Third Galois cohomology group of function fields of curves over number fields

TL;DR

This paper investigates the structure of the third Galois cohomology group of function fields of curves over totally imaginary number fields, establishing that all elements are symbols under certain conditions.

Contribution

It proves that when the function field contains a primitive p-th root of unity, all elements in the third Galois cohomology group are symbols, advancing understanding of cohomological properties of such fields.

Findings

All elements in the third Galois cohomology group are symbols under the given conditions.

The result applies specifically to function fields of curves over totally imaginary number fields containing primitive p-th roots of unity.

Abstract

Let F be the function field of a curve over totally imaginary number field. Let p be a prime. If F contains a primitive p th root of unity, then every element in the third Galois cohomology group of F with values in the group of p th roots of unity, is a symbol.

Equations

$$\begin{array}[]{rcl}(1+d)^{\ell}=1+\ell dy+d^{\ell}&=&1+\ell w(\rho-1)y+w^{\ell}(\rho-1)^{\ell}\\ &=&1+\ell w(\rho-1)y+w(\rho-1)^{\ell}-u(\rho-1)^{\ell}\\ &=&1+xyw(\rho-1)^{\ell}+w(\rho-1)^{\ell}-u(\rho-1)^{\ell}\\ &=&1+w(\rho-1)^{\ell}(xy+1)-u(\rho-1)^{\ell}.\end{array}$$

$$A\to \underset{i}{⨁}A/I_i\to \underset{i<j}{⨁}A/I_{ij}$$

$$H^{3}(F,\mathbb{Z}/\ell(2))\buildrel\partial\over{\to}\oplus_{x\in\mathscr{X}_{(1)}}H^{2}(\kappa(x),\mathbb{Z}/\ell(1))\buildrel\partial\over{\to}\oplus_{P\in\mathscr{X}_{(0)}}H^{1}(\kappa(P),\mathbb{Z}/\ell).$$

$${\partial }_P({\partial }_{C_1}(\zeta ))={\partial }_P({\partial }_{C_2}(\zeta ){)}^{-1}.$$

$$a_s\equiv u_1^{-1}(\prod _{t\ne i,j}{\pi }_t){\pi }_i{\pi }_j=u_1^{-1}(\prod _{t\ne i,j}{\pi }_t){\pi }_i(-\frac{v_i}{v_j}{\pi }_i^{\ell -1})=u_1^{-1}(\prod _{t\ne i,j}{\pi }_t)(-\frac{v_i}{v_j}){\pi }_i^{\ell }\text{modulo}{\delta }_s.$$

$$0\to H^{3}(F,\mathbb{Z}/\ell(2))\buildrel\partial\over{\to}\oplus_{x\in\mathscr{X}_{(1)}}H^{2}(\kappa(x),\mathbb{Z}/\ell(1))\buildrel\partial\over{\to}\oplus_{P\in\mathscr{X}_{(2)}}H^{1}(\kappa(P),\mathbb{Z}/\ell).$$

$$\begin{array}[]{rcl}<\!-a,-b,ab,c,d\!>\otimes<\!1,-\lambda\!>&=&-cd<\!1,-\lambda\!>\otimes<\!1,\mu\!>\otimes<\!1,\mu^{\prime}\!>+<\!1,-\lambda\!>\\ &=&-cd<\!1-\lambda\!>\otimes<\!\mu,\mu^{\prime},\mu\mu^{\prime}\!>.\end{array}$$

Full text

1. Introduction

Let $F$ be a field and $\ell$ a prime not equal to the characteristic of $F$. For $n\geq 1$, let $H^{n}(F,\mu_{\ell}^{\otimes n})$ be the $n^{\rm th}$ Galois cohomology group with coefficients in $\mu_{\ell}^{\otimes n}$. We have $F^{*}/F^{*\ell}\simeq H^{1}(F,\mu_{\ell})$. For $a\in F^{*}$, let $(a)\in H^{1}(F,\mu_{\ell})$ denote the image of the class of $a$ in $F^{*}/F^{*\ell}$. Let $a_{1},\cdots,a_{n}\in F^{*}$. The cup product $(a_{1})\cdots(a_{n})\in H^{n}(F,\mu_{\ell}^{\otimes n})$ is called a symbol. A theorem of Voevodsky ([30]) asserts that every element in $H^{n}(F,\mu_{\ell}^{\otimes n})$ is a sum of symbols. Let $\alpha\in H^{n}(F,\mu_{\ell}^{\otimes n})$. The symbol length of $\alpha$ is defined as the smallest $m$ such that $\alpha$ is a sum of $m$ symbols in $H^{n}(F,\mu_{\ell}^{\otimes n})$.

A consequence of class field theory is that if $K$ is a global field or a local field, then every element in $H^{n}(K,\mu_{\ell}^{\otimes n})$ is a symbol.

Let $K$ be a $p$-adic field and $F$ the function of a curve over $K$. Suppose that $K$ contains a primitive $\ell^{\rm th}$ root of unity. If $\ell\neq p$, then it was proved in ([29], cf. [3]) that the symbol length of every element in $H^{2}(F,\mu^{\otimes 2}_{\ell})$ is at most 2. If $p\neq\ell$, then it was proved in ([18], cf. [20]) that every element in $H^{3}(F,\mu_{\ell}^{\otimes 3})$ is a symbol. If $\ell=p$, then it was proved in ([19]) that for every central simple algebra $A$ over $F$, the index of $A$ divides the square of the period of $A$. In particular if $p=2$, then the symbol length of every element in $H^{2}(F,\mu^{\otimes 2}_{2})$ is at most 2. Since $u(F)=8$ ([9], [12], cf. [19]), it follows that every element in $H^{3}(F,\mu_{2}^{\otimes 3})$ is a symbol.

If $F$ is the function field of a curve over a global field of positive characteristic $p$, $\ell\neq p$ and $F$ contains a primitive $\ell^{\rm th}$ root of unity, then it was proved in ([20]) that every element in $H^{3}(F,\mu_{\ell}^{\otimes 3})$ is a symbol.

Let $K$ be a number field and $F$ the function field of a curve over $K$. In ([28]), it was proved that given finitely any elements $\alpha_{1},\cdots,\alpha_{r}\in H^{3}(F,\mu_{2}^{\otimes 3})$, there exists $f\in F^{*}$ such that $\alpha_{i}=(f)\cdot\beta_{i}$ for some $\beta_{i}\in H^{2}(F,\mu_{2}^{\otimes 2})$. In particular if there exists an integer $N$ such that the symbol length of every element in $H^{2}(F,\mu_{2}^{\otimes 2})$ is bounded by $N$, then the symbol length of every element in $H^{3}(F,\mu_{2}^{\otimes 3})$ is bounded by $N$. In ([13]), it was proved that such an integer $N$ exists under the hypothesis that a conjecture of Colliot-Thélène on the existence of 0-cycles of degree 1 holds. However it is still open whether such $N$ exists.

In this paper we prove the following (see 7.8)

Theorem 1.1**.**

Let $K$ be a global field or a local field and $F$ the function field of a curve over $K$. Let $\ell$ be a prime not equal to char$(K)$. Suppose that $K$ contains a primitive $\ell^{\rm th}$ root of unity. If $\ell\neq 2$ or $K$ is a local field or $K$ is a global field with no real orderings, then every element in $H^{3}(F,\mu_{\ell}^{\otimes 3})$ is a symbol.

The above theorem for $K$ a $p$-adic field and $\ell\neq p$ is proved in ([18], cf. [20]). Our method in this paper is uniform, it covers both global and local fields at the same time, without the assumption $\ell\neq p$.

We have the following (see 8.3)

Corollary 1.2**.**

Let $K$ be a totally imaginary number field and $F$ the function field of a curve over $K$. Let $q$ be a quadratic form over $F$ and $\lambda\in F^{*}$. If the dimension of $q$ is at least 5, then $q\otimes<\!1,-\lambda\!>$ is isotropic.

For a smooth integral variety $X$ over a field $k$, let $CH_{0}(X)$ be the Chow group of 0-cycles modulo rational equivalence. If $k$ is a number field and $X$ a smooth projective geometrically integral curve, then Mordell-Weil theorem asserts that $CH_{0}(X)$ is finitely generated.

Let $C$ be a smooth projective geometrically integral curve over a field $k$. Let $X\to C$ be a (admissible) quadric fibration (cf. [5]). Let $CH_{0}(X/C)$ be the kernel of the natural homomorphism $CH_{0}(X)\to CH_{0}(C)$. If char$(k)\neq 2$, Colliot-Thélène and Skorobogatov identified $CH_{0}(X/C)$ with a certain sub-quotient of $k(C)^{*}$ ([5]). From this identification it follows that $CH_{0}(X/C)$ is a 2-torsion group. Thus $CH_{0}(X/C)$ is finitely generated if and only if it is finite. Suppose that $k$ is a number field. If dim$(X)\leq 2$, then the finiteness of $CH_{0}(X/C)$ is a result of Gros ([8]). If dim$(X)=3$, then it was proved in ([5], [16]) that $CH_{0}(X/C)$ is finite. Thus for dim$(X)\leq 3$, $CH_{0}(X)$ if finitely generated. Colliot-Thélène and Skorobogatov conjectured that if dim$(X)\geq 4$ and $k$ is a totally imaginary number field, then $CH_{0}(X/C)=0$. As a consequence of our main theorem, we prove the following (see 8.4).

Corollary 1.3**.**

Let $K$ be a totally imaginary number field, $C$ a smooth projective geometrically integral curve over $K$. Let $X\to C$ be an admissible quadric fibration. If dim$(X)\geq 4$, then $CH_{0}(X/C)=0$. In particular $CH_{0}(X)$ is finitely generated.

Let $K$ be a local field and $p$ the characteristic of the residue field of $K$ or a global field of positive characteristic $p$. Let $F$ be the function field of a curve over $k$ and $\ell$ a prime not equal to $p$. Let us recall that the main ingredient in the proof of the fact that every element in $H^{3}(F,\mu_{\ell}^{\otimes 3})$ is a symbol ([18]) is a certain local-global principle for divisibility of an element of $H^{3}(F,\mu_{\ell}^{\otimes 3})$ by a symbol in $H^{2}(F,\mu_{\ell}^{\otimes 2})$ ([18], [20]). In fact it was proved that for a given $\zeta\in H^{3}(F,\mu_{\ell}^{\otimes 3})$ and a symbol $\alpha\in H^{2}(F,\mu_{\ell}^{\otimes 2})$ if for every discrete valuation $\nu$ of $F$ there exists $f_{\nu}\in F^{*}$ such that $\zeta-\alpha\cdot(f_{\nu})$ is unramified at $\nu$, then there exists $f\in F^{*}$ such that $\zeta=\alpha\cdot(f)$. In the proof of this local-global principle, the existence of residue homomorphisms on $H^{2}(F,\mu_{\ell}^{\otimes 2})$ and $H^{3}(F,\mu_{\ell}^{\otimes 3})$ is used. However note that if $K$ is a global field or a $p$-adic field with $\ell=p$, then there is no ‘residue homomorphism’ on $H^{2}(F,\mu_{\ell}^{\otimes 2})$ which describes the unramified Brauer group.

We now briefly explain the main ingredient of our result. Let $K$ be a global field or a local field and $F$ the function field of a curve over $K$. Let $\ell$ be a prime not equal to characteristic of $K$. Suppose that $K$ contains a primitive $\ell^{\rm th}$ root of unity. Let $\nu$ be a discrete valuation on $F$ and $\kappa(\nu)$ the residue field at $\nu$. Then there is a residueue homomorphism $H^{3}(F,\mu_{\ell}^{\otimes 3})\to{}_{\ell}\mathrm{Br}(\kappa(\nu))$ ([11, §1]). Let $\zeta\in H^{3}(F,\mu_{\ell}^{\otimes 3})$ and $\alpha=(a,b)\in H^{2}(F,\mu_{\ell}^{\otimes 2})$. First we show that if there is a regular proper model $\mathscr{X}$ of $F$ such that the triple $(\zeta,\alpha,\mathscr{X})$ satisfies certain assumptions, then there is a local global principle for the divisibility of $\zeta$ by $\alpha$ (cf. 6.5). One of the key assumptions is that $a\in F^{*}$ has some ‘nice’ properties at closed points of $\mathscr{X}$ which are on the support of the prime $\ell$ and in the ramification of $\zeta$ or $\alpha$ (cf. 5.1, 6.3). These assumptions on $a$ enable us to avoid the residue homomorphisms on $H^{2}(F,\mu_{\ell}^{\otimes 2})$. Let $\zeta\in H^{3}(F,\mu_{\ell}^{\otimes 3})$. Suppose that $\ell\neq 2$ or $K$ is a local field or $K$ is a global field without real orderings. Then we show that there exist a symbol $\alpha=[a,b)\in H^{2}(F,\mu_{\ell}^{\otimes 2})$ and a regular proper model $\mathscr{X}$ of $F$ satisfying the assumption of §6. Thus, by the local global principle for the divisibility, there exists $f\in F^{*}$ such that $\zeta-\alpha\cdot(f)$ is unramified on $\mathscr{X}$. Then, using a result of Kato ([11]), we arrive at the proof of our main result (7.7).

2. Preliminaries

Let $K$ be a field and $\ell$ a prime. Then every non-trivial element in $H^{1}(K,\mathbb{Z}/\ell)$ is represented by a pair $(L,\sigma)$, where $L/K$ is a cyclic field extension of degree $\ell$ and $\sigma$ a generator of Gal$(L/K)$.

Suppose $\ell\neq$ char$(K)$ and $K$ contains a primitive $\ell^{\rm th}$ root of unity. Fix a primitive $\ell^{\rm th}$ root of unity $\rho\in K$. Let $L/K$ be a cyclic extension of degree $\ell$. Then, by Kummer theory, we have $L=K(\sqrt[\ell]{a})$ for some $a\in K^{*}$ and $\sigma\in$ Gal$(L/K)$ given by $\sigma(\sqrt[\ell]{a})=\rho\sqrt[\ell]{a}$ is a generator of Gal$(L/K)$. Thus we have an isomorphism $K^{*}/K^{*\ell}\to H^{1}(K,\mathbb{Z}/\ell\mathbb{Z})$ given by sending the class of $a$ in $K^{*}/K^{*\ell}$ to the pair $(L,\sigma)$, where $L=K[X]/(X^{\ell}-a)$ and $\sigma(\sqrt[\ell]{a})=\rho\sqrt[\ell]{a}$. Let $a\in K^{*}$. If the image of the class of $a$ in $H^{1}(F,\mathbb{Z}/\ell\mathbb{Z})$ is $(L,\sigma)$ and $i$ is coprime to $\ell$, then the image of $a^{i}$ is $(L,\sigma^{i})$. In particular $(L,\sigma)^{i}=(L,\sigma^{i})$ for all $i$ coprime to $\ell$.

Suppose char$(K)=\ell$ and $L/K$ is a cyclic extension of degree $\ell$. Then, by Artin-Schreier theory, $L=K[X]/(X^{\ell}-X+a)$ for some $a\in K$. The element $\sigma\in$ Gal$(L/K)$ given by $\sigma(x)=x+1$, where $x\in L$ is the image of $X$ in $L$, is a generator of Gal$(L/K)$. Let $\wp:K\to K$ be the Artin-Schreier map $\wp(b)=b^{\ell}-b$. We have an isomorphism $K/\wp(K)\to H^{1}(K,\mathbb{Z}/\ell\mathbb{Z})$ given by sending the class of $a$ to the pair $(L,\sigma)$, where $L=K[X]/(X^{\ell}-X+a)$ and $\sigma(x)=x+1$. We note that if the image the class of $a$ is $(L,\sigma)$, then the image of the class of $ia$ is $(L,\sigma^{i})$ for all $1\leq i\leq\ell-1$.

In either case (char$(K)\neq\ell$ or char$(K)=\ell$), for $a\in K^{*}$ (or $K$), the pair $(L,\sigma)$ is denoted by $[a)$. Some times, by abuse of the notation, we also denote the cyclic extension $L$ by $[a)$.

Let $R$ be a regular local ring with field of fractions $K$, maximal ideal $m_{R}$ and residue field $\kappa$. Let $L/K$ be a finite separable extension and $S$ the integral closure of $R$ in $S$. We say that $L/K$ is unramified at $R$ if $S/m_{R}S$ is a separable $\kappa$-algebra. Let $R$ be a regular ring with field of fractions $K$ and $L/K$ a finite separable extension. We say that $L/K$ is unramified at a prime ideal $P$ of $R$ if $L/K$ is unramified at the local ring $R_{P}$ of $R$ at $P$. We say that $L/K$ is unramified on $R$ if $L/K$ is unramified at every prime ideal of $R$. If $L/K$ is unramified at a prime ideal $P$ of $R$, $S_{P}$ denotes the integral closure of $R_{P}$ and the separable $R_{P}/PR_{P}$-algebra $S_{P}/PS_{P}$ is called the residue field of $L$ at $P$. Note that $S_{P}/PS_{P}$ is a product of separable field extensions of $R_{P}/PR_{P}$. If $R$ is a regular local ring, then $L/K$ is unramified at $R$ if and only if the discriminant of $L/K$ is a unit in $R$ (cf. [14, Exercise 3.9, p.24]). Thus in particular, $L/K$ is unramified on $R$ if and only is $L/K$ is unramified at all height one prime ideals of $R$. If $L$ is a product of fields $L_{i}$ with $K\subset L_{i}$, then we say that $L/K$ is unramified on $R$ if each $L_{i}/K$ is unramified on $R$.

Lemma 2.1**.**

([6, Proposition 4.2.1]) Let $K$ be a field with a discrete valuation $\nu$ and $\kappa$ the residue field at $\nu$. Let $m$ be the maximal ideal of the valuation ring $R$ at $\nu$. Suppose that char$(K)=0$ and char$(\kappa)=\ell>0$. Suppose that $K$ contains a primitive $\ell^{\rm th}$ root of unity $\rho$. Then $\ell=x(\rho-1)^{\ell-1}$ for some unit $x$ at $\nu$ with $x\equiv-1$ modulo $m$. In particular $\nu(\rho-1)=\frac{\nu(\ell)}{\ell-1}$.

Proof.

Let $\eta=\rho-1$. Since $\eta$ is a zero of the polynomial $(X+1)^{\ell-1}+(X+1)^{\ell-2}+\cdots+1$ and the binomial coefficients ${}^{\ell}C_{i}$, $1\leq i\leq\ell-1$ are divisible by $\ell$, we have $\eta^{\ell-1}=-\ell(\eta^{\ell-2}+b_{\ell-3}\eta^{\ell-3}+\cdots+b_{1}\eta+1)$ for some $b_{i}\in R$. Since $\rho\equiv 1$ modulo $m$, $\eta\in m$ and hence $y=\eta^{\ell-2}+b_{\ell-3}\eta^{\ell-3}+\cdots+b_{1}\eta+1$ is a unit in $R$ and $y\equiv 1$ modulo $m$. Then $x=-y^{-1}$ has the required property. ∎

Lemma 2.2**.**

Suppose $R$ is a discrete valuation ring with field of fractions $K$ and residue field $\kappa$. Suppose that char$(K)=0$, char$(\kappa)=\ell>0$ and $K$ contains a primitive $\ell^{\rm th}$ root of unity $\rho$. Let $u\in R$ and $\overline{u}\in\kappa$ the image of $u$. If $1-u(\rho-1)^{\ell}\in R^{\ell}$, then $X^{\ell}-X+\bar{u}$ has a root in $\kappa$.

