 (a)] once again, we can deduce

[TABLE]

So, the inequality $\mathrm{ind}(\alpha_{L})<\mathrm{ind}(\alpha)$ holds in all cases. This shows that $(L,\,\mu)$ is an inductive pair, and the theorem is thus proved.

4 Proofs of technical lemmas

Our goal in this section is to prove Lemma 3.5.

We begin with the following observation.

Lemma 4.1**.**

Let $K$ be a field of characteristic $p>0$ and $\gamma\in K^{*}$. Suppose that $p\nmid v(\gamma)$ for some normalized discrete valuation $v$ on $K$.

Then there exists a cyclic extension $L/K$ of degree $p$ such that $\gamma\in N_{L/K}(L^{*})$.

Proof.

Choose $c,\,d\in\mathbb{Z}$ such that $pc+dv(\gamma)=1$ and put $\rho=\gamma^{d}\varpi^{pc}$, where $\varpi\in K$ is a uniformizer for $v$. Then $\rho$ is another uniformizer for $v$. If $L/K$ is a cyclic degree $p$ extension such that $\rho\in N_{L/K}(L^{*})$, then $\rho^{d}=\rho\varpi^{-pc}\in N_{L/K}(L^{*})$ and this implies $\gamma\in N_{L/K}(L^{*})$, since the group $K^{*}/N_{L/K}(L^{*})$ is $p$-torsion and $p\nmid d$. Therefore, by replacing $\gamma$ with $\rho$ if necessary, we may assume $v(\gamma)=1$.

We shall construct $L/K$ as an Artin–Schrier extension $L=K[X]/(X^{p}-X-\alpha)$, where $\alpha\in K$ and $\alpha\notin\wp(K):=\{a^{p}-a\,|\,a\in K\}$. We denote by $[\alpha,\,\lambda)$ the Brauer class of the cyclic $p$-algebra generated by two symbols $x,\,y$ over $K$ subject to the relations

[TABLE]

Then we have

[TABLE]

(for the second equivalence see e.g. [KK86, p.234, Prop. 2 (2)] or [AB10]). We choose

[TABLE]

Then the last condition in (4.1.1) holds for

[TABLE]

Moreover, we have $v(\alpha)=1-p<0$ and $p\nmid v(\alpha)$. This implies that $X^{p}-X-\alpha=0$ has no solution in $K$. Hence $\alpha\notin\wp(K)$ and $L:=K[X]/(X^{p}-X-\alpha)$ is a cyclic extension with the required property. ∎

Now we prove a characteristic $p$ version of [PPS18, Lemma 3.1].

Lemma 4.2**.**

Suppose given the following data:

• $(\mathrm{i})$

A global field $k$ of characteristic $p>0$ and a cyclic extension $K/k$ of $p$-power degree.

• $(\mathrm{ii})$

A Brauer class $\beta\in\mathrm{Br}(k)[p^{\infty}]$, an integer $r\in\mathbb{Z}$ and an element $\theta\in k^{*}$ such that

[TABLE]

• $(\mathrm{iii})$

An integer $d\geq 2$.

• $(\mathrm{iv})$

A finite set $S$ of places of $k$ including all the places $v$ such that $\beta\otimes k_{v}\neq 0\in\mathrm{Br}(k_{v})$.

• $(\mathrm{v})$

For each $v\in S$, a (connected or split) $(\mathbb{Z}/p\mathbb{Z})$-Galois cover $L_{v}$ of $k_{v}$ and an element $\xi_{v}\in L_{v}^{*}$ such that

1. (a)

$N_{L_{v}/k_{v}}(\xi_{v})=\theta$; 2. (b)

$r\beta\otimes L_{v}=(KL_{v}/L_{v},\,\xi_{v})\in\mathrm{Br}(L_{v})$; 3. (c)

$\mathrm{ind}(\beta\otimes KL_{v})<d$.

Here, by a $(\mathbb{Z}/p\mathbb{Z})$-Galois cover $L_{v}/k_{v}$ we mean that $L_{v}$ is either a cyclic field extension of degree $p$ over $k_{v}$ or the direct product of $p$ copies of $k_{v}$. In the latter case, we identity $\mathrm{Br}(L_{v})=\prod_{1\leq i\leq p}\mathrm{Br}(k_{v})$ and let $(KL_{v}/L_{v},\,\xi_{v})$ denote $(Kk_{v}/k_{v},\,\xi_{v,\,i})\in\prod_{1\leq i\leq p}\mathrm{Br}(k_{v})$, where $(\xi_{v,\,1},\dotsc,\xi_{v,\,p})=\xi_{v}\in L_{v}=\prod_{1\leq i\leq p}k_{v}$.

Then, there exists a separable field extension $L/k$ of degree $p$ and an element $\xi\in L^{*}$ such that

$N_{L/k}(\xi)=\theta$; 2. 2.

$r\beta\otimes L=(KL/L\,,\,\xi)\in\mathrm{Br}(L)$; 3. 3.

