2.1.1. Let $K$ be a discrete valuation field with residue field  $F$ , valuation ring $O_{K}$ and maximal ideal  $\mathfrak{m}_{K}$ . Assume $F$ is of characteristic $p>0$ .

Let $e_{K}=\text{ord}_{K}(p)$ be the absolute ramification index of $K$ ( $e_{K}=\infty$ if $K$ is of characteristic  $p$ ).

2.1.2. Let $\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log )=\unicode[STIX]{x1D6FA}_{O_{K}/\mathbb{Z}}^{1}(\log )$ be the module of differential forms with log poles of $O_{K}$ over $\mathbb{Z}$ with respect to the standard log structure of $O_{K}$ [Reference Kato21]. It is the $O_{K}$ -module defined by the following generators and relations.

Generators. $dx$ for $x\in O_{K}$ and $d\log (x)$ for $x\in K^{\times }$ .

Relations. $d(x+y)=dx+dy$ and $d(xy)=x\,dy+y\,dx$ for $x,y\in O_{K}$ . $d\log (xy)=d\log (x)+d\log (y)$ for $x,y\in K^{\times }$ . $dx=x\,d\log (x)$ for $x\in O_{K}\smallsetminus \{0\}$ .

Let

$$\begin{eqnarray}\hat{\unicode[STIX]{x1D6FA}}_{O_{K}}^{1}(\log )=\mathop{\varprojlim }_{n}\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log )/\mathfrak{m}_{K}^{n}\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log ).\end{eqnarray}$$

Note that $\hat{\unicode[STIX]{x1D6FA}}_{O_{K}}^{1}(\log )/\mathfrak{m}_{K}^{n}\hat{\unicode[STIX]{x1D6FA}}_{O_{K}}^{1}(\log )=\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log )/\mathfrak{m}_{K}^{n}\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log )$ for any $n\geqslant 1$ [Reference Fujiwara and Kato9, Chapter 0, 7.2.8, 7.2.16].

Proof. We may assume that $K$ is complete (and so we assume it).

(1) follows from the fact that $O_{K}\cong F[[T]]$ in this case.

Assume $K$ is of characteristic 0. Then $K$ is a finite extension of a complete discrete valuation field $K_{0}$ such that $p$ is a prime element in $K_{0}$ and the residue field of $K_{0}$ coincides with that of $K$ . We may assume $\tilde{b}_{\unicode[STIX]{x1D706}}\in O_{K_{0}}$ for all  $\unicode[STIX]{x1D706}$ . As is easily seen, there is an isomorphism between the $\mathfrak{m}_{K_{0}}$ -adic completion $\hat{\unicode[STIX]{x1D6FA}}_{O_{K_{0}}}^{1}$ of the differential module $\unicode[STIX]{x1D6FA}_{O_{K_{0}}/\mathbb{Z}}^{1}$ (without log) and $\hat{\bigoplus }_{\unicode[STIX]{x1D6EC}}O_{K_{0}}$ which sends the $\unicode[STIX]{x1D706}$ th base of $\bigoplus _{\unicode[STIX]{x1D6EC}}O_{K_{0}}$ to $d\tilde{b}_{\unicode[STIX]{x1D706}}$ . Let $\unicode[STIX]{x1D70B}$ be a prime element of $K$ and let $f(T)=\sum _{i=0}^{e}a_{i}T^{i}$ be the irreducible polynomial of $\unicode[STIX]{x1D70B}$ over $K$ such that $a_{e}=1$ . Then $a_{0}$ is a prime element of $K_{0}$ and $p|a_{i}$ for all $0\leqslant i<e$ . We have $O_{K_{0}}[T]/(f(T))\overset{\cong }{\rightarrow }O_{K}\;;\;T\mapsto \unicode[STIX]{x1D70B}$ . From this, we see that $\hat{\unicode[STIX]{x1D6FA}}_{O_{K}}^{1}(\log )$ has the presentation with generators the $O_{K}$ -module $O_{K}\otimes _{O_{K_{0}}}\hat{\unicode[STIX]{x1D6FA}}_{O_{K_{0}}}^{1}$ and $d\log (\unicode[STIX]{x1D70B})$ and with the relation $f^{\prime }(\unicode[STIX]{x1D70B})\unicode[STIX]{x1D70B}\,d\log (\unicode[STIX]{x1D70B})+\sum _{i=0}^{e-1}\unicode[STIX]{x1D70B}^{i}da_{i}=0$ . This proves (2).

We prove (3). The last relation is written as

$$\begin{eqnarray}\left(\mathop{\sum }_{i=1}^{e}ia_{i}\unicode[STIX]{x1D70B}^{i}\,d\log (\unicode[STIX]{x1D70B})\right)+\mathop{\sum }_{i=0}^{e-1}(\unicode[STIX]{x1D70B}^{i}a_{0}d(a_{i}/a_{0})+\unicode[STIX]{x1D70B}^{i}a_{i}\,d\log (a_{0}))=0\end{eqnarray}$$

which is trivial mod $p$ .◻

Proof. Assume first $K$ is of characteristic $p$ . Let $\unicode[STIX]{x1D70B}_{K}$ be a prime element of  $K$ . By (1) of 2.1.3, $\hat{\unicode[STIX]{x1D6FA}}_{O_{K}}^{1}(\log )$ is a free $O_{K}$ -module of rank 1 with base $d\log (\unicode[STIX]{x1D70B}_{K})$ . As is easily seen, if $f\in O_{L}$ and $df=0$ in $\hat{\unicode[STIX]{x1D6FA}}_{O_{L}}^{1}(\log )$ , then $f$ is a $p$ th power in  $O_{L}$ . Hence by the assumption that $L/K$ is separable, we have $d\unicode[STIX]{x1D70B}_{K}\neq 0$ in $\hat{\unicode[STIX]{x1D6FA}}_{O_{L}}^{1}(\log )$ . By 2.1.3, this proves the injectivity in question.