Proof.

Let $m$ be the maximal ideal of $R$. Suppose that $u\in m$. Then $\bar{u}=0$ and $X^{\ell}-X$ has a root in $\kappa$.

Suppose that $u\in R$ is a unit. Suppose $1-u(\rho-1)^{\ell}\in R^{\ell}$. Let $z\in R$ with $z^{\ell}=1-u(\rho-1)^{\ell}\in R$. Since $\rho-1\in m$, $1-u(\rho-1)^{\ell}$ is a unit in $R$ and hence $z$ is a unit in $R$ with $z^{\ell}\equiv 1$ modulo $m$. Since char$(\kappa)=\ell$, $z\equiv 1$ modulo $m$. Thus $z=1+d$ for some $d\in m$. Since $z^{\ell}=(1+d)^{\ell}=1+\ell d+\cdots+d^{\ell}$, all the binomial coefficients are divisible by $\ell$ and $d\in m$, we have $z^{\ell}=1+\ell dy+d^{\ell}$ for some unit $y\in R$ with $y\equiv 1$ modulo $m$. Since $z^{\ell}=1-u(\rho-1)^{\ell}$, we have $\ell dy+d^{\ell}=-u(\rho-1)^{\ell}$.

We claim that $\nu(d)=\nu(\rho-1)$. Suppose that $\nu(\ell d)=\nu(d^{\ell})$. Then $\nu(\ell)+\nu(d)=\ell\nu(d)$ and hence $\nu(d)=\frac{\nu(\ell)}{\ell-1}=\nu(\rho-1)$ (2.1). Suppose that $\nu(\ell d)<\nu(d^{\ell})$. Then $\nu(\ell dy+d^{\ell})=\nu(\ell d)=\nu(\ell)+\nu(d)$. Since $\ell dy+d^{\ell}=-u(\rho-1)^{\ell}$, $\nu(\ell)+\nu(d)=\ell\nu(\rho-1)$ and hence $\nu(d)=\ell\nu(\rho-1)-\nu(\ell)=\ell\frac{\nu(\ell)}{\ell-1}-\nu(\ell)=\frac{\nu(\ell)}{\ell-1}=\nu(\rho-1)$. Suppose that $\nu(\ell dy)>\nu(d^{\ell})$. Then $\ell\nu(\rho-1)=\nu(d^{\ell})=\ell\nu(d)$ and hence $\nu(d)=\nu(\rho-1)$.

Since $\nu(d)=\nu(\rho-1)$, we have $d=w(\rho-1)$ for some unit $w\in R$. By (2.1), we have $\ell=x(\rho-1)^{\ell-1}$ with $x\equiv-1$ modulo $m$. Thus $-u(\rho-1)^{\ell}=\ell dy+d^{\ell}=xyw(\rho-1)^{\ell}+w^{\ell}(\rho-1)^{\ell}$ and hence $-u=w^{\ell}+xyw$. Since $x\equiv-1$ modulo $m$ and $y\equiv 1$ modulo $m$, we have $\overline{w}^{\ell}-\overline{w}+\overline{u}=0$. In particular $X^{\ell}-X+\bar{u}$ has a root in $\kappa$. ∎

We have the following (cf. [7, Proposition 1.4])

Proposition 2.3**.**

*Suppose $R$ is a discrete valuation ring with field of fractions $K$ and residue field $\kappa$. Suppose that char$(K)=0$, char$(\kappa)=\ell>0$ and $K$ contains a primitive $\ell^{\rm th}$ root of unity $\rho$. Let $u\in R$ be a unit and $L=K[X]/(X^{\ell}-(1-u(\rho-1)^{\ell}))$. Let $S$ be the integral closure of $R$ in $L$. Then $L/K$ is unramified at $R$ and

$\bullet$ if $X^{\ell}-X+\overline{u}$ is irreducible in $\kappa[X]$, then there is a unique maximal ideal $m_{S}$ of $S$ with $S/m_{S}\simeq\kappa[X]/(X^{\ell}-X+\overline{u})$, where $\overline{u}$ is the image of $u$ in $\kappa$

$\bullet$ if $X^{\ell}-X+\overline{u}$ is reducible in $\kappa[X]$, then the maximal ideal $m_{R}$ of $R$ splits into product of $\ell$ maximal ideals of $R$.*

Proof.

Without loss of generality we assume that $R$ is complete and $L$ is a field. Then $S$ is a complete discrete valuation ring. Let $m_{R}$ be the maximal ideal of $R$ and $m_{S}$ the maximal ideal of $S$. Then $m_{R}S=m_{S}$.

Suppose that $X^{\ell}-X+\overline{u}$ is irreducible in $\kappa[X]$. Since $1-u(\rho-1)^{\ell}\in S$, by (2.2), $X^{\ell}-X-\overline{u}$ has a root in $S/m_{S}$. Since $[S/m_{S}:\kappa]\leq\ell$, $S/m_{S}\simeq\kappa[X]/(X^{\ell}-x+\overline{u})$ and hence $m_{S}$ is the unique maximal ideal of $S$ with $L/K$ unramified at $R$.

Suppose that $X^{\ell}-X+\overline{u}$ is reducible in $\kappa[X]$. Since char$(\kappa)=\ell$, $X^{\ell}-X+\overline{u}$ has $\ell$ distinct roots in $\kappa$. Since $R$ is complete, $X^{\ell}-X+\overline{u}$ has a root $w$ in $R$. Let $d=w(\rho-1)\in R$. Then, as in the proof of (2.2), we have $(1+d)^{\ell}=1+\ell dy+d^{\ell}$ for some $y\in R$ with $y\equiv 1$ modulo $m_{R}$. By (2.1), we have $\ell=x(\rho-1)^{\ell-1}$ for some $x\in R$ with $x\equiv-1$ modulo $m_{R}$. Since $w^{\ell}=w-u$ and $d=w(\rho-1)$, we have

[TABLE]

Since $xy+1\equiv 0$ modulo $m$, we have $(1+d)^{\ell}=1-u(\rho-1)^{\ell}$ modulo $(\rho-1)^{\ell}m$ and hence $1-u(\rho-1)^{\ell}\in R^{*\ell}$ (cf. [7, §0.3]) ∎

Corollary 2.4**.**

Suppose that $A$ is a regular local ring of dimension two with field of fractions $F$, maximal ideal $m$ and residue field $\kappa$. Suppose that char$(F)=0$, char$(\kappa)=\ell>0$ and $F$ contains a primitive $\ell^{\rm th}$ root of unity $\rho$. Let $u\in A$ be a unit and $L=F[X]/(X^{\ell}-(1-u(\rho-1)^{\ell}))$. Suppose that $L$ is a field. Let $S$ be the integral closure of $A$ in $L$. Then $L/F$ is unramified on $A$ and $S/mS\simeq\kappa[X]/(X^{p}-X+\overline{u})$, where $\overline{u}$ is the image of $u$ in $\kappa$.

Proof.

Since char$(\kappa)=\ell$ and $\rho^{\ell}=1$, $1-\rho$ is in the maximal ideal of $A$ and hence $1-u(\rho-1)^{\ell}$ is a unit in $A$. Let $P$ be a prime ideal of $A$ of height one. Suppose char$(A/P)\neq\ell$. Since $1-u(\rho-1)^{\ell}$ is a unit in $A$, $L/K$ is unramified at $P$. If char$(R/P)=\ell$, then by (2.3), $L/K$ is unramified at $P$. Thus $L/K$ is unramified on $A$.

Let $m=(\pi,\delta)$ be the maximal ideal of $A$. Since $L/K$ is unramified on $A$, $S/\pi S$ is a regular semi-local ring (cf. [14, Proposition 3.17, p.27]). Suppose that char$(A/(\pi))\neq\ell$. Since $1-u(\rho-1)^{\ell}$ is a unit at $\pi$, $L/K$ is unramified at $\pi$ and $S\otimes_{A}A_{(\pi)}/(\pi)\simeq(A_{(\pi)}/(\pi))[X]/(X^{\ell}-(1-\overline{u}(\overline{\rho}-1)^{\ell}))$, where $\bar{~{}}$ denotes the image modulo $(\pi)$. Hence by (2.3), $S/(\pi,\delta)S=\kappa[X]/(X^{\ell}-X+\overline{u})$. Suppose that char$(A/(\pi))=\ell$. Then, by (2.3), the field of fractions of $S/\pi S$ is the field of fractions of $(A/(\pi))[X]/(X^{\ell}-X+\overline{u})$. Since $u$ is a unit in $A/(\pi)$, $A/(\pi)[X]/(X^{\ell}-X+\overline{u})$ is a regular local ring and hence $S/\pi S\simeq A/(\pi)[X]/(X^{\ell}-X+\overline{u})$. Hence $S/(\pi,\delta)S=\kappa[X]/(X^{\ell}-X+\overline{u})$. ∎

Let $R$ be a regular ring of dimension at most 2 with field of fractions $K$ and $\ell$ a prime. If $\ell$ is not equal to char$(K)$, then assume that $K$ contains a primitive $\ell^{\rm th}$ root of unity $\rho$. Suppose $L=[a)$ is a cyclic extension of $K$ of degree $\ell$. Let $P$ be a prime ideal of $R$, $\kappa(P)=R_{P}/PR_{P}$ and $S_{P}$ the integral closure of $R_{P}$ in $L$. Suppose char$(\kappa(P))\neq\ell$. Then $L=K[X]/(X^{\ell}-a)$ and hence $S_{P}/PS_{P}\simeq\kappa(P)[X]/(X^{\ell}-\overline{a})$ where $\overline{a}$ is the image of $a$ in $\kappa(P)$. Suppose char$(\kappa(P))=\ell$, char$(K)\neq\ell$ and $a=1-u(\rho-1)^{\ell}$ for some $u\in R_{P}$. Then, by (2.3, 2.4), $S_{P}/PS_{P}\simeq\kappa(P)[X]/(X^{\ell}-X+\overline{u})$. Suppose char$(\kappa(P))$ = char$(K)=\ell$ and $a\in R_{P}$. Then $L=K[X]/(X^{\ell}-X+a)$ and hence $S_{P}/PS_{P}\simeq\kappa(P))[X]/(X^{\ell}-X+\overline{a})$. Thus, in either case, $S_{P}/PS_{P}$ is either a cyclic field extension of degree $\ell$ over $\kappa(P)$ or the split extension of degree $\ell$ over $\kappa(P)$ and we denote these $S_{P}/PS_{P}$ by $[a(P))$. If $P=(\pi)$ for some $\pi\in R$, then we also denote $[a(P))$ by $[a(\pi))$. If $P$ induces a discrete valuation $\nu$ on $K$, then we also denote $[a(P))$ by $[a(\nu))$. For an element $b\in R$, we also denote the image of $b$ in $R/P$ by $b(P)$. If $b\in R$ and $c\in R/P$, we write $b=c\in R/P$ for $b\equiv c$ modulo $P$.

Lemma 2.5**.**

*Let $A$ be a semi local regular ring of dimension at most two with field of fractions $F$. Let $\ell$ be a prime not equal to the characteristic of $F$. Suppose that $F$ contains a primitive $\ell^{\rm th}$ root of unity. For each maximal ideal $m$ of $A$, let $[u_{m})$ be a cyclic extension of $A/m$ of degree $\ell$. Then there exists $a\in A$ such that

$\bullet$ $[a)$ is unramified on $A$ with residue field $[u_{m})$ at each maximal ideal $m$ of $A$

$\bullet$ if $\ell=2$ and $A/m$ is finite for all maximal ideals $m$ of $A$, then $a$ can be chosen to be a sum of two squares in $A$.*

Proof.

Let $\rho\in F$ be a primitive $\ell^{\rm th}$ root of unity. Let $m$ be a maximal ideal of $A$. If char$(A/m)\neq\ell$, then let $b_{m}=\frac{1-u_{m}}{(\rho-1)^{\ell}}\in A/m$. If char$(A/m)=\ell$, then let $b_{m}=u_{m}\in A/m$. Chose $b\in A$ with $b=b_{m}\in A/m$ for all maximal ideals $m$ of $A$ and $a=1-b(\rho-1)^{\ell}$. Let $m$ be a maximal ideal of $A$. Suppose that char$(A/m)\neq\ell$. Then, by the choice of $a$ and $b$, we have $a=1-b_{m}(\rho-1)^{\ell}=u_{m}\in A/m$. Thus $[a)$ is unramified at $m$ with the residue field $[u_{m})$ at $m$. Suppose that char$(A/m)=\ell$. Then, by (2.3, 2.4), $[a)$ is unramified at $m$ with the residue field $[\overline{b})$. Since $b=b_{m}=u_{m}\in A/m$, the residue field of $[a)$ at $m$ is $[u_{m})$.

Suppose $\ell=2$ and $A/m$ is a finite field for all maximal ideals $m$ of $A$. Let $m$ be a maximal ideal of $A$. Suppose that char$(A/m)\neq 2$. Since every element of $A/m$ is a sum of two squares in $A/m$ ([25, p. 39, 3.7]), there exist $x_{m},y_{m}\in A/m$ such that $x_{m}^{2}+y_{m}^{2}=1-4u_{m}$. Suppose that char$(A/m)=2$. Since $A/m$ is a finite field, every element in $A/m$ is a square. Let $y_{m}\in A/m$ be such that $y_{m}^{2}=u_{m}$. Let $x,y\in A$ be such that for every maximal ideal $m$ of $A$,

$\bullet$ if char$(A/m)\neq 2$, then $x=\frac{x_{m}-1}{4}\in A/m$ and $y=\frac{y_{m}}{2}\in A/m$

$\bullet$ if char$(A/m)=2$, then $x=0\in A/m$ and $y=y_{m}\in A/m$.

Let $a=(1+4x)^{2}+(2y)^{2}\in A$. Let $m$ be a maximal ideal of $A$. Suppose char$(A/m)\neq 2$. Then $a=x_{m}^{2}+y_{m}^{2}=u_{m}\in A/m$ and hence $[a)$ is unramified at $m$ with residue field at $m$ equal to $[u_{m})$. Suppose that char$(A/m)=2$. Then $\frac{1-a}{4}=u_{m}\in A/m$ and hence $[a)$ is unramified at $m$ with residue field $[u_{m})$ (2.3, 2.4).

∎

Lemma 2.6**.**

Let $R$ be a regular ring of dimension at most 2 and $K$ its field of fractions. Let $\ell$ be a prime not equal to char$(K)$. Suppose that $K$ contains a primitive $\ell^{\rm th}$ root of unity $\rho$. Let $L=K(\sqrt[\ell]{u})$ for some $u\in R$. Let $m_{1},\cdots,m_{r},m_{r+1},\cdots,m_{n}$ be maximal ideals of $R$. Suppose that $\kappa(m_{j})=\ell$ and $L/K$ is unramified at $m_{j}$ for all $r+1\leq j\leq n$. Then there exists $v\in R$ such that $L=K(\sqrt[\ell]{v})$, $v\equiv u$ modulo $m_{i}$ for all $1\leq i\leq r$ and $\frac{1-v}{(\rho-1)^{\ell}}$ is a unit at $m_{j}$ for all $r+1\leq j\leq n$.

Proof.

For a maximal ideal $m$ of $R$, let $K_{m}$ denote the field of factions of the completion of $R$ at $m$.

Let $r+1\leq j\leq n$. Since char$(\kappa(m_{j}))=\ell$ and $L/K$ unramified at $m_{j}$, the residue field of $L$ at $m_{j}$ is $\kappa(m_{j})[X]/(X^{\ell}-X+\overline{w_{j}})$ for some $w_{j}\in R_{m_{j}}$. Since the residue field of $K[X]/(X^{\ell}-(1-w_{j}(\rho-1)^{\ell}))$ is isomorphic to $\kappa(m_{j})[X]/(X^{\ell}-X+\overline{w_{j}})$ (2.3, 2.4), $L\otimes K_{m_{j}}\simeq K_{m_{j}}[X]/(X^{\ell}-(1-w_{j}(\rho-1)^{\ell}))$. Since char$(K)\neq\ell$ and $L=K(\sqrt[\ell]{u})$, there exists $\theta_{j}\in K_{m_{j}}$ such that $u\theta_{j}^{\ell}=1-w_{j}(\rho-1)^{\ell}$. Let $\theta\in R$ be such that $\theta\equiv 1$ modulo $m_{i}$ for $1\leq i\leq r$ and $\theta\equiv w_{j}$ modulo $m_{j}$ for $r+1\leq j\leq n$. Then $v=u\theta^{\ell}$ has the required properties. ∎

The following is a generalization of a result of Saltman ([24, Proposition 0.3]).

Lemma 2.7**.**

Let $A$ be a UFD. For $1\leq i\leq n$, let $I_{i}=(a_{i})\subset A$ with gcd$(a_{i},a_{j})=1$ for all $i\neq j$. For each $i<j$, let $I_{ij}=I_{i}+I_{j}$. Suppose that the ideals $I_{i,j}$ are comaximal. Then

[TABLE]

is exact, where for $i<j$, the map from $A/I_{i}\oplus A/I_{j}\to A/I_{ij}$ is given by $(x,y)\mapsto x-y$.

Proof.