$\mathrm{ind}(\alpha_{L})<d$; 4. 4.

$L\otimes_{k}k_{v}\cong L_{v}$* for all $v\in S$;* 5. 5.

$\xi$* is sufficiently close to $\xi_{v}$ in $L_{v}$ in the $v$-adic topology, for all $v\in S$.*

Proof.

First of all, we may further assume $S$ contains the following sets of places:

• •

$S_{0}:=\{v\,|\,v(\theta)\neq 0\}$;

• •

$S_{1}:=\{v\,|\,K/k\text{ is ramified at }v\}$;

• •

at least one place $v$ such that $L_{v}/k_{v}$ is a (cyclic) field extension.

In fact, for any $v\notin S$, we have $\beta\otimes k_{v}=0$ by assumption (iv), whence $\mathrm{ind}(\beta\otimes KL_{v})=\mathrm{ind}(\beta\otimes Kk_{v})=1<2\leq d$. On the other hand, Lemma 4.1 tells us that there is a cyclic field extension $L_{v}/k_{v}$ of degree $p$ such that $\theta\in N_{L_{v}/k_{v}}(L_{v}^{*})$. We choose any $\xi_{v}\in L_{v}^{*}$ such that $\theta=N_{L_{v}/k_{v}}(\xi_{v})$. Then

[TABLE]

by assumption (ii). Since the corestriction map $\mathrm{Cor}:\mathrm{Br}(L_{v})\to\mathrm{Br}(k_{v})$ is injective (in fact an isomorphism) by local class field theory, we have $(KL_{v}/L_{v},\,\xi_{v})=0=r\beta\otimes L_{v}$. This means that condition (v) can still be satisfied when we add a new place $v$ for which $L_{v}$ is a field.

Now consider an arbitrary $v\in S$. By [PPS18, Lemma 2.2], there exists $\rho_{v}\in L_{v}$ sufficiently close to $1$ such that $N_{L_{v}/k_{v}}(\rho_{v})=1$ and $L_{v}=k_{v}(\xi_{v}\rho_{v})$. We may assume $\rho_{v}$ is a norm from $KL_{v}/L_{v}$, by the fact that for any finite abelian extension $M/H$ of local fields, the norm subgroup $N_{M/H}(M^{*})$ is open in $H^{*}$, even in the characteristic $p$ case (see e.g. [Ser79, p.219, $\S$XIV.6, Cor. 1]). Thus, it follows that $r\beta\otimes L_{v}=(KL_{v}/L_{v},\,\xi_{v})=(KL_{v}/L_{v},\,\rho_{v}\xi_{v})$. Replacing $\xi_{v}$ with $\xi_{v}\rho_{v}$ if necessary, we may assume $L_{v}=k_{v}(\xi_{v})$.

As in the proof of [PPS18, Lemma 3.1], by using Chebotarev’s density theorem and the strong approximation theorem, we can choose a place $v_{0}$ outsider $S$ such that $K/k$ splits completely at $v_{0}$ and we can find a monic polynomial

[TABLE]

with constant coefficient $f(0)=(-1)^{p}\theta$, which is close to the minimal polynomial of $\xi_{v}$ over $k_{v}$ for each $v\in S$. Then $L:=k[X]/(f)$ is a field extension of $k$, since we assumed that $L\otimes_{k}k_{v}=L_{v}$ is a field for at least one $v\in S$. The canonical image $\xi$ of $X$ in $L$ is an element with $N_{L/k}(\xi)=\theta$.

The work of checking that the pair $(L,\,\xi)$ has all the required properties can be done in the same way as in [PPS18, p.416]. The following fact will be used once again in this verification: For any finite abelian extension of local fields, any element close to 1 is a norm. ∎

**(4.3) ** Proof of Lemma 3.5. Now we can prove Lemma 3.5, which is essential in our proof of Theorem 1.1. Only three cases require consideration.

Case 1. $r\bar{\alpha}^{\prime}_{E_{0}}=0$ and $E_{0}\neq k$.

Case 2. $r\bar{\alpha}^{\prime}_{E_{0}}=0$ and $E_{0}=k$.

Case 3. $r\bar{\alpha}^{\prime}_{E_{0}}\neq 0$.

In Case 1, we choose $L_{0}/k$ to be the unique degree $p$ subextension of the cyclic extension $E_{0}/k$. So there is no need to check (3.5) (3) and condition (2) of the lemma holds by the choice of $L_{0}$. It remains to show that one can find an element $\xi_{0}\in L_{0}$ satisfying

[TABLE]

Indeed, from the assumption $r\bar{\alpha}^{\prime}_{E_{0}}=0$ we know that

[TABLE]

Hence $(E_{0}/k,\,x^{p})=rp\bar{\alpha}^{\prime}=(E_{0}/k,\,\theta)$, which implies $\bar{\theta}x^{-p}=N_{E_{0}/k}(y)$ for some $y\in E_{0}$. Let $\xi_{0}=N_{E_{0}/L_{0}}(y)x$. Then

[TABLE]

and

[TABLE]

This completes the proof in Case 1.