Proof by induction on $n$. The case $n=2$ is in ([24, Lemma 0.2]). Assume that $n\geq 3$. Let $a_{i}\in A/I_{i}$ maps to zero in $\oplus A/I_{ij}$. By induction, there exists $b\in A$ such that $b=a_{i}\in A/I_{i}$ for $1\leq i\leq n-1$. We claim that $I_{1}\cap\cdots\cap I_{n-1}+I_{n}=(I_{1}+I_{n})\cap\cdots\cap(I_{n-1}+I_{n})$. Since both sides contain $I_{n}$, it is enough to prove the equality after going modulo $I_{n}$. Since gcd$(a_{i},a_{j})=1$ for all $i\neq j$, we have $I_{1}\cap\cdots\cap I_{n-1}=Aa_{1}\cdots a_{n-1}$ and hence $I_{1}\cap\cdots\cap I_{n-1}+I_{n}/I_{n}=(A/I_{n})\overline{a}_{1}\cdots\overline{a}_{n-1}$. Since $I_{ij}$ are comaximal, $I_{in}/I_{n}=(A/I_{n})\overline{a}_{i}$ are comaximal for $1\leq i\leq n-1$ and hence $(A/I_{n})\overline{a}_{1}\cdots\overline{a}_{n-1}=(A/I_{n})\overline{a}_{1}\cap\cdots\cap(A/I_{n})\overline{a}_{n-1}$. Let $b_{1}\in A/(I_{1}\cap\cdots\cap I_{n-1})$ be the image of $b$. Then, by the case $n=2$, there exists $a\in A$ such that $a=b_{1}\in A/I_{1}\cap\cdots\cap I_{n-1}$ and $a=a_{n}\in A/I_{n}$.Thus $a$ has the required properties. ∎

3. Central simple algebras

Let $K$ be a field, $L/K$ a cyclic extension of degree $n$ with $\sigma\in$ Gal$(L/K)$ a generator and $b\in K^{*}$. Let $(L,\sigma,b)$ denote the cyclic algebra $L\oplus Lx\oplus\cdots\oplus Lx^{n-1}$ with relations $x^{n}=b$, $x\lambda=\sigma(\lambda)x$ for all $\lambda\in L$. Then $(L,\sigma,b)$ is a central simple algebra over $K$ and represents an element in the $n$-torsion subgroup ${}_{n}\mathrm{Br}(K)$ of the Brauer group $\operatorname{\mathrm{Br}}(K)$ ([1, Theorem 18, p. 98]). Suppose that $n$ is coprime to char$(K)$ and $K$ contains a primitive $n^{\rm th}$ root of unity. Then $L=K(\sqrt[n]{a})$ for some $a\in K^{*}$. Fix a primitive $n^{\rm th}$ root of unity $\rho$ in $K$. Let $\sigma$ be the generator of Gal$(L/K)$ given by $\sigma(\sqrt[n]{a})=\rho\sqrt[n]{a}$. Then, the cyclic algebra $(L,\sigma,b)$ is denoted by $[a,b)$. Suppose that $n$ is prime and equal to char$(K)$. Then, $L=K[X]/(X^{n}-X+a)$ for some $a\in K$. If $\sigma$ is the generator of Gal$(L/K)$ given by $\sigma(x)=x+1$, then the cyclic algebra $(L,\sigma,b)$ is also denoted by $[a,b)$.

For any Galois module $M$ over $K$, let $H^{n}(K,M)$ denote the Galois cohomology of $K$ with coefficients in $M$. Let $K$ be a field and $\ell$ a prime. Let $\mathbb{Z}/\ell(i)$ be the Galois modules over $K$ as in ([11, §0]). We have a canonical isomorphisms $H^{1}(K,\mathbb{Z}/\ell)\simeq{\rm Hom}_{\rm cont}({\rm Gal}(K^{\rm ab}/K),\mathbb{Z}/\ell)$ and ${}_{\ell}\mathrm{Br}(K)\simeq H^{2}(K,\mathbb{Z}/\ell(1))$, where $K^{\rm ab}$ is the maximal abelian extension of $K$ ([11, §0]).

Suppose $A$ is a regular domain with field of fractions $F$. We say that an element $\alpha\in H^{2}(F,\mathbb{Z}/\ell(1))$ is unramified on $A$ if $\alpha$ is represented by a central simple algebra over $F$ which comes from an Azumaya algebra over $A$. If it is not unramified, then we say that $\alpha$ is ramified on $A$. Suppose $P$ is a prime ideal of $A$ and $\alpha\in H^{2}(F,\mathbb{Z}/\ell(1))$. We say that $\alpha$ is unramified at $P$ if $\alpha$ is unramified at $A_{P}$. If $\alpha$ is not unramified at $P$, then we say that $\alpha$ is ramified at $P$. Suppose that $\alpha$ is unramified at $P$. Let $\mathscr{A}$ be an Azumaya algebra over $A_{P}$ with the class of $\mathscr{A}\otimes_{A_{P}}F$ equal to $\alpha$. The algebra $\overline{\alpha}=\mathscr{A}\otimes_{A_{P}}(A_{P}/PA_{P})$ is called the specialization of $\alpha$ at $P$. Since $A_{P}$ is a regular local ring, the class of $\overline{\alpha}$ is independent of the choice of $\mathscr{A}$. Let $a,b\in F$ and $\alpha=[a,b)\in H^{2}(F,\mathbb{Z}/\ell(1))$. If the cyclic extension $[a)$ is unramified at $P$ and $b$ is a unit at $P$, then $\alpha$ is unramified at $P$ and the specialization of $\alpha$ at $P$ is $[a(P),b(P))$, where $[a(P))$ is the residue field of $[a)$ at $P$ and $b(P)$ is the image of $b$ in $A_{P}/PA_{P}$.

Suppose that $R$ is a discrete valuation ring with field of fractions $K$ and residue field $\kappa$. Let $\ell$ be a prime not equal to char$(K)$. Suppose that char$(\kappa)\neq\ell$ or char$(\kappa)=\ell$ with $\kappa=\kappa^{\ell}$. Then there is a *residue homomorphism * $\partial:H^{2}(K,\mathbb{Z}/\ell(1))\to H^{1}(\kappa,\mathbb{Z}/\ell)$ ([11, §1]). Further a class $\alpha\in H^{2}(K,\mathbb{Z}/\ell(1))$ is unramified at $R$ if and only if $\partial(\alpha)=0$. Let $a,b\in K^{*}$. If $[a)$ is unramified at $R$, then $\partial([a,b))=[a(\nu))^{\nu(b)}$, where $\nu$ is the discrete valuation on $K$. In particular if $[a)$ is unramified on $R$ and $\ell$ divides $\nu(b)$, then $[a,b)$ is unramified on $R$.

Lemma 3.1**.**

(cf. [13, Lemma 3.1]) Let $A$ be a regular ring of dimension 2 and $F$ its field of fractions. Let $\ell$ be a prime not equal to char$(F)$ and $\alpha\in H^{2}(F,\mathbb{Z}/\ell(1))$. If $\alpha$ is unramified at all height one prime ideals of $A$, then $\alpha$ is unramified on $A$.

Lemma 3.2**.**

Let $R$ be a complete discrete valued field with field of fractions $K$ and residue field $\kappa$. Let $\ell$ be a prime not equal to char$(\kappa)$. Let $D$ be a central simple algebra of index $\ell$ over $K$. Suppose that $D$ is ramified at $R$. If $L/K$ is the unramified extension of $K$ with residue field equal to the residue of $D$ at $R$, then $D\otimes L$ is a split algebra.

Proof.

We have $D=D_{0}\otimes(L,\sigma,\pi)$ for some generator of Gal$(L/K)$, $\pi$ a parameter in $R$ and $D_{0}$ unramified at $R$ (cf. [15, Lemma 4.1]). Further $\ell=$ ind$(D)=$ ind$(D_{0}\otimes L)[L:K]$ (cf. [15, Lemma 4.2]). Since $D$ is ramified at $R$, $[L:K]=\ell$ and hence $D_{0}\otimes L=0$. Hence $D_{0}=(L,\sigma,u)$ for some $u\in K$ and $D=(L,\sigma,u\pi)$. Thus $D\otimes L$ is a split algebra. ∎

Lemma 3.3**.**

*Let $A$ be a complete regular local ring of dimension 2 with field of fractions $F$ and residue field $\kappa$. Suppose that $\kappa$ is a finite field. Let $m=(\pi,\delta)$ be the maximal ideal of $A$. Let $\ell$ be a prime not equal to char$(F)$ and $\alpha=[a,b)\in H^{2}(F,\mathbb{Z}/\ell(1))$ for some $a,b\in F^{*}$. Suppose that

$\bullet$ if char$(\kappa)=\ell$, then the cyclic extension $[a)$ is unramified on $A$

$\bullet$ $\alpha$ is unramified on $A$ except possibly at $\delta$

$\bullet$ the specialization of $\alpha$ at $\pi$ is unramified on $A/(\pi)$.

Then $\alpha=0$.*

Proof.

Suppose that char$(\kappa)\neq\ell$. Then, it follows from ([27, Proposition 3.4]) that $\alpha=0$ (cf. [15, Corollary 5.5]).

Suppose that char$(\kappa)=\ell$. Since $F$ is the field of fractions of $A$, without loss of generality, we assume that $b\in A$ and not divisible by $\theta^{\ell}$ for any prime $\theta\in A$. Write $b=v\delta^{n}\theta_{1}^{n_{1}}\cdots\theta_{r}^{n_{r}}$ for some distinct primes $\theta_{i}\in A$ with $(\delta)\neq(\theta_{i})$ for all $i$, $1\leq n_{i}\leq\ell-1$, $0\leq n\leq\ell-1$ and $v\in A$ a unit. Since $\kappa$ is a finite field, $A$ is complete and $[a)$ is unramified on $A$, we have $[a,v)=0$ and hence $\alpha=[a,b)=[a,\delta^{n}\theta_{1}^{n_{1}}\cdots\theta_{r}^{n_{r}})$.

Since $[a)$ is unramified on $A$, for any prime $\theta\in A$, $[a,\theta)$ is unramified on $A$ except possibly at $\theta$. Let $1\leq j\leq r$. Since $\alpha=[a,b)=[a,\delta^{n})\prod[a,\theta_{i}^{n_{i}})$, $[a,\delta^{n})$ and $[a,\theta_{i}^{n_{i}})$ are unramified at $\theta_{j}$ for all $i\neq j$, $[a,\theta_{j}^{n_{j}})$ is unramified at $\theta_{j}$ and hence $[a,\theta_{j}^{n_{j}})$ is unramified on $A$ (cf. 3.1). Since $\kappa$ is a finite field and $A$ is complete, $[a,\theta_{j}^{n_{j}})=0$. Thus, we have $\alpha=[a,\delta^{n})$.

If $n=0$, then $\alpha=0$. Suppose $1\leq n\leq\ell-1$. Let $\overline{\alpha}$ be the specialization of $\alpha$ at $\pi$. Since $\alpha=[a,\delta^{n})$ and $[a)$ is unramified at $\pi$, we have $\overline{\alpha}=[a(\pi),\overline{\delta}^{n})$, where $[a(\pi))$ is the residue field of $[a)$ at $\pi$ and $\overline{\delta}$ is the image of $\delta$ in $A_{P}/(\pi)$. Since $\overline{\alpha}$ is unramified on $A/(\pi)$, $A$ is complete and $\kappa$ is a finite field, $\overline{\alpha}=[a(\pi),\overline{\delta}^{n})=0$. Since $\partial(\overline{\alpha})=[a(m))^{n}=1$ and $n$ is coprime to $\ell$, $[a(m))=0$. Since $A$ is complete, $[a)$ is trivial and hence $\alpha=0$. ∎

We now recall the chilly points and the chilly loops associated to a central simple algebra, due to Saltman ([23], [24]). Let $\mathscr{X}$ be a regular integral excellent scheme of dimension 2 and $F$ its field of fractions. Let $\ell$ be a prime which is not equal to char$(F)$. Suppose that $F$ contains a primitive $\ell^{\rm th}$ root of unity. Let $\alpha\in H^{2}(F,\mathbb{Z}/\ell(1))$. Suppose that ram${}_{\mathscr{X}}(\alpha)\subset\{D_{1},\cdots,D_{n}\}$ for some regular irreducible curves $D_{i}$ on $\mathscr{X}$ with normal crossings. Let $P\in D_{i}\cap D_{j}$ be a closed point. If char$(\kappa(P))\neq\ell$, then, we say that $P$ is a chilly point of $\alpha$ if $\alpha=\alpha_{0}+(u,\pi_{i}\pi_{j}^{s})$ for some $\alpha_{0}$ unramified at $P$, $u$ a unit at $P$ and $\pi_{i},\pi_{j}$ primes at $P$ defining $D_{i}$ and $D_{j}$ at $P$ respectively.

Let $\Gamma$ be a graph with vertices $D_{i}$’s and edges as chilly points,i.e. two distinct vertices $D_{i}$ and $D_{j}$ have an edge between them if there is a chilly point in $D_{i}\cap D_{j}$. A loop in this graph is called a chilly loop on $\mathscr{X}$. Let $\mathscr{X}[\frac{1}{\ell}]$ be the open subscheme of $\mathscr{X}$ obtained by inverting $\ell$. Since, by the definition of chilly point, char$(\kappa(P))\neq\ell$ for any chilly point $P$, we have the following

Proposition 3.4**.**

([23, Corollary 2.9]) There exists a sequence of blow-ups $\mathscr{X}^{\prime}\to\mathscr{X}$ centered at closed points $P\in\mathscr{X}[\frac{1}{\ell}]$ such that $\alpha$ has no chilly loops on $\mathscr{X}^{\prime}$.

Let $K$ be a global field and $\ell$ a prime. Let $\beta\in{}_{\ell}\mathrm{Br}(K)$. Let $\nu$ be a discrete vacation of $K$, $K_{\nu}$ the completion of $K$ at $\nu$ and $\kappa(\nu)$ the residue field at $\nu$. Since $K_{\nu}$ is a local field, the invariant map gives an isomorphism $\partial_{\nu}:{}_{\ell}\mathrm{Br}(K_{\nu})=H^{2}(K_{\nu},\mathbb{Z}/\ell(1))\to H^{1}(\kappa(\nu),\mathbb{Z}/\ell)$.

Proposition 3.5**.**

*Let $K$ be a global field and $\ell$ a prime. If $\ell$ is not equal to char$(K)$, then assume that $K$ contains a primitive $\ell^{\rm th}$ root of unity $\rho$. Let $\beta\in{}_{\ell}\mathrm{Br}(K)$. Let $S$ be a finite set of discrete valuations of $K$ containing all the discrete valuations $\nu$ of $K$ with $\partial_{\nu}(\beta)\neq 0$. Let $S^{\prime}$ be a finite set of discrete valuations of $K$ with $S\cap S^{\prime}=\emptyset$. Let $a\in K^{*}$ and for each $\nu\in S^{\prime}$, let $n_{\nu}\geq 2$ be an integer. Suppose that for every $\nu\in S$, $[a)$ is unramified at $\nu$ with $\partial_{\nu}(\beta)=[a(\nu))$. Further assume that if $\ell=2$, then $\beta\otimes K_{\nu}(\sqrt{a})=0$ for all real places $\nu$ of $K$. Then there exists $b\in K^{*}$ such that

$\bullet$ $\beta=[a,b)$

$\bullet$ if $\nu\in S$, then $\nu(b)=1$

$\bullet$ if $\nu\in S^{\prime}$, then $\nu(b-1)\geq n_{\nu}$.*

Proof.

Let $L=[a)$. Let $\nu\in S$. If $\partial_{\nu}(\beta)=0$, then $\beta\otimes K_{\nu}=0$ ([4, p. 131]). Suppose that $\partial_{\nu}(\beta)\neq 0$. Then $[a(\nu))$ is a field extension of $\kappa(\nu)$ of degree $\ell$ and hence $L\otimes_{K}K_{\nu}$ is a degree $\ell$ field extension of $K_{\nu}$. Thus $\beta\otimes_{K}(L\otimes K_{\nu})=0$ (cf. [4, p. 131]). Suppose $\nu$ is a real place of $K$. Then, by the assumption on $a$, $\beta\otimes_{K}(L\otimes_{K}K_{\nu})=0$. Thus $\beta\otimes L=0$ (cf. [4, p. 187]) and hence there exists $c\in K^{*}$ such that $\beta=[a,c)$ (cf. [1, p. 94]).

Let $R$ be the semi local ring at the discrete valuations in $S\cup S^{\prime}$. Replacing $c$ by $c\theta^{n\ell}$ for some $\theta\in K^{*}$ and $n\geq 1$, we assume that $c\in R$. For $\nu\in S\cup S^{\prime}$, let $\pi_{\nu}\in R$ be a parameter at $\nu$. Let $\nu\in S$. Since $[a)$ is unramified at $\nu$, $\partial_{\nu}(\beta)=\partial_{\nu}([a,c))=[a(\nu))^{\nu(c)}$. Suppose $[a(\nu))$ is non-trivial. Since, by the hypothesis, $\partial_{\nu}(\beta)=[a(\nu))$, $\nu(c)-1$ is divisible by $\ell$. Since $[L:K]=\ell$, $\pi_{\nu}^{\nu(c)-1}$ is a norm from $L\otimes_{K}K_{\nu}/K_{\nu}$. Suppose that $[a(\nu))$ is trivial. Then $L\otimes_{K}K_{\nu}$ is the split extension and hence every element of $K_{\nu}$ is a norm from $L\otimes_{K}K_{\nu}/K_{\nu}$. Thus for each $\nu\in S$, there exists $x_{\nu}\in L\otimes_{K}K_{\nu}$ with norm $\pi_{\nu}^{\nu(c)-1}$. Let $\nu\in S^{\prime}$. Then $\partial_{\nu}(\beta)=0$ and we have $\beta\otimes K_{\nu}=[a,c)\otimes K_{\nu}=0$ ([4, p. 131]). Hence $c$ is a norm from $L\otimes_{K}K_{\nu}$. For each $\nu\in S^{\prime}$, $x_{\nu}\in L\otimes_{K}K_{\nu}$ with norm $c$. Let $z\in L$ be sufficiently close to $x_{\nu}$ such that $\nu(N_{L\otimes_{K}K_{\nu}}(z)-\pi_{\nu}^{\nu(c)-1})\geq\nu(c)$ for all $\nu\in S$ and $\nu(N_{L\otimes_{K}K_{\nu}}(z)-c)\geq\nu(c)+n_{\nu}$ for all $\nu\in S^{\prime}$.

Let $d$ be the norm of $z$ and $b=cd^{-1}$. Then $\beta=[a,cd^{-1})=[a,b)$. Let $\nu\in S$. Since $\nu(d-\pi_{\nu}^{\nu(c)-1})\geq\nu(c)$, we have $\nu(d)=\nu(c)-1$ and hence $\nu(b)=\nu(cd^{-1})=1$. Let $\nu\in S^{\prime}$. Since $\nu(d-c)\geq\nu(c)+n_{\nu}\geq 2$, $\nu(d)=\nu(c)$ and $\nu(b-1)=\nu(cd^{-1}-1)\geq n_{\nu}$. ∎

4. A complex of Kato

Let $K$ be a complete discrete valued field with residue field $\kappa$. Let $\ell$ be a prime not equal to characteristic of $K$. If $\ell=$ char$(\kappa)$, then assume that $[\kappa:\kappa^{\ell}]\leq\ell$. Then, there is a residue homomorphism $\partial:H^{3}(K,\mathbb{Z}/\ell(2))\to H^{2}(\kappa,\mathbb{Z}/\ell(1))$ ([11, §1]). We say that an element $\zeta\in H^{3}(K,\mathbb{Z}/\ell(2))$ is *unramified * at the discrete valuation of $F$ if $\partial(\zeta)=0$.

Let $\mathscr{X}$ be a two-dimensional regular integral excellent scheme and $F$ the function field of $\mathscr{X}$. For $x\in\mathscr{X}$, let $F_{x}$ be the field of fractions of the completion $\hat{A}_{x}$ of the local ring $A_{x}$ at $x$ on $\mathscr{X}$ and $\kappa(x)$ the residue field at $x$. Let $x\in\mathscr{X}$ and $C$ be the closure of $\{x\}$ in $\mathscr{X}$. Then, we also denote $F_{x}$ by $F_{C}$. If the dimension of $C$ is one, then $C$ defines a discrete valuation $\nu_{C}$ (or $\nu_{x}$) on $F$. Let $\mathscr{X}_{(i)}$ be the set of points of $\mathscr{X}$ with the dimension of the closure of $\{x\}$ equal to $i$. Let $\ell$ be a prime not equal to char$(F)$. Suppose that $F$ contains a primitive $\ell^{\rm th}$ root of unity. If $P\in\mathscr{X}_{(0)}$ is a closed point of $\mathscr{X}$ with char$(\kappa(P))=\ell$, then we assume $\kappa(P)=\kappa(P)^{\ell}$. Let $x\in\mathscr{X}_{(1)}$. We have a residue homomorphism $\partial_{x}:H^{3}(F,\mathbb{Z}/\ell(2))\to H^{2}(\kappa(x),\mathbb{Z}/\ell(1))$ ([11, §1]). We say that an element $\zeta\in H^{3}(F,\mathbb{Z}/\ell(2))$ is unramified at $x$ (or $C$) if $\zeta$ is unramified at $\nu_{x}$. Further if $P\in\mathscr{X}_{(0)}$ is in the closure of $\{x\}$, then we have a *residue homomorphism * $\partial_{P}:H^{2}(\kappa(x),\mathbb{Z}/\ell(1))\to H^{1}(\kappa(P),\mathbb{Z}/\ell)$ ([11, §1]). For $x\in\mathscr{X}_{(1)}$, if $C$ is the closure of $\{x\}$, we also denote $\partial_{x}$ by $\partial_{C}$. An element $\alpha\in H^{2}(\kappa(x),\mathbb{Z}/\ell(1))\simeq{}_{\ell}\mathrm{Br}(\kappa(x))$ is unramified at $P$ if and only if $\partial_{P}(\alpha)=0$. We use the additive notation for the group operations on $H^{2}(F,\mathbb{Z}/\ell(1))$ and $H^{3}(F,\mathbb{Z}/\ell(2))$ and multiplicative notation for the group operation on $H^{1}(F,\mathbb{Z}/\ell)$.

Proposition 4.1**.**

([11, Proposition 1.7]) Then

[TABLE]

is a complex, where the maps are given by the residue homomorphism.

Lemma 4.2**.**

([10, §3.2, Lemma 3], [11, Lemma 1.4(3)]) Let $x\in\mathscr{X}_{(1)}$ and $\nu_{x}$ be the discrete valuation on $F$ at $x$. Then $\partial_{x}:H^{3}(F_{x},\mathbb{Z}/\ell(2))\to H^{2}(\kappa(x),\mathbb{Z}/\ell(1))$ is an isomorphism. Further if $\alpha\in H^{2}(F,\mathbb{Z}/\ell(1))$ is unramified at $x$ and $f\in F^{*}$, then $\partial_{x}(\alpha\cdot(f))=\overline{\alpha}^{\nu_{x}(f)}$.

The following is a consequence of (4.1).

Corollary 4.3**.**

Let $C_{1}$ and $C_{2}$ be two irreducible regular curves in $\mathscr{X}$ interesting transversely at a closed point $P$. Let $\zeta\in H^{3}(F,\mathbb{Z}/\ell(2))$. Suppose that $\zeta$ is unramified at all codimension one points of $\mathscr{X}$ passing through $P$ except possibly at $C_{1}$ and $C_{2}$. Then

[TABLE]

Corollary 4.4**.**

Let $C$ be an irreducible curve on $\mathscr{X}$ and $P\in C$ with $C$ regular at $P$. Let $\zeta\in H^{3}(F,\mathbb{Z}/\ell(2))$. Suppose that $\zeta$ is unramified at all codimension one points of $\mathscr{X}$ passing through $P$ except possibly at $C$. If $\kappa(P)$ is finite, then $\zeta\otimes F_{P}=0$. In particular if $\kappa(P)$ is finite, then $\zeta$ is unramified at every discrete valuation of $F$ centered at $P$.

Proof.

Since $C$ is regular at $P$, there exists an irreducible curve $C^{\prime}$ passing through $P$ and intersecting $C$ transversely at $P$. Then, by (4.3), we have $\partial_{P}(\partial_{C}(\zeta))=\partial_{P}(\partial_{C^{\prime}}(\zeta))^{\ell-1}$. Since, by assumption, $\partial_{C^{\prime}}(\zeta)=0$, we have $\partial_{P}(\partial_{C}(\zeta))=1$.

Let $\pi\in A_{P}$ be a prime defining $C$ at $P$. Since $C$ is regular at $P$, $A_{P}/(\pi)$ is a discrete valued ring with residue field $\kappa(P)$ and $\kappa(C)$ is the field of fractions of $A_{P}/(\pi)$. Further $\pi$ remains a regular prime in $\hat{A}_{P}$ and $\hat{A}_{P}/(\pi)$ is the completion of $A_{P}/(\pi)$. In particular the field of fractions of $\hat{A}_{P}/(\pi)$ is the completion $\kappa(C)_{P}$ of the field $\kappa(C)$ at the discrete valuation given by the discrete valuation ring $A_{P}/(\pi)$. Let $\tilde{\nu}$ be the discrete valuation on $F_{P}$ given by the height one prime ideal $(\pi)$ of $\hat{A}$ and $\nu$ the discrete valuation of $F$ given by the height one prime ideal $(\pi)$ of $A$. Then the restriction of $\tilde{\nu}$ to $F$ is $\nu$ and the residue field $\kappa(\tilde{\nu})$ at $\tilde{\nu}$ is $\kappa(C)_{P}$.

Since $\partial_{P}(\partial_{C}(\zeta))=1$, we have $\partial_{C}(\zeta)\otimes\kappa(C)_{P}=0$ ( [11, Lemma 1.4(3)]). Since $\partial_{\tilde{\nu}}(\zeta\otimes F_{P})=\partial_{C}(\zeta)\otimes\kappa(C)_{P}=0$. Let $F_{P,\tilde{\nu}}$ be the completion of $F_{P}$ at $\tilde{\nu}$. Since $\partial_{\tilde{\nu}}:H^{3}(F_{P,\tilde{\nu}},\mathbb{Z}/\ell(2))\to H^{2}(\kappa(C)_{P},\mathbb{Z}/\ell(2))$ is an isomorphism ([11, Lemma 1.4(3)]), $\zeta\otimes F_{P,\tilde{\nu}}=0$.

Let $\nu^{\prime}$ be a discrete valuation of $F_{P}$ given by a hight one prime ideal of $\hat{A}$ not equal to $(\pi)$. Then, by the assumption on $\zeta$, $\partial_{\nu^{\prime}}(\zeta\otimes F_{P})=0$ and hence $\zeta\otimes F_{P,\nu^{\prime}}=0$ ([11, Lemma 1.4(3)]), where $F_{P,\nu^{\prime}}$ is the completion of $F_{P}$ at $\nu^{\prime}$. Hence, by ([21, Lemma 6.2]), $\zeta\otimes F_{P}=0$. ∎

5. A local global principle

Let $\mathscr{X}$, $F$ and $\ell$ be as in §4. Let $\zeta\in H^{3}(F,\mathbb{Z}/\ell(1))$. Let $\alpha=[a,b)\in H^{2}(F,\mathbb{Z}/\ell(1))$. In this section we show that under some additional assumptions on $\mathscr{X}$, $\zeta$ and $\alpha$, there exists $f\in F^{*}$ such that $\partial_{x}(\zeta-\alpha\cdot(f))$ is unramified at all the discrete valuations of $\kappa(x)$ centered at closed points of $\overline{\{x\}}$ for all $x\in\mathscr{X}_{(1)}$ (see 5.7).

Let $\zeta\in H^{3}(F,\mathbb{Z}/\ell(1))$ and $\alpha=[a,b)\in H^{2}(F,\mathbb{Z}/\ell(1))$. For the rest of this section, we assume the following.

Assumptions 5.1**.**

Suppose ($\mathscr{X}$, $\zeta$, $\alpha$) satisfy the following conditions.

1. A1)

ram${}_{\mathscr{X}}(\zeta)=\{C_{1},\cdots,C_{r}\}$, $C_{i}$’s are regular irreducible curves with normal crossings 2. A2)

ram${}_{\mathscr{X}}(\alpha)=\{D_{1},\cdots,\cdots,D_{n}\}$, $D_{j}$’s are regular curves with normal crossings and $D_{i}\neq C_{j}$ for all $i,j$

By reindexing, we have ram${}_{\mathscr{X}}(\alpha)=\{D_{1},\cdots,D_{m},\cdots,D_{n}\}$, with char$(\kappa(D_{i}))=\ell$ for $1\leq i\leq m$ and char$(\kappa(D_{j}))\neq\ell$ for $m+1\leq j\leq n$.

1. A3)

and $D_{i}\cap D_{j}=\emptyset$ for all $1\leq i\leq m$ and $m+1\leq j\leq n$ 2. A4)

if $P\in D_{i}\cap D_{j}$ for some $m+1\leq i<j\leq n$, then char$(\kappa(P))\neq\ell$ 3. A5)

there are no chilly loops (see §3) for $\alpha$ on $\mathscr{X}$ 4. A6)

$\partial_{C_{i}}(\zeta)$ is the specialization of $\alpha$ at $C_{i}$ for all $i$ 5. A7)

$C_{i}\cap D_{j}=\emptyset$ for all $i$ and $1\leq j\leq m$ 6. A8)

if $P\in C_{i}\cap D_{s}$ for some $i$ and $s$, then $P\in C_{i}\cap C_{j}$ for some $i\neq j$ 7. A9)

for every $i\neq j$, there is at most one $D_{t}$ such that $C_{i}\cap C_{j}\cap D_{t}\neq\emptyset$ 8. A10)

if $P\in\mathscr{X}_{(2)}$ with char$(\kappa(P))=\ell$ and $P\in D_{i}$ for some $i$, then $\frac{1-a}{(\rho-1)^{\ell}}\in A_{P}$. 9. A11)

if $P\in C_{i}\cap C_{j}\cap D_{t}$ for some $i<j$ and for some $t$, then $D_{t}$ is given by a regular prime $u\pi_{i}^{\ell-1}+v\pi_{j}$ at $P$, for some prime $\pi_{i}$ (resp. $\pi_{j}$) defining $C_{i}$ (resp. $C_{j}$) at $P$ and units $u,v$ at $P$

Let $\mathscr{P}$ be a finite set of closed points of $\mathscr{X}$ containing $C_{i}\cap C_{j}$, $D_{i}\cap D_{j}$ for all $i\neq j$, $C_{i}\cap D_{j}$ for all $i,j$ and at least one point from each $C_{i}$ and $D_{j}$. Let $A$ be the regular semi local ring at $\mathscr{P}$ on $\mathscr{X}$. For every $P\in\mathscr{P}$, let $M_{P}$ be the maximal ideal of $A$ at $P$. For $1\leq i\leq r$ and $1\leq j\leq n$, let $\pi_{i}\in A$ be a prime defining $C_{i}$ on $A$ and $\delta_{j}\in A$ a prime defining $D_{j}$ on $A$.

Since char$(F)\neq\ell$, $[a)$ is the extension $F(\sqrt[\ell]{a})$. Since $\alpha=[a,b)$, without loss of generality, we assume that $a,b\in A$ and both $a$ and $b$ are $\ell^{\rm th}$ power free.

Lemma 5.2**.**

For $1\leq j\leq n$, let $n_{j}=\ell\nu_{D_{j}}(\ell)+1$. Then there exists a unit $u\in A$ such that $u\prod\pi_{i}$ is an $\ell^{\rm th}$ power modulo $\delta_{j}^{n_{j}}$ for all $1\leq j\leq n$. In particular $u\prod\pi_{i}\in F_{D_{j}}^{\ell}$ for all $j$.

Proof.

Let $\pi=\prod_{1}^{r}\pi_{i}$ and $\delta=\prod_{1}^{m}\delta_{j}^{n_{j}}$. Since, by the assumption A7), $C_{i}\cap D_{j}=\emptyset$ for all $i$ and $1\leq j\leq m$, the ideals $A\pi$ and $A\delta$ are comaximal in $A$. In particular the image of $\pi$ in $A/(\delta)$ is a unit. By the Chinese remainder theorem, there exists $u_{1}\in A$ be such that $u_{1}=\pi\in A/(\delta)$, $u_{1}=1\in A/(\pi)$ and $u_{1}=1\in A/M_{P}$ for all $P\in\mathscr{P}\setminus((\cup_{1}^{r}C_{i})\cup(\cup_{1}^{m}D_{j}))$. Since the image of $\pi$ in $A/(\delta)$ is a unit, $u_{1}$ is a unit in $A$. Let $\pi^{\prime}=u_{1}^{-1}\pi$.

Let $m+1\leq s\leq n$ and $a_{s}$ be the image of $\pi^{\prime}$ in $A/(\delta_{s})$. We claim that $a_{s}=w_{s}b_{s}^{\ell}$ for some $w_{s},b_{s}\in A/(\delta_{s})$ with $w_{s}$ a unit in $A/(\delta_{s})$. Let $M$ be a maximal ideal of $A/(\delta_{s})$. Then $M=M_{P}/(\delta_{s})$ for some $P\in D_{s}\cap\mathscr{P}$. Suppose $P\not\in C_{i}$ for all $i$. Then $\pi^{\prime}$ is a unit at $P$ and hence $a_{s}$ is a unit at $M$. Suppose $P\in C_{i}$ for some $i$. Then $P\in C_{i}\cap D_{s}$. Thus, by the assumption A8), there exists $j\neq i$ such that $P\in C_{i}\cap C_{j}$. Suppose $i<j$. Then, by the assumption A11), $\delta_{s}=v_{i}\pi_{i}^{\ell-1}+v_{j}\pi_{j}$ for some units $v_{i}$ and $v_{j}$ at $P$. Hence

[TABLE]

Since $\pi_{t}$, $t\neq i,j$, is a unit at $P$ (assumption A1)), $a_{s}\equiv w_{P}\pi_{j}^{\ell}$ modulo $\delta_{s}$, for some $w_{P}\in A/(\delta_{s})$ a unit at $P$. Suppose $i>j$. Then $\delta_{s}=v_{j}\pi_{j}+v_{i}\pi_{i}^{\ell-1}$ for some units $v_{i}$ and $v_{j}$ at $P$. Hence, as above, $a_{s}\equiv w_{P}\pi_{i}^{\ell}$ modulo $\delta_{s}$, for some $w_{P}\in A/(\delta_{s})$ a unit at $P$. Hence at every maximal ideal of $A/(\delta_{s})$, $a_{s}$ is a product of a unit and an $\ell^{\rm th}$ power. Since $D_{s}$ is a regular curve on $\mathscr{X}$, $A/(\delta_{s})$ is a semi local regular ring and hence $A/(\delta_{s})$ is an UFD. In particular $a_{s}=w_{s}b_{s}^{\ell}$ for some $w_{s},b_{s}\in A/(\delta_{s})$ with $w_{s}$ a unit. Since $\pi^{\prime}=1\in A/(\delta)$, $a_{s}=\pi^{\prime}\in A/(\delta_{s})$ and $a_{s}=w_{s}b_{s}^{\ell}$, we have $w_{s}$ is an $\ell^{\rm th}$ power in $A/(\delta_{s},\delta)$. Hence $w_{s}=w_{s}^{\prime\ell}\in A/(\delta_{s},\delta)$ for some unit $w_{s}^{\prime}\in A/(\delta_{s},\delta)$. Since $D_{j}$’s have normal crossings, the image of the ideal $(\delta_{s},\delta)$ in $A/(\delta)$ is either the unit ideal or a maximal ideal. Thus, by the Chinese remainder theorem, there exists a unit $w_{\delta}\in A/(\delta)$ such that $w_{s}=w_{\delta}^{\ell}\in A/(\delta_{s},\delta)$ for all $s$. By (2.7), there exists $w^{\prime}\in A$ such that $w^{\prime}=w_{s}\in A/(\delta_{s})$, $w^{\prime}=w_{\delta}^{\ell}\in A/(\delta)$.

Then, by the Chinese remainder theorem, there exists $w\in A$ such that $w=1\in\kappa(P)$ for all $P\in\mathscr{P}\setminus(\cup_{1}^{n}D_{i})$ and $w=w^{\prime}\in A/(\delta\prod_{m+1}^{n}\delta_{s})$. Since $w_{s}\in A/(\delta_{s})$, $w_{\delta}\in A/(\delta)$ are units, $w$ is a unit in $A$.

Let $u=w^{-1}u_{1}^{-1}$. Since $u_{1}$ and $w$ are units in $A$, $u\in A$ is a unit. We have $u\prod\pi_{i}=w^{-1}\pi^{\prime}\equiv w_{s}^{-1}a_{s}=b_{s}^{\ell}$ modulo $\delta_{s}$ for $m+1\leq s\leq n$ and $u\prod\pi_{i}=w^{-1}\pi^{\prime}=w_{\delta}^{-\ell}\in A/(\delta)$. Since $\nu_{D_{j}}(\ell)=0$ for $m+1\leq j\leq n$ (assumption A2)), $u\prod\pi_{i}$ is an $\ell^{\rm th}$ power in $A/(\delta_{j}^{n_{j}})$ for $1\leq j\leq n$. Since $n_{j}=\ell\nu_{D_{j}}(\delta_{j})+1$, $u\prod\pi_{i}\in F_{D_{j}}^{\ell}$ for all $j$ (cf. [7, §0.3]). ∎

Let $u\in A$ be a unit as in (5.2) and $\pi=u\prod_{1}^{r}\pi_{i}\in A$. Then $div_{\mathscr{X}}(\pi)=\sum C_{i}+\sum_{1}^{d}t_{s}E_{s}$ for some irreducible curves $E_{s}$ with $E_{s}\cap\mathscr{P}=\emptyset$. In particular $C_{i}\neq E_{s}$, $D_{j}\neq E_{s}$ for all $i$, $j$ and $s$. Let $\mathscr{P}^{\prime}$ be a finite set of points of $\mathscr{X}$ containing $\mathscr{P}$, $C_{i}\cap E_{s}$, $D_{j}\cap E_{s}$ for all $i$, $j$ and $s$ and at least one point from each $E_{s}$. Let $A^{\prime}$ be the semi local ring at $\mathscr{P}^{\prime}$. For $1\leq i\leq n$, let $\delta_{i}^{\prime}\in A^{\prime}$ be a prime defining $D_{i}$ on $A^{\prime}$. Note that $\delta_{i}^{\prime}A^{\prime}\cap A=\delta_{i}A$ for all $i$.

Lemma 5.3**.**

*There exists $v\in A^{\prime}$ such that

$\bullet$ $v$ is a unit and $F(\sqrt[\ell]{v})/F$ is unramified at all the points $P\in\mathscr{P}^{\prime}\cap(\cup_{1}^{n}D_{i})$ except possible at the points $P$ in $D_{i}\cap D_{j}$ for all $i\neq j$ with char$(\kappa(P))\neq\ell$

$\bullet$ if char$(\kappa(D_{j}))\neq\ell$, then the extension $F(\sqrt[\ell]{v})/F$ is unramified at $D_{j}$ with the reside field of $F(\sqrt[\ell]{v})$ at $D_{j}$ is equal to $\partial_{D_{j}}(\alpha)$

$\bullet$ if char$(\kappa(D_{j}))=\ell$, then $F_{D_{i}}(\sqrt[\ell]{v})\simeq F_{D_{i}}(\sqrt[\ell]{a})$. In particular $\alpha\otimes F_{D_{j}}(\sqrt[\ell]{v})$ is trivial*

Proof.