Now consider Case 2, where $E_{0}=k$ and $r\bar{\alpha}^{\prime}=r\alpha^{\prime}_{E_{0}}=0$.

In this case we have $\alpha^{\prime}=\alpha\neq 0$ and $\bar{\alpha}^{\prime}_{E_{0}}=\bar{\alpha}^{\prime}\neq 0$. So (3.5) (2) is needless to check. To prove the desired result, we shall apply Lemma 4.2 with

[TABLE]

and we will take the pair $(L,\,\xi)$ obtained in (4.2) to be the pair $(L_{0},\,\xi_{0})$ asserted in (3.5).

Note that conditions (1) in (3.5) is guaranteed by assertions (1) and (2) in Lemma 4.2, and (3.5) (3) is part of (4.2) (3). The only remaining job is to take $S$ to be the finite set of places $v$ such that $\bar{\alpha}^{\prime}\otimes k_{v}\neq 0$ and to provide for each $v\in S$ a pair $(L_{v},\,\xi_{v})$ satisfying assumption (v) in (4.2).

If $p\nmid v(\bar{\theta})$, we can find a cyclic extension $L_{v}/k_{v}$ of degree $p$ and $\xi_{v}\in L_{v}^{*}$ such that $\bar{\theta}=N_{L_{v}/k_{v}}(\xi_{v})$, whence condition (a) in (4.2) (v). Our assumption in the present case says $E_{0}=k$ and $r\beta=r\bar{\alpha}^{\prime}_{E_{0}}=0$. So condition (b) in (4.2) (v) holds as well. For condition (c), it suffices to observe that

[TABLE]

since $k_{v}$ is a local field.

Next assume $p\,|\,v(\bar{\theta})$. We now choose $L_{v}/k_{v}$ to be the unique unramified extension of degree $p$. Then $\bar{\theta}$ is a norm from $L_{v}/k_{v}$. Conditions (b) and (c) can be checked as in the previous paragraph.

Case 2 is thus solved.

In the third case, $r\bar{\alpha}^{\prime}_{E_{0}}\neq 0$. As in Case 2, we will use (4.2) by taking $\beta=\bar{\alpha}^{\prime}$, $K=E_{0}$ and $d=\mathrm{ind}(\bar{\alpha}^{\prime}_{E_{0}})$. Again, it is sufficient to construct, for each place $v$ such that $\bar{\alpha}^{\prime}\otimes k_{v}\neq 0$, a pair $(L_{v},\,\xi_{v})$ satisfying (4.2) (v).

Indeed, as in Case 2 we can find a cyclic extension $L_{v}/k_{v}$ and an element $\xi_{v}\in L_{v}$ satisfying condition (a) in (4.2) (v). Condition (b) can be deduced from (a) by using the corestriction map $\mathrm{Cor}:\mathrm{Br}(L_{v})\to\mathrm{Br}(k_{v})$ and its injectivity. So it remains to check condition (c) of (4.2) (v).

Note that $E_{0}k_{v}\cap L_{v}$ is either $k_{v}$ or $L_{v}$.

If $E_{0}k_{v}\cap L_{v}=k_{v}$, we have $[E_{0}L_{v}:E_{0}k_{v}]=p=[L_{v}:k_{v}]$ and

[TABLE]

since $E_{0}k_{v}$ is a local field.

If $E_{0}k_{v}\cap L_{v}=L_{v}$, then $E_{0}k_{v}=E_{0}L_{v}$. Writing $rp=mp^{t}$ with $m$ coprime to $p$ and $t\geq 1$, we have

[TABLE]

Since

[TABLE]

we have

[TABLE]

Therefore, from (4.3.1) we deduce that

[TABLE]

To finish the proof, it remains to check that $d=\mathrm{ind}(\bar{\alpha}^{\prime}_{E_{0}})=p^{t}$. Indeed, from the fact that $mp^{t}\bar{\alpha}^{\prime}=rp\bar{\alpha}^{\prime}=(E_{0}/k,\,\bar{\theta})$ we see that $d=\mathrm{ind}(\bar{\alpha}^{\prime}_{E_{0}})=\mathrm{per}(\bar{\alpha}^{\prime}_{E_{0}})$ divides $p^{t}$ (noticing that $E_{0}$ is a global field). On the other hand, our hypothesis in the present case (Case 3) is that $mp^{t-1}\bar{\alpha}^{\prime}_{E_{0}}=r\bar{\alpha}^{\prime}_{E_{0}}\neq 0$. Hence $d=\mathrm{per}(\bar{\alpha}^{\prime}_{E_{0}})>p^{t-1}$. So we have $d=p^{t}$ as desired.

This completes the proof of Lemma 3.5.

Acknowledgements. This work is supported by a grant from the National Natural Science Foundation of China (Project No. 11801260). The author thanks Zhengyao Wu for helpful discussions.