Let $1\leq i\leq n$. Suppose char$(\kappa(P))=\ell$ for all $P\in D_{i}$. If char$(\kappa(D_{i}))\neq\ell$, let $w_{i}\in\kappa(D_{i})$ be such that $\partial_{D_{i}}(\alpha)=[w_{i})$. Suppose cjar$(\kappa(D_{i}))=\ell$. By the assumption A10), $\frac{a-1}{(\rho-1)^{\ell}}\in A_{P}$ for all $P\in D_{i}$. In particular $\frac{a-1}{(\rho-1)^{\ell}}$ is regular at $D_{i}$ and the image of $\frac{a-1}{(\rho-1)^{\ell}}$ in $\kappa(D_{i})$ is in $A^{\prime}/(\delta^{\prime}_{i})$. Let $w_{i}$ be the image of $\frac{a-1}{(\rho-1)^{\ell}}$ in $A^{\prime}/(\delta^{\prime}_{i})$.

Suppose there exists $P\in D_{i}$ with char$(\kappa(P))\neq\ell$. Then char$(\kappa(D_{i}))\neq\ell$. By ([24, Proposition 7.10]), there exists $w_{i}\in\kappa(D_{i})^{*}$ such that

$\bullet$ $\partial_{D_{j}}(\alpha)=\kappa(D_{i})(\sqrt[\ell]{w_{i}})$,

$\bullet$ $w_{i}$ is defined at all $P\in\mathscr{P}\cap D_{i}$ with char$(\kappa(P))\neq\ell$,

$\bullet$ $w_{i}$ is a unit at all $P\in(\mathscr{P}\cap D_{i})\setminus(\cup_{j\neq i}D_{j})$ with char$(\kappa(P))\neq\ell$

$\bullet$ $w_{i}(P)=w_{j}(P)$ for all $P\in D_{i}\cap D_{j}$, $i\neq j$ with char$(\kappa(P))\neq\ell$.

Let $P\in\mathscr{P}\cap D_{i}$. Suppose char$(\kappa(P))=\ell$. Then, by the assumptions A10), $[a)$ is unramified at $P$ (cf. 2.3). Since $\alpha=[a,b)$, $\partial_{D_{i}}(\alpha)=[a(D_{i}))^{\nu_{D_{i}}(b)}$. In particular $\partial_{D_{i}}(\alpha)=\kappa(D_{i})(\sqrt[\ell]{w_{i}})$ is unramified at $P$. Thus, by (2.6), we assume that $\frac{1-w_{i}}{(\rho-1)^{\ell}}$ is unit at all $P\in\mathscr{P}\cap D_{i}\setminus(\cap_{j\neq i}D_{j})$ with char$(\kappa(P))=\ell$. Since char$(\kappa(D_{i}))\neq\ell$, by assumptions A3) and A4), if $P\in D_{i}\cap D_{j}$ for some $j\neq i$, then char$(\kappa(P))\neq\ell$. Thus $\frac{1-w_{i}}{(\rho-1)^{\ell}}\in A^{\prime}/(\delta_{i})$.

Let $P\in D_{i}\cap D_{j}$ for some $i\neq j$. Suppose char$(\kappa(P))=\ell$. Then, by the assumption A3) and A4), char$(\kappa(D_{i}))=$ char$(\kappa(D_{j}))=\ell$ and by the choice of $w_{i}$, we have $w_{i}(P)=w_{j}(P)\in\kappa(P)$. Suppose char$(\kappa(P))\neq\ell$. Then, by the choice of $w_{i}$, we have $w_{i}=w_{j}\in\kappa(P)$.

Let $u_{i}=w_{i}\in A^{\prime}/(\delta^{\prime}_{i})$ if char$(\kappa(D_{i}))=\ell$ and $u_{i}=(1-w_{i})/(\rho-1)^{\ell}\in A^{\prime}/(\delta^{\prime}_{i})$ if char$(\kappa(D_{i}))\neq\ell$. By the assumptions A3) and A4), $u_{i}$ is defined at all $P\in\mathscr{P}\cap D_{i}$ for all $i$ and $u_{i}(P)=u_{j}(P)\in\kappa(P)$ for all $P\in D_{i}\cap D_{j}$ for $i\neq j$. Thus, by (2.7), there exists $u^{\prime}\in A^{\prime}$ such that $u^{\prime}=u_{i}$ modulo $(\delta^{\prime}_{i})$ for all $i$. By the Chinese remainder theorem, we get $v^{\prime}\in A^{\prime}$ such that $v^{\prime}=u^{\prime}\in A^{\prime}/(\prod\delta^{\prime}_{i})$ and $v^{\prime}=0\in\kappa(P)$ for all $P\in\mathscr{P}^{\prime}$ with $P\not\in D_{i}$ for all $i$.

We now show that $v=1-(\rho-1)^{\ell}v^{\prime}$ has all the required properties.

Let $P\in\mathscr{P}^{\prime}$. Suppose char$(\kappa(P))=\ell$. Then $\rho-1\in M_{P}$. Since $v^{\prime}\in A^{\prime}$, $v$ is a unit at $P$ and $F(\sqrt[\ell]{v})$ is unramified at $P$ (2.4). Suppose char$(\kappa(P))\neq\ell$. Suppose that $P\not\in D_{i}$ for all $i$. Then, by the choice of $v^{\prime}$, $v^{\prime}\in M_{P}$ and hence $v$ is a unit at $P$ and $F(\sqrt[\ell]{v})/F$ is unramified at $P$. Suppose that $P\in D_{i}$ for some $i$. Since char$(\kappa(P))\neq\ell$, char$(\kappa(D_{i}))\neq\ell$. Thus, by the choice of $v^{\prime}$, we have $v^{\prime}=u^{\prime}=u_{i}=(1-w_{i})/(\rho-1)^{\ell}\in A^{\prime}/(\delta^{\prime}_{i})$. Hence $v=w_{i}\in A^{\prime}/(\delta^{\prime}_{i})$. Suppose $P\not\in D_{j}$ for all $j\neq i$. Then, by the choice $w_{i}$ is a unit at $P$ and hence $v$ is a unit at $P$. In particular $F(\sqrt[\ell]{v})/F$ is unramified at $P$. Thus $v$ is a unit and $F(\sqrt[\ell]{v})/F$ is unramified at all $P\in\mathscr{P}^{\prime}$ except possibly at $P\in D_{i}\cap D_{j}$ with char$(\kappa(P))\neq\ell$.

Suppose char$(\kappa(D_{i}))\neq\ell$. Then, by the choice of $v$, we have $v=1-(\rho-1)^{\ell}v^{\prime}=1-(\rho-1)^{\ell}u_{i}=w_{i}\in A^{\prime}/(\delta^{\prime}_{i})\subset\kappa(D_{i})$. Since $w_{i}\neq 0$, $v$ is a unit at $\delta_{i}$ and $F(\sqrt[\ell]{v})$ is unramified at $D_{i}$ with residue field $\kappa(D_{i})(\sqrt[\ell]{w_{i}})=\partial_{D_{i}}(\alpha)$.

Suppose that char$(\kappa(D_{i}))=\ell$. Since $v=1-(\rho-1)^{\ell}v^{\prime}$ and $v^{\prime}=u_{i}=w_{i}\in A^{\prime}/(\delta^{\prime}_{i})$, $F(\sqrt[\ell]{v})$ is unramified at $D_{i}$ with residue field equal to $\kappa(D_{i})[X]/(X^{\ell}-X+w_{i})$ (2.3). Since $w_{i}$ is the image of $\frac{a-1}{(\rho-1)^{\ell}}$ in $A^{\prime}/(\delta^{\prime}_{i})$, the residue field of $F(\sqrt[\ell]{a})$ at $\delta^{\prime}_{i}$ is $\kappa(D_{i})[X]/(X^{\ell}-X+w_{i})$ (2.3). Hence $F_{D_{i}}(\sqrt[\ell]{v})\simeq F_{D_{i}}(\sqrt[\ell]{a})$. Since $\alpha=[a,b)$, $\alpha\otimes F_{\delta^{\prime}_{i}}(\sqrt{v})$ is trivial. ∎

Remark 5.4**.**

If $\ell$ is a unit in $A^{\prime}$, then the extension $F(\sqrt[\ell]{v})/F$ given in the above lemma is the lift of the residues of $\alpha$ which is proved in ([24, Proposition 7.10]).

Choose $u\in A$ as in (5.2) and $\pi=u\prod\pi_{i}\in A$ and $div_{\mathscr{X}}(\pi)=\sum C_{i}+\sum_{1}^{d}t_{s}E_{s}$ for some irreducible curves $E_{s}$ with $E_{s}\cap\mathscr{P}=\emptyset$ (see the paragraph before 5.3). Let $v\in A^{\prime}$ as in (5.3). Let $V_{1},\cdots,V_{q}$ be the irreducible curves in $\mathscr{X}$ where $F(\sqrt[\ell]{v\pi})$ is ramified. Since $\pi\in F_{D_{j}}^{\ell}$ (5.2) and $F(\sqrt{v})$ is unramified at $D_{j}$ (5.3) for all $j$, $V_{i}\neq D_{j}$ for all $i$ and $j$. Let $\mathscr{P}^{\prime\prime}=\mathscr{P}\cup(\cup(D_{i}\cap E_{s}))\cup(\cup(D_{i}\cap V_{j})$. After reindexing $E_{s}$, we assume that there exists $d_{1}\leq d$ such that $E_{s}\cap\mathscr{P}^{\prime\prime}\neq\emptyset$ for $1\leq s\leq d_{1}$ and $E_{s}\cap\mathscr{P}^{\prime\prime}=\emptyset$ for $d_{1}+1\leq s\leq d$.

Lemma 5.5**.**

Let $A^{\prime\prime}$ be the regular semi local ring at $\mathscr{P}^{\prime\prime}$. Then there exists $h\in F^{*}$ which is a norm from the extension $F(\sqrt[\ell]{v\pi})$ such that div${}_{\mathscr{X}}(h)=-\sum_{1}^{d_{1}}t_{i}E_{i}+\sum r_{i}E_{i}^{\prime}$, where $E_{j}^{\prime}\cap\mathscr{P}^{\prime\prime}=\emptyset$ for all $j$.

Proof.

Let $L=F(\sqrt[\ell]{v\pi})$ and $T$ be the integral closure of $A^{\prime\prime}$ in $L$.

Let $1\leq s\leq d_{1}$ and $P\in\mathscr{P}^{\prime\prime}\cap E_{s}$. Since $E_{s}\cap\mathscr{P}=\emptyset$, $P\in D_{i}\cap E_{s}$ for some $i$. Since $v$ is a unit at all $P\in(\mathscr{P}^{\prime}\setminus\mathscr{P})$ (5.3) and $D_{i}\cap E_{s}\subset\mathscr{P}^{\prime}$, $v$ is a unit at $P$ and hence $v$ is a unit at $E_{s}$.

Suppose that $t_{s}$ is coprime to $\ell$. Since $div_{\mathscr{X}}(\pi)=\sum C_{i}+\sum_{1}^{d}t_{s}E_{s}$ and $v$ is a unit at $E_{s}$, there is a unique curve $\tilde{E}_{s}$ in $T$ lying over $E_{s}$. Let $t^{\prime}_{s}=t_{s}$.

Suppose that $t_{s}=\ell t^{\prime}_{s}$. Let $\tilde{E}_{s}=t^{\prime}_{s}\sum E_{s,i}$, where $E_{s,i}$ are the irreducible divisors in $T$ which lie over $E_{s}$. Let $\tilde{E}=-\sum t^{\prime}_{s}\tilde{E}_{s}$.

We claim that $\tilde{E}$ is a principal divisor on $T$. Since $T$ is normal it is enough to check this at every maximal ideal of $T$. Let $M$ be a maximal ideal of $T$. Then $M\cap A^{\prime\prime}=M_{P}$ for some $P\in\mathscr{P}^{\prime\prime}$. Suppose $P\not\in E_{s}$ for all $1\leq s\leq d_{1}$. Then $\tilde{E}$ is trivial at $M$. Suppose that $P\in E_{s}$ for some $s$ with $1\leq s\leq d_{1}$. Then, as we have seen above, $P\in D_{i}\cap E_{s}$ for some $i$. Since $D_{i}\cap C_{j}\in\mathscr{P}$ for all $i$ and $j$ and $\mathscr{P}\cap E_{s}=\emptyset$, $P\not\in C_{i}$ for all $i$. Hence div${}_{A_{P}}(\pi)=\sum_{P\in E_{i}}t_{i}E_{i}$. Since $v$ is a unit at $P$ (5.3), div${}_{A_{P}}(v\pi)=$ div${}_{A_{P}}(\pi)$ and hence $\tilde{E}=$ $-$div$(\sqrt[\ell]{v\pi})$ at $M$. In particular $\tilde{E}$ is principal at $M$. Hence $\tilde{E}=div_{T}(g)$ for some $g\in L$. Let $h=N_{L/F}(g)$. Then div${}_{A^{\prime\prime}}(h)=-\sum_{1}^{d_{1}}t_{i}E_{i}$ and hence $h$ has the required properties. ∎

Lemma 5.6**.**

Let $h\in F^{*}$ be as in (5.5) with div${}_{\mathscr{X}}(h)=-\sum_{1}^{d_{1}}t_{i}E_{i}+\sum r_{j}E_{j}^{\prime}$. Then $\alpha$ is unramified at $E_{j}^{\prime}$. Further, if $r_{j}$ is coprime to $\ell$ for some $j$, then the specialization of $\alpha$ at $E_{j}^{\prime}$ is unramified at every discrete valuation of $\kappa(E_{j}^{\prime})$ which is centered on $E_{j}^{\prime}$.

Proof.

Since $E_{j}^{\prime}\cap\mathscr{P}^{\prime\prime}=\emptyset$ and $D_{i}\cap\mathscr{P}^{\prime\prime}\neq\emptyset$ for all $i$, $E_{j}^{\prime}\neq D_{i}$ for all $i$. Hence, by the assumption A2), $\alpha$ is unramified at $E_{j}^{\prime}$.

Let $P$ be a closed point of $E_{j}^{\prime}$ for some $j^{\prime}$ with $r_{j}$ coprime to $\ell$. Let $L=F(\sqrt[\ell]{v\pi})/F$ and $B_{P}$ be the integral closure of $A_{P}$ in $L$. We first show that $\alpha\otimes_{F}L$ is unramified on $B_{P}$.

Suppose $P\not\in D_{i}$ for all $i$. Then $\alpha$ is unramified at $P$ (assumption A2)). Hence there exists an Azumaya algebra $\mathscr{A}_{P}$ over $A_{P}$ such that the class of $\mathscr{A}_{P}\otimes_{A_{P}}F$ is $\alpha$ (cf. 3.1). In particular $\alpha\otimes_{F}L$ is the class of $\mathscr{A}_{P}\otimes_{A_{P}}B_{P}$ and hence $\alpha\otimes_{F}L$ is unramified on $B_{P}$.

Suppose $P\in D_{i}$ for some $i$. Since $E_{j}^{\prime}\cap\mathscr{P}^{\prime\prime}=\emptyset$ (5.5), $P\not\in\mathscr{P}^{\prime\prime}$. Since $\cup(V_{i^{\prime}}\cap D_{i})\subset\mathscr{P}^{\prime\prime}$, $P\not\in\cup V_{i^{\prime}}$ for all $i^{\prime}$ and hence $L$ is unramified at $P$. Hence $B_{P}$ is a regular semi local domain. Let $Q\subset B_{P}$ be a hight one prime ideal and $Q_{0}=Q\cap A_{P}$. Then $Q$ is a height one prime ideal of $A_{P}$. If $\alpha$ is unramified at $Q_{0}$, then $\alpha\otimes_{F}L$ is unramified at $Q$. Suppose that $\alpha$ is ramified at $Q_{0}$. Since $P\not\in D_{j}$ for $j\neq i$, $Q_{0}$ is the prime ideal corresponding to $D_{i}$. Since $\pi\in F_{D_{i}}^{\ell}$ (5.2), $F_{D_{i}}(\sqrt[\ell]{v\pi})=F_{D_{i}}(\sqrt[\ell]{v})$. Suppose that char$(\kappa(D_{i}))\neq\ell$. Since $L/F$ is unramified at $D_{i}$ with residue field equal to $\partial_{D_{i}}(\alpha)$ (5.3), $\alpha\otimes L/F$ is unramified at $Q$ (cf. [15, Lemma 4.1]). Suppose that char$(\kappa(D_{i}))=\ell$. Since $\alpha\otimes F_{D_{i}}(\sqrt[\ell]{v})$ is trivial (5.3), $\alpha\otimes_{F}L$ is unramified at $Q$. Since $B_{P}$ is a regular semi local ring of dissension two, $\alpha\otimes F(\sqrt[\ell]{v\pi})$ is unramified at $B_{P}$ (cf. [13, Lemma 3.1]). Let $\mathscr{A}_{P}$ be an Azumaya algebra over $B_{P}$ such that $\alpha\otimes_{F}L$ is the class of $\mathscr{A}_{P}\otimes_{B_{P}}L$.

Since $E_{j}$ is in the support of $h$, $r_{j}$ is coprime to $\ell$ and $h$ is a norm from $F(\sqrt[\ell]{v\pi})$, $v\pi\in F_{E_{j}}^{\ell}$. Let $\theta\in A_{P}$ be a prime defining $E_{j}$ at $P$. Then $\theta=\theta_{1}\cdots\theta_{\ell}$ for distinct primes $\theta_{i}\in B_{P}$ and $A_{P}/(\theta)\subset B_{P}/(\theta_{s})\subset\kappa(E_{j})$ for all $s$. Let $\beta\in H^{2}(\kappa(E_{j}^{\prime}),\mathbb{Z}/\ell(1))$ be the specialization of $\alpha$ at $E_{j}^{\prime}$. Then $\beta$ is the class of $\mathscr{A}_{P}\otimes_{B_{P}/(\theta_{i})}\kappa(E_{j})$. Hence $\beta$ is unramified at $B_{P}/(\theta_{s})$. Since $B_{P}/(\theta_{s})$ is integral over $A/(\theta)$, $\beta$ is unramified at all discrete valuations of $\kappa(E_{j})$ which are centered at $A_{P}/(\theta)$. ∎

Theorem 5.7**.**

Suppose $(\mathscr{X},\zeta,\alpha)$ satisfies the assumption in 5.1. Then there exists $f\in K^{*}$ such that for every $x\in\mathscr{X}_{(1)}$, $\partial_{x}(\zeta-\alpha\cdot(f))$ is unramified at every discrete valuation of $\kappa(x)$ centered on the closure of $\{x\}$.

Proof.

Let $\mathscr{P}$ be a finite set of closed points of $\mathscr{X}$ containing $C_{i}\cap C_{j}$ for all $i\neq j$, $D_{i}\cap D_{j}$ for all $i\neq j$, $C_{i}\cap D_{j}$ for all $i$ and $j$ and at least one point from each $C_{i}$ and $D_{j}$. Let $A$ be the semi local ring at $\mathscr{P}$. Let $\pi_{i}\in A$ be primes defining $C_{i}$.

Let $u\in A$ be a unit in $A$ as in (5.2) and $\pi=u\prod_{1}^{r}\pi_{i}\in A$. Then $div_{\mathscr{X}}(\pi)=\sum C_{i}+\sum_{i}^{d}t_{i}E_{i}$ with $E_{s}\cap\mathscr{P}=\emptyset$. In particular $C_{i}\neq E_{s}$, $D_{j}\neq E_{s}$ for all $i$, $j$ and $s$. Let $\mathscr{P}^{\prime}$ be a finite set of points containing $\mathscr{P}$, $C_{i}\cap E_{s}$, $D_{j}\cap E_{s}$ for all $i$, $j$ and $s$ and at least one point from each $E_{s}$. Let $A^{\prime}$ be the semi local ring at $\mathscr{P}^{\prime}$. Let $v\in A^{\prime}$ be as in (5.3).

Let $V_{1},\cdots,V_{q}$ be the irreducible curves in $\mathscr{X}$ where $F(\sqrt[\ell]{v\pi})$ is ramified. Let $\mathscr{P}^{\prime\prime}=\mathscr{P}\cup(\cup(D_{j}\cap E_{s}))\cup(\cup(D_{j}\cap E_{s}))$. After reindexing $E_{s}$, we assume that there exists $d_{1}\leq d$ such that $E_{s}\cap\mathscr{P}^{\prime\prime}\neq\emptyset$ for $1\leq s\leq d_{1}$ and $E_{s}\cap\mathscr{P}^{\prime\prime}=\emptyset$ for $d_{1}+1\leq s\leq d$. Let $A^{\prime\prime}$ be the regular semi local ring at $\mathscr{P}^{\prime\prime}$. Let $h\in F^{*}$ be as in (5.5).

We claim that $f=h\pi$ has the required properties, i.e. $\partial_{x}(\zeta-\alpha\cdot(f))$ is unramified at every discrete valuation of $\kappa(x)$ for all $x\in\mathscr{X}_{(1)}$.

Let $x\in\mathscr{X}_{(1)}$ and $D$ be the closure of $\{x\}$. Suppose $D=C_{i}$ for some $i$. Then $h$ is a unit at $C_{i}$ (5.5), $\alpha$ is unramified at $C_{i}$ (assumption A2)) and $\pi$ is a parameter at $C_{i}$, we have $\partial_{C_{i}}(\alpha\cdot(f))$ is the specialization of $\alpha$ at $C_{i}$ (4.2). Hence, by the assumption A6), $\partial_{C_{i}}(\zeta-\alpha\cdot(f))=0$.

Suppose that $D=D_{j}$ for some $j$. By the assumption A2), $\partial_{D_{j}}(\zeta)=0$. Suppose $\alpha$ is unramified at $D_{j}$. Since $\pi$ and $h$ are units at $D_{j}$, $\partial_{D_{j}}(\alpha\cdot(f))=0$ (4.2). Suppose $\alpha$ is ramified at $D_{j}$. If char$(\kappa(D_{j}))=\ell$, then by the choice $\alpha\otimes F_{D_{j}}(\sqrt[\ell]{v})=0$ (5.3). Suppose that char$(\kappa(D_{j}))\neq\ell$. Since $F_{D_{j}}(\sqrt[\ell]{v})$ is unramified with residue field equal to $\partial_{D_{j}}(\alpha)$ (5.3), we have $\alpha\otimes F_{D_{j}}(\sqrt[\ell]{v})=0$ (3.2). In particular, in either case, $\alpha\cdot(g)=0\in H^{3}(F_{D_{j}}(\sqrt{v}),\mathbb{Z}/\ell(2))$. Since $\pi\in F_{D_{i}}^{\ell}$ (5.2), $L\otimes F_{D_{j}}=F_{D_{j}}(\sqrt[\ell]{v})$ and $\alpha\cdot(\pi)=0\in H^{3}(F_{D_{j}},\mathbb{Z}/\ell(2))$. Thus $\alpha\cdot(h)=$ cor${}_{L/F}(\alpha\cdot(g))=0\in H^{3}(F_{D_{j}},\mathbb{Z}/\ell(2))$ and $\partial_{D_{j}}(\alpha\cdot(h))=0$. Hence $\partial_{D_{j}}(\zeta-\alpha\cdot(f))=0$.

Suppose $D\neq C_{i}$ and $D_{j}$ for all $i$ and $j$. Then $\partial_{D}(\zeta)=0$ and $\alpha$ is unramified at $D$. If $\nu_{D}(f)$ is a multiple of $\ell$, then $\partial_{D}(\alpha\cdot(f))=0$. Suppose that $\nu_{D}(f)$ is coprime to $\ell$. Since $div_{\mathscr{X}}(\pi)=\sum C_{i}+\sum_{1}^{d}t_{i}E_{i}$ (5.2), div${}_{\mathscr{X}}(h)=-\sum_{1}^{d_{1}}t_{s}E_{s}+\sum r_{i}E^{\prime}_{i}$ (5.5) and $f=h\pi$, we have div${}_{\mathscr{X}}(f)=\sum C_{i}+\sum_{d_{1}+1}^{d}t_{s}E_{s}+\sum r_{i}E_{i}^{\prime}$. Since $\nu_{D}(f)$ is coprime to $\ell$ and $D\neq C_{i}$ for all $i$, $D=E_{s}$ for some $d_{1}+1\leq s\leq d$ or $D=E^{\prime}_{i}$ for some $i$.

If $D=E^{\prime}_{i}$, then by (5.6), $\overline{\alpha}$ is unramified at every discrete valuation of $\kappa(D)$ centered on $D$. Suppose $D=E_{s}$ for some $d_{1}+1\leq s\leq d$. Then by the choice of $d_{1}$, $E_{s}\cap\mathscr{P}^{\prime\prime}=\emptyset$ and hence $E_{s}\cap D_{j}=\emptyset$ for all $j$. Let $P\in E_{s}$. Then $\alpha$ is unramified at $P$ (assumption A2)) and hence $\overline{\alpha}$ is unramified at $P$. In particular $\overline{\alpha}$ is unramified at every discrete valuation of $\kappa(E_{s})$ centered at $P$. Since $\alpha$ is urnamified at $E_{s}$, $\partial_{E_{s}}(\alpha\cdot(f))=\overline{\alpha}^{\nu_{E_{s}}(f)}$ (4.2). Since $\overline{\alpha}$ is unramified at every discrete valuation of $\kappa(E_{s})$ centered on $E_{s}$, $\partial_{E_{s}}(\alpha\cdot(f))$ is unramified at every discrete valuation of $\kappa(E_{s})$ centered on $E_{s}$. Hence $f$ has the required property. ∎

6. Divisibility of elements in $H^{3}$ by symbols in $H^{2}$

Let $K$ be a global field or a local field and $F$ the function field of a curve over $K$. If $K$ is a number field or a local field, let $R$ be the ring of integers in $K$. If $K$ is a global field of positive characteristic, let $R$ be the field of constants of $K$. Let $\mathscr{X}$ be a regular proper model of $F$ over Spec$(R)$. Let $\ell$ be a prime not equal to char$(K)$. Suppose that $K$ contains a primitive $\ell^{\rm th}$ root of unity $\rho$. Then for any $P\in\mathscr{X}_{(2)}$, $\kappa(P)$ is a finite field. Hence if char$(\kappa(P))=\ell$, then $\kappa(P)=\kappa(P)^{\ell}$.

Thus we have a complex (cf. 4.1)

[TABLE]

Let $\zeta\in H^{3}(F,\mathbb{Z}/\ell(2))$ and $\alpha=[a,b)\in H^{2}(F,\mathbb{Z}/\ell(1))$. In this section we prove (see 6.5) a certain local global principle for divisibility of $\zeta$ by $\alpha$ if $(\mathscr{X},\zeta,\alpha)$ satisfies certain assumptions (see 6.3).

For a sequence of blow-ups $\eta:\mathscr{Y}\to\mathscr{X}$ and for an irreducible curve $C$ in $\mathscr{X}$, we denote the strict transform of $C$ in $\mathscr{Y}$ by $C$ itself.

We begin with the following

Lemma 6.1**.**

Suppose $(\mathscr{X},\zeta,\alpha)$ satisfies the assumption A1) of 5.1. Let $\mathscr{Y}\to\mathscr{X}$ be a sequence of blow ups centered on closed points of $\mathscr{X}$ which are not in $C_{i}\cap C_{j}$ for all $i\neq j$. Let $1\leq I\leq 11$ with $I\neq 3,5,7$. If $(\mathscr{X},\zeta,\alpha)$ satisfies the assumption AI) of 5.1, then $(\mathscr{Y},\zeta,\alpha)$ also satisfies the assumption AI).

Proof.

Let $Q$ be a closed point of $\mathscr{X}$ which is not in $C_{i}\cap C_{j}$ for $i\neq j$ and $\eta:\mathscr{Y}\to\mathscr{X}$ a simple blow-up at $Q$. It is enough to prove the lemma for $(\mathscr{Y},\zeta,\alpha)$.

Let $E$ be the exceptional curve in $\mathscr{Y}$. Since $Q\not\in C_{i}\cap C_{j}$ for $i\neq j$ and $(\mathscr{X},\zeta,\alpha)$ satisfies A1) of 5.1, by (4.4), $\zeta$ is unramfied at $E$.

Let $1\leq I\leq 11$ with $I\neq 3,5,7$. Suppose further $I\neq 4,10$. Since the exceptional curve $E$ is not in ram${}_{\mathscr{Y}}(\zeta)$, if $(\mathscr{X},\zeta,\alpha)$ satisfies the assumption AI) of 5.1, then $(\mathscr{Y},\zeta,\alpha)$ also satisfies the same assumption.

Suppose $(\mathscr{X},\zeta,\alpha)$ satisfies the assumption A4) of 5.1. Suppose char$(\kappa(Q))=\ell$. Then char$(\kappa(E))=\ell$ and hence $(\mathscr{Y},\zeta,\alpha)$ also satisfies the assumption A4) of 5.1. Suppose char$(\kappa(Q))\neq\ell$. Then char$(\kappa(P))\neq\ell$ for all $P\in E$ and hence $(\mathscr{Y},\zeta,\alpha)$ also satisfies the assumption A4) of 5.1.

Suppose $(\mathscr{X},\zeta,\alpha)$ satisfies the assumption A10) of 5.1. If char$(\kappa(Q))\neq\ell$, then char$(\kappa(P))\neq\ell$ for all $P\in E$ and hence $(\mathscr{Y},\zeta,\alpha)$ also satisfies the assumption A10) of 5.1. Suppose that char$(\kappa(Q))=\ell$. If $Q\not\in D_{i}$ for any $i$, then $\alpha$ is unramified at $Q$ and hence $\alpha$ is unramified at $E$. In particular $E\not\in$ ram${}_{\mathscr{Y}}(\alpha)$ and hence $(\mathscr{Y},\zeta,\alpha)$ also satisfies the assumption A10) of 5.1. Suppose $Q\in D_{i}$ for some $i$. Since $(\mathscr{X},\zeta,\alpha)$ satisfies A10) of 5.1, $\frac{1-a}{(\rho-1)^{\ell}}\in A_{Q}$. Let $P\in E$. Since $A_{Q}\subset A_{P}$, $\frac{1-a}{(\rho-1)^{\ell}}\in A_{P}$. Hence $(\mathscr{Y},\zeta,\alpha)$ also satisfies the assumption A10) of 5.1. ∎

Lemma 6.2**.**

Let $\mathscr{Y}\to\mathscr{X}$ be a sequence of blow ups centered on closed points $Q$ of $\mathscr{X}$ with char$(\kappa(Q))\neq\ell$. Suppose $(\mathscr{X},\zeta,\alpha)$ satisfy the assumptions A1) and A2). If $(\mathscr{X},\zeta,\alpha)$ satisfies the assumption A3) or A7) of 5.1, then $(\mathscr{Y},\zeta,\alpha)$ also satisfies the same assumption.

Proof.

Let $Q$ be a closed point of $\mathscr{X}$ with char$(\kappa(Q))\neq\ell$ and $E$ the exceptional curve in $\mathscr{Y}$. Since char$(\kappa(E))\neq\ell$ and for any closed point $P$ of $E$ char$(\kappa(P))\neq\ell$, the lemma follows. ∎

Assumptions 6.3**.**

Suppose $(\mathscr{X},\zeta,\alpha)$ satisfies the following.

1. B1)

ram${}_{\mathscr{X}}(\zeta)=\{C_{1},\cdots,C_{r}\}$, $C_{i}$’s are irreducible regular curves with normal crossings 2. B2)

ram${}_{\mathscr{X}}(\alpha)=\{D_{1},\cdots,D_{n}\}$ with $D_{j}$’s irreducible curves such that $C_{i}\neq D_{j}$ for all $i$ and $j$ 3. B3)

if $D_{s}\cap C_{i}\cap C_{j}\neq\emptyset$ for some $s$, $i\neq j$, then char$(\kappa(D_{s}))\neq\ell$ 4. B4)

if $P\in D_{j}$ for some $1\leq j\leq n$ with char$(\kappa(P))=\ell$, then $\frac{1-a}{(\rho-1)^{\ell}}\in A_{P}$ 5. B5)

$\partial_{C_{i}}(\zeta)$ is the specialization of $\alpha$ at $C_{i}$ for all $i$ 6. B6)

if $\ell=2$, then $\zeta\otimes F\otimes K_{\nu}$ is trivial for all real places $\nu$ of $K$ 7. B7)

if $\ell=2$, then $a$ is a sum of two squares in $F$ 8. B8)

for $1\leq i<j\leq r$, there exists at most one $D_{s}$ with $D_{s}\cap C_{i}\cap C_{j}\neq\emptyset$ and if $P\in D_{s}\cap C_{i}\cap C_{j}$, then $D_{s}$ is defined by $u\pi_{i}^{\ell-1}+v\pi_{j}$ at $P$ for some units $u$ and $v$ at $P$ and $\pi_{i},\pi_{j}$ primes defining $C_{i}$ and $C_{j}$ at $P$.

Let $\mathscr{P}$ be a finite set of closed points of $\mathscr{X}$ containing $C_{i}\cap C_{j}$, $D_{i}\cap D_{j}$ for all $i\neq j$, $C_{i}\cap D_{j}$ for all $i,j$ and at least one point from each $C_{i}$ and $D_{j}$. Let $A$ be the regular semi local ring at $\mathscr{P}$ on $\mathscr{X}$. For every $P\in\mathscr{P}$, let $M_{P}$ be the maximal ideal of $A$ at $P$. For $1\leq i\leq r$ and $1\leq j\leq n$, let $\pi_{i}\in A$ be a prime defining $C_{i}$ on $A$ and $\delta_{j}\in A$ a prime defining $D_{j}$ on $A$.

Lemma 6.4**.**

Suppose $(\mathscr{X},\zeta,\alpha)$ satisfies the assumptions 6.3. Let $\mathscr{Y}\to\mathscr{X}$ be a sequence of blow ups centered on closed points of $\mathscr{X}$ which are not in $C_{i}\cap C_{j}$ for $i\neq j$. Then $(\mathscr{Y},\zeta,\alpha)$ also satisfies the assumptions 6.3.

Proof.

Let $Q$ be a closed point of $\mathscr{X}$ which is not in $C_{i}\cap C_{j}$ for $i\neq j$ and $\eta:\mathscr{Y}\to\mathscr{X}$ a simple blow-up at $Q$. It is enough to show that $(\mathscr{Y},\zeta,\alpha)$ satisfies the assumptions 6.3.

Let $E$ be the exceptional curve in $\mathscr{Y}$. Since $Q\not\in C_{i}\cap C_{j}$ for $i\neq j$, by (4.4), $\zeta$ is unramfied at $E$. Hence ram${}_{\mathscr{Y}}(\zeta)=\{C_{1},\cdots,C_{r}\}$ and $(\mathscr{Y},\zeta,\alpha)$ satisfies B1). We have ram${}_{\mathscr{Y}}(\alpha)\subset\{D_{1},\cdots,D_{n},E\}$ and hence $(\mathscr{Y},\zeta,\alpha)$ satisfies B2). Since $E\cap C_{i}\cap C_{j}=\emptyset$ for all $i\neq j$, $(\mathscr{Y},\zeta,\alpha)$ satisfies B3) and B8).

Suppose $Q\in D_{i}$ for some $i$ with char$(\kappa(Q))=\ell$. Then, by B4), $\frac{1-a}{(\rho-1)^{\ell}}\in A_{Q}$ and hence $\frac{1-a}{(\rho-1)^{\ell}}\in A_{P}$ for all closed points $P$ of $E$. Suppose that $Q\not\in D_{i}$ of any $i$. Then $\alpha$ is unramified at $Q$. In particular $\alpha$ is unramified at $E$ and hence $E\not\in$ ram${}_{\mathscr{Y}}(\alpha)$. Thus and $(\mathscr{Y},\zeta,\alpha)$ satisfies B4).

Since $E$ is not in ram${}_{\mathscr{Y}}(\alpha)$, $(\mathscr{Y},\zeta,\alpha)$ satisfies B5).

Since B6) and B7) do not depend on the model, $(\mathscr{Y},\zeta,\alpha)$ satisfies all the assumptions 6.3. ∎

Theorem 6.5**.**

Let $K$, $F$ and $\mathscr{X}$ be as above. Let $\zeta\in H^{3}(F,\mathbb{Z}/\ell(2))$ and $\alpha=[a,b)\in H^{2}(F,\mathbb{Z}/\ell(1))$. Suppose that $F$ contains a primitive $\ell^{\rm th}$ root of unity. If $(\mathscr{X},\zeta,\alpha)$ satisfies the assumptions 6.3, then there exists $f\in F^{*}$ such that $\zeta=\alpha\cdot(f)$.

Proof.

Suppose $(\mathscr{X},\zeta,\alpha)$ satisfies the assumptions 6.3. First we show that there exists a sequence of blow ups $\eta:\mathscr{Y}\to\mathscr{X}$ such that $(\mathscr{Y},\zeta,\alpha)$ satisfies the assumptions 5.1.

Let $P\in\mathscr{X}_{(2)}$. Suppose $P\in D_{s}$ for some $s$ and $D_{s}$ is not regular at $P$ or $P\in D_{s}\cap D_{t}$ for some $s\neq t$. Then, by the assumption B8), $P\not\in C_{i}\cap C_{j}$ for all $i\neq j$. Thus, there exists a sequence of blow ups $\mathscr{X}^{\prime}\to\mathscr{X}$ at closed points which are not in $C_{i}\cap C_{j}$ for all $i\neq j$ such that ram${}_{\mathscr{X}^{\prime}}(\alpha)$ is a union of regular with normal crossings. By (6.4), $\mathscr{X}^{\prime}$ also satisfies the assumptions 6.3. Thus, replacing $\mathscr{X}$ by $\mathscr{X}^{\prime}$ we assume that $(\mathscr{X},\zeta,\alpha)$ satisfies the assumptions 6.3 and $D_{i}$’s are regular with normal crossings. In particular $(\mathscr{X},\zeta,\alpha)$ satisfies the assumptions A1) and A2) of 5.1.

Suppose there exists $i\neq j$ and $P\in D_{i}\cap D_{j}$ such that char$(\kappa(D_{i}))\neq\ell$, char$(D_{j})\neq\ell$ and char$(\kappa(P))=\ell$. Let $\mathscr{X}^{\prime}\to\mathscr{X}$ be the blow-up at $P$ and $E$ the exceptional curve in $\mathscr{X}^{\prime}$. Then char$(\kappa(E))=$ char$(\kappa(P))=\ell$ and $D_{i}\cap D_{j}\cap E=\emptyset$ in $\mathscr{X}^{\prime}$. By the assumption B8), $P\not\in C_{i^{\prime}}\cap C_{j^{\prime}}$ for all $i^{\prime}\neq j^{\prime}$ and hence $\mathscr{X}^{\prime}$ satisfies assumptions of 6.3 (cf. 6.4) and assumptions A1) and A2) of 5.1 (cf. 6.1). Thus replacing $\mathscr{X}$ by a sequence of blow-ups at closed points in $D_{i}\cap D_{j}$ for $i\neq j$, we assume that $\mathscr{X}$ satisfies assumptions of 6.3 and assumptions A1), A2) and A4) of 5.1.

Since $(\mathscr{X},\zeta,\alpha)$ satisfies the assumptions B4), B5) and B8) of 6.3 $(\mathscr{X},\zeta,\alpha)$ satisfies the assumptions A6), A9), A10) and A11) of 5.1.

Suppose $P\in C_{i}\cap D_{s}$ for some $i,s$ and $P\not\in C_{j}$ for all $j\neq i$. Since $\zeta$ is unramified at $P$ except at $C_{i}$, $\partial_{C_{i}}(\zeta)$ is zero over $\kappa(C_{i})_{P}$ (4.4). By the assumption B5), we have $\partial_{C_{i}}(\zeta)=\overline{\alpha}$. Thus, by (3.3), $\alpha\otimes F_{P}=0$. Let $\mathscr{X}^{\prime}\to\mathscr{X}$ be the blow-up at $P$ and $E$ the exceptional curve in $\mathscr{X}^{\prime}$. Since $\alpha\otimes F_{P}=0$ and $F_{P}\subset F_{E}$, $\alpha$ is unramified at $E$ and hence ram${}_{\mathscr{X}^{\prime}}(\alpha)=\{D_{1},\cdots,D_{n}\}$. Note that $C_{i}\cap D_{s}=\emptyset$ in $\mathscr{X}^{\prime}$. Hence $(\mathscr{X}^{\prime},\zeta,\alpha)$ satisfies the assumption A8) of 5.1. Since $P\not\in C_{j}$ for all $j\neq i$, $(\mathscr{X}^{\prime},\zeta,\alpha)$ satisfies the assumptions 6.3 (6.4) and assumptions 5.1 except possibly A3), A5) and A7) (6.1). Thus, replacing $\mathscr{X}$ by $\mathscr{X}^{\prime}$ we assume that $(\mathscr{X},\zeta,\alpha)$ satisfies assumptions 6.3 and the assumptions of 5.1 except possibly A3), A5) and A7).

Let ram${}_{\mathscr{X}}(\alpha)=\{D_{1},\cdots,D_{m},D_{m+1},\cdots,D_{n}\}$ with char$(\kappa(D_{s}))=\ell$ for $1\leq s\leq m$ and char$(\kappa(D_{t}))\neq\ell$ for $m+1\leq t\leq n$. Suppose $D_{s}\cap D_{t}\neq\emptyset$ for some $1\leq s\leq m$ and $m+1\leq t\leq n$. Let $P\in D_{s}\cap D_{t}$. Then char$(\kappa(P))=\ell$ and hence $\frac{a-1}{(\rho-1)^{\ell}}\in A_{P}$ (assumption B4)). In particular $[a)$ is unramified at $P$ (cf. 2.3). Since $\alpha$ is ramified at $D_{t}$, $\nu_{D_{t}}(b)$ is coprime to $\ell$ and hence there exists $i$ such that $\nu_{D_{s}}(b)+i\nu_{D_{t}}(b)$ is divisible by $\ell$. Let $\mathscr{X}_{1}\to\mathscr{X}$ be the blow-up at $P$ and $E_{1}$ the exceptional curve in $\mathscr{X}_{1}$. We have $\nu_{E_{1}}(b)=\nu_{D_{s}}(b)+\nu_{D_{t}}(b)$. Let $Q_{1}$ be the point in $E_{1}\cap D_{t}$ and $\mathscr{X}_{2}\to\mathscr{X}_{1}$ be the blow-up at $Q_{1}$. Let $E_{2}$ be the exceptional curve in $\mathscr{X}_{2}$. We have $\nu_{E_{2}}(b)=\nu_{E_{1}}(b)+\nu_{D_{t}}(b)=\nu_{D_{s}}(b)+2\nu_{D_{t}}(b)$. Continue this process $i$ times and get $\mathscr{X}_{i}\to\mathscr{X}_{i-1}$ and $E_{i}$ the exceptional curve in $\mathscr{X}_{i}$. Then $\nu_{E_{i}}(b)=\nu_{D_{t}}(b)+i\nu_{D}(b)$ is divisible by $\ell$. Since $[a)$ is unramified at $P$, $\alpha$ is unramified at $E_{i}$. Since char$(\kappa(E_{j}))=\ell$ for all $j$, $E_{i-1}\cap D_{t}=\emptyset$ in $\mathscr{X}_{i}$ and $E_{i}$ is in not in ram${}_{\mathscr{X}_{i}}(\alpha)$. Since $P\not\in C_{i}\cap C_{j}$ for all $i\neq j$ (assumption B4)), $\mathscr{X}_{i}$ satisfies assumptions 6.3 (cf. 6.4). Thus, replacing $\mathscr{X}$ by $\mathscr{X}_{i}$, we assume that $D_{s}\cap D_{t}=\emptyset$ for all $1\leq s\leq m$ and $m+1\leq t\leq n$ and $\mathscr{X}$ satisfies assumptions 6.3. Thus $\mathscr{X}$ satisfies all the assumptions of 5.1 expect possibly A5) and A7) (cf. 6.1 ).

Suppose $C_{i}\cap D_{t}\neq\emptyset$ for some $i$ and $t$. Since $(\mathscr{X},\zeta,\alpha)$ satisfies the assumptions A8) and A9) of 5.1, there exists $j\neq i$ such that $C_{i}\cap C_{j}\cap D_{t}\neq\emptyset$. Since $(\mathscr{X},\zeta,\alpha)$ satisfies the assumption B3) of 6.3, char$(\kappa(D_{t}))\neq\ell$. Hence $C_{i}\cap D_{t}=\emptyset$ for all $i$ and $1\leq t\leq m$. In particular $(\mathscr{X},\zeta,\alpha)$ satisfies the assumption A7) of 5.1 and hence $(\mathscr{X},\zeta,\alpha)$ satisfies all the assumptions of 5.1 except possibly A5).

Let $P\in\mathscr{X}_{(2)}$. Suppose that $P$ is a chilly point for $\alpha$. Then $P\in D_{s}\cap D_{t}$ for some $D_{s},D_{t}\in$ ram${}_{\mathscr{X}}(\alpha)$ with $D_{s}\neq D_{t}$ with char$(\kappa(P))\neq\ell$. In particular $P\not\in C_{i}\cap C_{j}$ for all $i\neq j$ (assumption B8)). Since there is a sequence of blow-ups $\mathscr{Y}\to\mathscr{X}$ centered on chilly points of $\alpha$ on $\mathscr{X}$ with no chilly loops on $\mathscr{Y}$ (3.4), by (6.1, 6.2), replacing $\mathscr{X}$ by $\mathscr{Y}$, we assume that $(\mathscr{X},\zeta,\alpha)$ satisfies assumptions 6.3 and 5.1.

Thus, by (5.7), there exists $f\in F^{*}$ such that for every $x\in\mathscr{X}_{(1)}$, $\partial_{x}(\zeta-\alpha\cdot(f))$ is unramified at every discrete valuation of $\kappa(x)$ centered at a closed point of the closure $\overline{\{x\}}$ of $\{x\}$. Since $\kappa(x)$ is a global field or a local field, every discrete valuation of $\kappa(x)$ is centered on a closed point of $\overline{\{x\}}$. Hence $\partial_{x}(\zeta-\alpha\cdot(f))$ is unramified at every discrete valuation of $\kappa(x)$.

For place $\nu$ of $K$, let $K_{\nu}$ be the completion of $K$ at $\nu$ and $F_{\nu}=F\otimes_{K}K_{\nu}$.

Let $\nu$ be a real place of $K$. Since $a$ is a sum of two squares in $F$, $a$ is a norm from the extension $F_{\nu}(\sqrt{-1})$. Let $\tilde{a}\in F_{\nu}(\sqrt{-1})$ with norm equal to $a$. Since $H^{2}(F_{\nu}(\sqrt{-1}),\mathbb{Z}/2(2))=0$ ([26, p. 80]) and cor${}_{F_{\nu}(\sqrt{-1})/F_{\nu}}[\tilde{a},b)=[a,b)\otimes F_{\nu}$, $\alpha=[a,b)=0\in H^{2}(F_{\nu},\mathbb{Z}/2(2))$. Since, by assumption $\zeta\otimes F_{\nu}=0$, $\zeta-\alpha\cdot(f)=0\in H^{3}(F_{\nu},\mathbb{Z}/2(2))$.

Let $x\in\mathscr{X}_{(1)}$. Since $\zeta-\alpha\cdot(f)=0\in H^{3}(F_{\nu},\mathbb{Z}/2(2))$ for all real places $\nu$ of $K$, it follows that $\partial_{x}(\zeta-\alpha\cdot(f))=0\in H^{2}(\kappa(x)_{\nu^{\prime}},\mathbb{Z}/2(1))$ for all real places $\nu^{\prime}$ of $\kappa(x)$. Since $\partial_{x}(\zeta-\alpha\cdot(f))$ is unramified at every discrete valuation of $\kappa(x)$, $\partial_{x}(\zeta-\alpha\cdot(f))=0$ (cf. [4, p. 130]). Hence $\zeta-\alpha\cdot(f)$ is unramified on $\mathscr{X}$.

Let $\nu$ be a finite place of $K$. Since $\zeta-\alpha\cdot(f)$ is unramified on $\mathscr{X}$, $(\zeta-\alpha\cdot(f))\otimes_{F}F_{\nu})=0\in H^{3}(F_{\nu},\mathbb{Z}/\ell(2))$ ([11, Corollary p. 145]). Hence $\zeta=\alpha\cdot(f)$ ([11, Theorem 0.8(2)]). ∎

7. Main theorem

In this section we prove our main result (7.7). Let $K$ be a global field or a local field and $F$ the function field of a curve over $K$. Let $\ell$ be a prime not equal to char$(K)$. Suppose that $F$ contains a primitive $\ell^{\rm th}$ root of unity $\rho$. If $K$ is a number field or a local field, let $R$ be the ring of integers in $K$. If $K$ is a global field of positive characteristic, let $R$ be the field of constants of $K$.

To prove our main result (7.7), we first show (7.6) that given $\zeta\in H^{3}(F,\mathbb{Z}/\ell(2))$ with $\zeta\otimes_{F}(F\otimes_{K}K_{\nu})=0$ for all real places $\nu$ of $K$, there exist $\alpha=[a,b)\in H^{2}(F,\mathbb{Z}/\ell(1))$ and a regular proper model $\mathscr{X}$ of $F$ over $R$ such that the triple $(\mathscr{X},\zeta,\alpha)$ satisfies the assumptions 6.3.

Let $\zeta\in H^{3}(F,\mathbb{Z}/\ell(2))$ be such that $\zeta\otimes_{F}(F\otimes_{K}K_{\nu})=0$ for all real places $\nu$ of $K$. Choose a regular proper model $\mathscr{X}$ of $F$ over $R$ (cf. [22, p. 38]) such that

$\bullet$ ram${}_{\mathscr{X}}(\zeta)\cup$ supp${}_{\mathscr{X}}(\ell)\subset\{C_{1},\cdots,C_{r_{1}},\cdots,C_{r}\}$, where $C_{i}$’s are irreducible regular curves with normal crossings

$\bullet$ for $i\neq j$, $C_{i}$ and $C_{j}$ intersect at most at one closed point

$\bullet$ $C_{i}\cap C_{j}=\emptyset$ if $i,j\leq r_{1}$ or $i,j>r_{1}$.

For $x\in\mathscr{X}_{(1)}$, let $\beta_{x}=\partial_{x}(\zeta)$. Let $\mathscr{P}_{0}\subset\cup C_{i}$ be a finite set of closed points of $\mathscr{X}$ containing $C_{i}\cap C_{j}$ for $1\leq i<j\leq r$, and at least one closed point from each $C_{i}$. Let $A$ be the regular semi local ring at the points of $\mathscr{P}_{0}$. Let $Q\in C_{i}$ be a closed point. Since $C_{i}$ is regular on $\mathscr{X}$, $Q$ gives a discrete valuation $\nu^{i}_{Q}$ on $\kappa(C_{i})$.

Lemma 7.1**.**

*There exists $a\in A$ such that

$\bullet$ $\frac{a-1}{(\rho-1)^{\ell}}\in A$ and $[a)$ is unramified on $A$

$\bullet$ for $1\leq i\leq r_{1}$ and $P\in C_{i}\cap\mathscr{P}_{0}$, $\partial_{P}(\beta_{x_{i}})=[a(P))$

$\bullet$ for $r_{1}+1\leq i\leq r$ and $P\in C_{i}\cap\mathscr{P}_{0}$, $\partial_{P}(\beta_{x_{i}})=[a(P))^{-1}$

$\bullet$ if $P\in\mathscr{P}_{0}$ and $P\not\in C_{i}\cap C_{j}$ for all $i\neq j$, then $[a(P))$ is the trivial extension

$\bullet$ if $\ell=2$, then $a$ is a sum of two squares in $A$.*

Proof.

Let $P\in\mathscr{P}_{0}$. Suppose $P\in C_{i}\cap C_{j}$ for some $i<j$. Then, by the choice of $\mathscr{X}$, the pair $(i,j)$ is uniquely determined by $P$. Let $u_{P}\in\kappa(P)$ be such that $\partial_{P}(\partial_{x_{i}}(\zeta))=[u_{P})$. If $P\not\in C_{i}\cap C_{j}$ for all $i\neq j$, let $u_{P}\in\kappa(P)$ with $[u_{P})$ the trivial extension.

Then, by (2.5), there exists $a\in A$ such that for every $P\in\mathscr{P}_{0}$, the cyclic extension $[a)$ over $F$ is unramified on $A$ with the residue field $[a(P))$ of $[a)$ at $P$ is $[u_{P})$. Further if $\ell=2$, choose $a$ to be a sum of two squares in $A$ (2.5). From the proof of (2.5), we have $\frac{a-1}{(\rho-1)^{\ell}}\in A$.

Let $P\in\mathscr{P}_{0}$. Suppose that $P\in C_{i}$ for some $i$ and $P\not\in C_{j}$ for all $i\neq j$. Then $\partial_{P}(\partial_{x_{i}}(\zeta))=1$ (4.3) and by the choice of $a$ and $u_{P}$, we have $[a(P))=[u_{P})=1$. Suppose that $P\in C_{i}\cap C_{j}$ for some $i\neq j$. Suppose $i<j$. Then by the choice of $a$ and $u_{P}$ we have $\partial_{P}(\partial_{x_{i}}(\zeta))=[u_{P})=[a(P))$. Suppose $i>j$. Then by the choice of $a$ and $u_{P}$ we have $\partial_{P}(\partial_{x_{j}}(\zeta))=[u_{P})=[a(P))$. Since $\partial_{P}(\partial_{x_{i}}(\zeta))=\partial_{P}(\partial_{x_{j}}(\zeta))^{-1}$ (4.3), we have $\partial_{P}(\partial_{x_{i}}(\zeta))=[a(P))^{-1}$. Thus $a$ has the required properties. ∎

Let $a\in A$ be as in (7.1). Let $L_{1},\cdots,L_{d}$ be the irreducible curves in $\mathscr{X}$ which are in the support of $a$ or $\frac{a-1}{(\rho-1)^{\ell}}$. Then the cyclic extension $L=[a)=F(\sqrt[\ell]{a})$ is unramified at any irreducible curve in $\mathscr{X}$ which is not equal to $L_{j}$ for any $j$ (cf. 2.3).

Lemma 7.2**.**

Then $L_{i}\cap\mathscr{P}_{0}=\emptyset$ for all $i$. In particular $L_{i}\neq C_{j}$ for all $i,j$ and char$(\kappa(L_{i}))\neq\ell$.

Proof.

By the choice of $a$, $a\in A$ and $\frac{a-1}{(\rho-1)^{\ell}}\in A$ (7.1). Hence $\mathscr{P}_{0}\cap L_{i}=\emptyset$ for all $i$. Since $\mathscr{P}_{0}$ contains at least one point from each $C_{j}$, $L_{i}\neq C_{j}$ for all $i$ and $j$. Since supp${}_{\mathscr{X}}(\ell)\subset\{C_{1},\cdots,C_{r}\}$, char$(\kappa(L_{i}))\neq\ell$ for all $i$. ∎

Let $\mathscr{P}_{1}\subset\cup_{j}L_{j}$ be a finite set of closed points of $\mathscr{X}$ consisting of $L_{i}\cap L_{j}$ for $i\neq j$, $L_{i}\cap C_{j}$, one point from each $L_{i}$. Since $L_{i}\cap\mathscr{P}_{0}=\emptyset$ for all $i$ (7.2), $\mathscr{P}_{0}\cap\mathscr{P}_{1}=\emptyset$.

Let $\mathscr{P}=\mathscr{P}_{0}\cup\mathscr{P}_{1}$ and $B$ be the semi local ring at $\mathscr{P}$ on $\mathscr{X}$. For each $i$ and $j$, let $\pi_{i}\in B$ be a prime defining $C_{i}$ and $\delta_{j}\in B$ a prime defining $L_{j}$.

Lemma 7.3**.**

*For each $P\in C_{i}\cap\mathscr{P}_{1}$, let $n_{i}^{P}$ be a positive integer. Then for each $i$, $1\leq i\leq r$, there exists $b_{i}\in B/(\pi_{i})\subset\kappa(C_{i})$ such that

$\bullet$ $\partial_{C_{i}}(\zeta)=[a(C_{i}),b_{i})$

$\bullet$ $\nu^{i}_{P}(b_{i})=1$ for all $P\in C_{i}\cap\mathscr{P}_{0}$, $1\leq i\leq r_{1}$

$\bullet$ $\nu^{i}_{P}(b_{i})=\ell-1$ for all $P\in C_{i}\cap\mathscr{P}_{0}$, $r_{1}+1\leq i\leq r$

$\bullet$ $\nu^{i}_{P}(b_{i}-1)\geq n^{P}_{i}$ for all $P\in\mathscr{P}_{1}\cap C_{i}$ for all $i$.*

Proof.

Let $1\leq i\leq r$. Let $\beta_{x_{i}}=\partial_{x_{i}}(\zeta)\in H^{2}(\kappa(C_{i}),\mathbb{Z}/\ell(1))$ and $a_{i}=a(C_{i})$.

Suppose $1\leq i\leq r_{1}$. By (7.1), $\partial_{P}(\beta_{x_{i}})=[a_{i}(P))$ for all $P\in C_{i}\cap\mathscr{P}_{0}$. If $P\not\in\mathscr{P}_{0}$, then $\partial_{P}(\beta_{x_{i}})=0$ for all $i$ (4.3). By the assumption, $\beta_{x_{i}}\otimes\kappa(C_{i})_{\nu}=0$ for all real places $\nu$ of $\kappa(C_{i})$. Thus, by (3.5), there exists $b_{i}\in\kappa(C_{i})^{*}$ such that $\beta_{x_{i}}=[a_{i},b_{i})$, with $\nu^{i}_{P}(b_{i})=1$ for all $P\in C_{i}\cap\mathscr{P}_{0}$ and $\nu^{i}_{P}(b_{i}-1)\geq n^{P}_{i}$ for all $P\in C_{i}\cap\mathscr{P}_{1}$. In particular $b_{i}$ is regular at all $P\in C_{i}\cap\mathscr{P}$ and hence $b_{i}\in B/(\pi_{i})$.

Suppose $r_{1}+1\leq i\leq r$. Let $P\in C_{i}\cap\mathscr{P}_{0}$. Since $\partial_{P}(\beta_{x_{i}})=[a(P))^{-1}$ for all $P\in C_{i}\cap\mathscr{P}_{0}$ (7.1), $\partial_{P}(\beta_{x_{i}}^{-1})=[a(P))$. Thus, as above, by (3.5), there exists $c_{i}\in B/(\pi_{i})$ such that $\beta_{x_{i}}^{-1}=[a_{i},c_{i})$, with $\nu^{i}_{P}(c_{i})=1$ for all $P\in C_{i}\cap\mathscr{P}_{0}$ and $\nu^{i}_{P}(c_{i}-1)\geq n^{P}_{i}$ for all $P\in C_{i}\cap\mathscr{P}_{1}$. Let $b_{i}=c_{i}^{\ell-1}\in B/(\pi_{i})$. Then $\beta_{x_{i}}=[a_{i},b_{i})$. Let $P\in C_{i}\cap\mathscr{P}_{1}$. Since $c_{i}\in B/(\pi_{i})$ and $\nu^{i}_{P}(c_{i}-1)\geq n^{P}_{i}$, it follows tat $\nu^{i}_{P}(b_{i}-1)\geq n^{P}_{i}$. Thus $b_{i}$ has the required properties. ∎

Let $\delta=\prod\delta_{j}\in B$. For $1\leq i\leq r$, let $\overline{\delta}(i)\in B/(\pi_{i})$ be the image of $\delta$. Let $d$ be an integer greater than $\nu_{P}^{i}(\overline{\delta}(i))+1$ for all $i$ and $P\in C_{i}\cap\mathscr{P}$.

Lemma 7.4**.**

*Let $b_{i}\in B/(\pi_{i})$ be as in (7.3) for $n_{i}^{P}=d$ for all $P\in C_{i}\cap\mathscr{P}$. Then there exists $b\in B$ such that

$\bullet$ $b=b_{i}$ modulo $\pi_{i}$ for all $i$

$\bullet$ $b=1$ modulo $\delta_{j}$ for all $j$

$\bullet$ $b$ is a unit at all $P\in\mathscr{P}_{1}$.*

Proof.

For $1\leq i\leq r$, let $I_{i}=(\pi_{i})\subset B$ and $I_{r+1}=(\delta)\subset B$. Clearly the gcd$(\pi_{i},\pi_{j})=1$ and gcd$(\pi_{i},\delta)=1$ for all $1\leq i<j\leq r$. For $1\leq i<j\leq r$, $I_{ij}=I_{i}+I_{j}$ is either maximal ideal or equal to $B$. For $1\leq i\leq r$, we have $I_{i(r+1)}=(\pi_{i},\delta)$. Since $L_{s}\cap\mathscr{P}_{0}=\emptyset$ for all $s$, $(\delta_{s},\pi_{i},\pi_{j})=A$ for all $1\leq i<j\leq r$ and for all $s$. Thus the ideals $I_{ij}$, $1\leq i<j\leq r+1$, are coprime. Let $b_{r+1}=1\in B/(I_{r+1})$.

Let $1\leq i<j\leq r$. Suppose $(\pi_{i},\pi_{j})\neq B$. Then $(\pi_{i},\pi_{j})$ is a maximal ideal of $B$ corresponding to a point $P\in C_{i}\cap C_{j}$. Since $P\in\mathscr{P}_{0}$, by the choice of $b_{i}$ and $b_{j}$ (cf. 7.4), we have $\nu^{i}_{P}(b_{i})=1$, $\nu^{i}_{P}(b_{j})=\ell-1$ and hence $b_{i}=b_{j}=0\in B/(\pi_{i},\pi_{j})=B/I_{ij}$.

Suppose $I_{i(r+1)}\neq B$ for some $1\leq i\leq r$. Then we claim that $b_{i}=1\in B/I_{i(r+1)}$. For each $P\in L_{j}\cap C_{i}$, let $M_{P}$ be the maximal ideal of $B$ at $P$. Since $\mathscr{X}$ is regular and $C_{i}$ is regular on $\mathscr{X}$, we have $M_{P}=(\pi_{i},\pi_{i,P})$ for some $\pi_{i,P}\in M_{P}$ and the image of $\pi_{i,P}$ in $B/(\pi_{i})$ is a parameter at the discrete valuation $\nu_{P}^{i}$. Since $d>\nu_{P}^{i}(\overline{\delta}(i))$, we have $(\pi_{i},\prod\pi_{i,P}^{d})\subset(\pi_{i},\delta)=I_{i(r+1)}$. Since $B/(\pi_{i},\prod\pi_{i,P}^{d})\simeq\prod_{P}B/(\pi_{i},\pi_{i,P}^{d})$ and $\nu_{P}^{i}(b_{i}-1)\geq d$, we have $b_{i}=1\in B/(\pi_{i},\prod\pi_{i,P}^{d})$. Since $B/I_{i}+I_{r+1}$ is a quotient of $B/I_{i}+(\prod_{P}\pi_{i,P})^{d}$, it follows that $b_{i}=b_{r+1}=1\in B/I_{i}+I_{r+1}=B/I_{i(r+1)}$.

Thus, by (2.7), there exists $b\in B$ such that $b=b_{i}\in B/(\pi_{i})$ for all $i$ and $b=1\in B/I_{r+1}$. Since $I_{r+1}=(\delta)\subset(\delta_{j})$ and $b=1\in B/(\delta)$, we have $b=1\in A/(\delta_{j})$ for all $j$. Let $P\in\mathscr{P}_{1}$. Then $P\in L_{j}$ for some $j$. Since $b=1\in B/(\delta_{j})$, $b$ is a unit at $P$. Thus $b$ has all the required properties. ∎

Lemma 7.5**.**

Let $a$ be as in (7.1) and $b$ as in (7.4) and $\alpha=[a,b)$. Then $\alpha$ is unramified at all $C_{i}$, $L_{j}$ and at all $Q\in\mathscr{P}_{1}$. Further $\partial_{C_{i}}(\zeta)$ is the specialization of $\alpha$ at $C_{i}$ for all $1\leq i\leq r$.

Proof.

Since $[a)$ is unramified at $C_{i}$ (7.1) and $b$ is a unit at $C_{i}$ for all $i$ (7.4), $\alpha$ is unramified at $C_{i}$ and the specialization of $\alpha$ at $C_{i}$ is $[a(C_{i}),b_{i})=\partial_{C_{i}}(\zeta)$ (7.3, 7.4). Since char$(\kappa(L_{j}))\neq\ell$ (7.2) and $b=1$ modulo $\delta_{j}$ (7.4), $b$ is an $\ell^{\rm th}$ power in $F_{L_{j}}$ and hence $\alpha\otimes F_{L_{j}}=0$. In particular $\alpha$ is unramified at $L_{j}$.

Let $Q\in\mathscr{P}_{1}$. Then $b$ is a unit at $Q$ (7.4). Let $x$ be a dimension one point of Spec$(B_{Q})$. Then $b$ is a unit at $x$. If $[a)$ is unramified at $x$, then $\alpha$ is unramified at $x$. Suppose $[a)$ is ramified at $x$. Then, by the choice of $L_{j}$’s, $x$ is the generic point of $L_{j}$ for some $j$ and hence $\alpha$ is unramified at $x$. Thus $\alpha$ is unramified at $Q$ (cf. 3.1). ∎

Proposition 7.6**.**

The triple $(\mathscr{X},\zeta,[a,b))$ satisfies the assumptions 6.3.

Proof.

By the choice of $\mathscr{X}$, B1) of 6.3 is satisfied. Let ram${}_{\mathscr{X}}(\alpha)=\{D_{1},\cdots,D_{n}\}$. Since $\alpha$ is unramified at all $C_{i}$ (7.5), B2) of 6.3 is satisfied. Since supp${}_{\mathscr{X}}(\ell)\subset\{C_{1},\cdots,C_{r}\}$ and $D_{i}\neq C_{j}$ for all $i$ and $j$, char$(\kappa(D_{i}))\neq\ell$ for all $i$ and hence B3) of 6.3 is satisfied.

Let $P\in D_{j}$ some $j$ with char$(\kappa(P))=\ell$. Since supp${}_{\mathscr{X}}(\ell)\subset\{C_{1},\cdots,C_{r}\}$, $P\in C_{i}$ for some $i$. Since $\alpha$ is unramified at all $Q\in\mathscr{P}_{1}$ (7.5), $P\not\in\mathscr{P}_{1}$. Since $C_{i}\cap L_{s}\subset\mathscr{P}_{1}$ for all $s$, $P\not\in L_{s}$ for all $s$ and hence $\frac{a-1}{(\rho-1)^{\ell}}\in A_{P}$. Thus B4) of 6.3 is satisfied.

Since $\partial_{C_{i}}(\zeta)$ is the specialization of $\alpha$ at $C_{i}$ (7.5), B5) of 6.3 is satisfied.

By the assumption on $\zeta$, B6) of 6.3 is satisfied. If $\ell=2$, then, by the choice of $a$ (7.1), B7) of 6.3 is satisfied.

Let $P\in C_{i}\cap C_{j}$ for some $i<j$. Then, by the choice of $b_{i}$ and $b_{j}$ (7.3), we have $b_{i}=\overline{u}_{j}\overline{\pi}_{j}$ for some unit $u_{j}$ at $P$ and $b_{j}=\overline{u}_{i}\overline{\pi}_{i}^{\ell-1}$ for some unit $u_{i}$ at $P$. Since $b=b_{i}$ modulo $\pi_{i}$ and $b=b_{j}$ modulo $\pi_{j}$, we have $b=v_{i}\pi_{i}^{\ell-1}+v_{j}\pi_{j}$ for some units $v_{i},v_{j}$ at $P$. In particular $b$ is a regular prime at $P$. Since $[a)$ is unramified at $P$ (7.1) and $b$ being a prime at $P$, $\alpha$ is unramified at $P$ except possibly at $b$. Thus there is at most one $D_{s}$ with $P\in D_{s}$ and such a $D_{s}$ is defined by $b=v_{i}\pi_{i}^{\ell-1}+v_{j}\pi_{j}$ for some units $v_{i},v_{j}$ at $P$. In particular B8) of 6.3 is satisfied.

∎

Theorem 7.7**.**

Let $K$ be a global field or a local field and $F$ the function field of a curve over $K$. Let $\ell$ be a prime not equal to the characteristic of $K$. Suppose that $K$ contains a primitive $\ell^{\rm th}$ root of unity. Let $\zeta\in H^{3}(F,\mathbb{Z}/\ell(2))$. Suppose that $\zeta\otimes_{F}(F\otimes_{K}K_{\nu})$ is trivial for all real places $\nu$ of $K$. Then there exist $a,b,f\in F^{*}$ such that $\zeta=[a,b)\cdot(f)$.

Proof.

By (7.6), there exist $a,b\in F^{*}$ and regular proper model $\mathscr{X}$ of $F$ such that the triple $(\mathscr{X},\zeta,\alpha)$ satisfy the assumption 6.3. Thus, by (6.5), there exists $f\in F^{*}$ such that $\zeta=\alpha\cdot(f)=[a,b)\cdot(f)$. ∎

Corollary 7.8**.**

Let $K$ be a global field or a local field and $F$ the function field of a curve over $K$. Let $\ell$ be a prime not equal to the characteristic of $K$. Suppose that $K$ contains a primitive $\ell^{\rm th}$ root of unity. Suppose that either $\ell\neq 2$ or $K$ has no real orderings. Then for every element $\zeta\in H^{3}(F,\mathbb{Z}/\ell(3))$, there exist $a,b,c\in F^{*}$ such that $\zeta=[a,b)\cdot(c)$.

8. Applications

In this section we given some applications of our main result to quadratic forms and Chow group of zero-cycles.

Let $K$ be a field of characteristic not equal to 2. Let $W(K)$ denote the Witt group of quadratic forms over $K$ and $I(K)$ the fundamental ideal of $W(K)$ consisting of classes of even dimensional forms (cf. [25, Ch. 2]). For $n\geq 1$, let $I^{n}(K)$ denote the $n^{\rm th}$ power of $I(K)$. For $a_{1},\cdots,a_{n}\in F^{*}$, let $<<a_{1},\cdots,a_{n}>>$ denote the $n$-fold Pfister form $<\!1,-a_{1}\!>\otimes\cdots\otimes<\!1,-a_{n}\!>$ (cf. [25, Ch. 4].

Theorem 8.1**.**

Let $k$ be a totally imaginary number field and $F$ the function field of a curve over $k$. Then every element in $I^{3}(F)$ is represented by a 3-fold Pfister form. In particular if the class of a quadratic form $q$ is in $I^{3}(F)$ and dimension of $q$ is at least 9, then $q$ is isotropic.

Proof.

Since every element in $H^{3}(F,\mathbb{Z}/2(3))$ is a symbol (7.8) and cd${}_{2}(F)\leq 3$, it follows from ([2, Theorem 2]) that every element in $I^{3}(F)$ is represented by a 3-fold Pfister form (see the proof of [17, Theorem 4.1]). ∎

Proposition 8.2**.**

Let $F$ be a field of characteristic not equal to 2 with cd${}_{2}(F)\leq 3$ Suppose that every element in $H^{3}(F,\mathbb{Z}/2(3))$ is a symbol. If $q$ is a quadratic form over $F$ of dimension at least 5 and $\lambda\in F^{*}$, then $q\otimes<\!1,-\lambda\!>$ is isotropic.

Proof.

Without loss of generality we assume that dimension of $q$ is 5. By scaling we also assume that $q=<\!-a,-b,ab,c,d\!>$ for some $a,b,c,d\in F^{*}$. Let $q^{\prime}=<\!-a,-b,ab,c,d,-cd\!>\otimes<\!1,-\lambda\!>$. Since $<\!-a,-b,ab,c,d,-cd\!>\in I^{2}(K)$ ([25, p. 82]), $q^{\prime}\in I^{3}(F)$. Hence, by (8.1), $q^{\prime}$ is represented by 3-fold Pfister form. Since $q^{\prime}\otimes F(\sqrt{\lambda})=0$, $q^{\prime}=<\!1,-\lambda\!>\otimes<\!1,\mu\!>\otimes<\!1,\mu^{\prime}\!>$ for some $\mu,\mu^{\prime}\in F^{*}$ (cf. [25, p. 45, Theorem 5.2], [25, p. 143, Corollary 1.5] and [25, p. 144, Theorem 1.4]). Since $H^{4}(F,\mathbb{Z}/2(4))=0$, $I^{4}(F)=0$ ([2, Corollary 2]), we have $q^{\prime}=-cd<\!1,-\lambda\!>\otimes<\!1,\mu\!>\otimes<\!1,\mu^{\prime}\!>$.

Thus we have

[TABLE]

In particular $<\!-a,-b,ab,c,d\!>\otimes<\!1,-\lambda\!>$ is isotropic ([25, p. 34]). ∎

Corollary 8.3**.**

Let $K$ be a totally imaginary number field and $F$ the function field a curve over $K$. Let $q$ be a quadratic forms over $F$ of dimension at least 5. Let $\lambda\in F^{*}$. Then the quadratic form $q\otimes<\!1,-\lambda\!>$ is isotropic.

Proof.

Since $K$ is a totally imaginary number field and $F$ is a function field of a curve over $k$, we have $H^{4}(F,\mathbb{Z}/2(4))=0$. Since every element in $H^{3}(F,\mathbb{Z}/2(3))$ is a symbol (7.8), $q\otimes<\!1,-\lambda\!>$ is isotropic (8.2) ∎

Theorem 8.4**.**

Let $k$ be a totally imaginary number field and $C$ a smooth projective geometrically integral curve over $K$. Let $\eta:X\to C$ be an admissible quadric fibration. If dim$(X)\geq 4$, then $CH_{0}(X)$ is a finitely generated abelian group.

Proof.

Let $q$ be a quadratic form over $k(C)$ defining the generic fibre of $\eta:X\to C$. Let $N_{q}(k(C))$ be the subgroup of $k(C)^{*}$ generated by $fg$ with $f,g\in k(C)^{*}$ represented by $q$. Let $\lambda\in k(C)^{*}$. Since dim$(X)\geq 4$, the dimension of $q$ is at least 5. Thus, by (8.3), $q\otimes<\!1,-\lambda\!>$ is isotropic. Hence $\lambda$ is a product of two values of $q$. In particular $\lambda\in N_{q}(k(C))$ and $k(C)^{*}=N_{q}(k(C))$.

Let $CH_{0}(X/C)$ be the kernel of the induced homomorphism $CH_{0}(X)\to CH_{0}(C)$. Then, by ([5]), $CH_{0}(X/C)$ is a sub quotient of the group $k(C)^{*}/N_{q}(k(C))$ and hence $CH_{0}(X/C)=0$. In particular $CH_{0}(X)$ is isomorphic to a subgroup of $CH_{0}(C)$. Since, by a theorem of Mordell-Weil, $CH_{0}(C)$ is finitely generated, $CH_{0}(X)$ is finitely generated. ∎