Assume next $K$ is of characteristic 0. We have $K\subset L^{\prime }\subset L$ where $L^{\prime }$ is a complete discrete valuation field such that $L$ is a finite extension of  $L^{\prime }$ , the residue field of $L^{\prime }$ coincides with that of  $L$ , and a prime element of $K$ is still a prime element of $L^{\prime }$ . Take a $p$ -base $(b_{\unicode[STIX]{x1D706}})_{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}}$ of the residue field of $L$ and let $\tilde{b}_{\unicode[STIX]{x1D706}}$ be a lifting of $b_{\unicode[STIX]{x1D706}}$ to $O_{L^{\prime }}$ for each  $\unicode[STIX]{x1D706}$ . Then the proof of (2) of 2.1.3 shows that $\hat{\unicode[STIX]{x1D6FA}}_{O_{L^{\prime }}}^{1}(\log )=O_{L^{\prime }}\otimes _{O_{K}}\hat{\unicode[STIX]{x1D6FA}}_{O_{K}}^{1}(\log )\oplus (\hat{\bigoplus }_{\unicode[STIX]{x1D6EC}}O_{L^{\prime }})$ . Let $\unicode[STIX]{x1D70B}$ be a prime element of $L$ and let $f(T)=\sum _{i=0}^{e}a_{i}T^{i}$ be the irreducible polynomial of $\unicode[STIX]{x1D70B}$ over $L^{\prime }$ with $a_{e}=1$ . Then $a_{0}$ is a prime element of $L^{\prime }$ and $a_{0}|a_{i}$ for $0\leqslant i\leqslant e-1$ . From the fact $O_{L^{\prime }}[T]/(f(T))\overset{\cong }{\rightarrow }O_{L}\;;\;T\mapsto \unicode[STIX]{x1D70B}$ , we have that $\hat{\unicode[STIX]{x1D6FA}}_{O_{L}}^{1}(\log )$ has a presentation with the generators the $O_{L}$ -module $O_{L}\otimes _{O_{L^{\prime }}}\hat{\unicode[STIX]{x1D6FA}}_{O_{L^{\prime }}}^{1}(\log )$ and $d\log (\unicode[STIX]{x1D70B})$ and with the relation

$$\begin{eqnarray}(f^{\prime }(\unicode[STIX]{x1D70B})\unicode[STIX]{x1D70B}/a_{0})\,d\log (\unicode[STIX]{x1D70B})+\mathop{\sum }_{i=0}^{e-1}\unicode[STIX]{x1D70B}^{i}(a_{i}\,d\log (a_{0})+a_{0}d(a_{i}/a_{0}))=0\end{eqnarray}$$

(note $f^{\prime }(\unicode[STIX]{x1D70B})\unicode[STIX]{x1D70B}/a_{0}\in O_{L}\smallsetminus \{0\}$ ). This proves 2.1.5.◻

2.1.6. Let $a$ be an integer such that $0\leqslant a\leqslant e_{K}$ . Take a ring homomorphism $\unicode[STIX]{x1D704}:F\rightarrow O_{K}/\mathfrak{m}_{K}^{a}$ which lifts the identity map of $F$ ( $\unicode[STIX]{x1D704}$ exists by a theorem of Cohen [Reference Grothendieck12, Chapitre 0, 19.6.1] and a prime element $\unicode[STIX]{x1D70B}$ of  $K$ ). Then we have an isomorphism

(5) $$\begin{eqnarray}F[T]/(T^{a})\overset{\cong }{\rightarrow }O_{K}/\mathfrak{m}_{K}^{a}\;;\;\mathop{\sum }_{i}a_{i}T^{i}\mapsto \mathop{\sum }_{i}\unicode[STIX]{x1D704}(a_{i})\unicode[STIX]{x1D70B}^{i}\end{eqnarray}$$

of rings with log structures given by $T$ and $\unicode[STIX]{x1D70B}$ , respectively, which sends $T$ to $\unicode[STIX]{x1D70B}$ in the log structures. (See [Reference Kato21] for log structures.) Since $O_{K}/\mathfrak{m}_{K}^{a}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log )$ coincides with the module of differential forms with log poles [Reference Kato21] of the ring $O_{K}/\mathfrak{m}_{K}^{a}$ with the log structure given by  $\unicode[STIX]{x1D70B}$ , it is isomorphic to the module of differential forms with log poles of the ring $F[T]/(T^{a})$ with the log structure given by  $T$ . In the case $K$ is of mixed characteristic and $a=e_{K}$ , this gives another proof of (3) of 2.1.3.

Proof. Note that the Cartier operator $C:\unicode[STIX]{x1D6FA}_{F}^{r}\rightarrow \unicode[STIX]{x1D6FA}_{F}^{r}$ is characterized by the properties $C(x^{p}\,d\log y_{1}\wedge \cdots \wedge d\log y_{r})=x\,d\log y_{1}\wedge \cdots \wedge d\log y_{r}$ for $x\in F$ and $y_{i}\in F^{\times }$ and $C(d\unicode[STIX]{x1D714})=0$ for $\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}_{F}^{r-1}$ . Using also the Cartier operator $C:\unicode[STIX]{x1D6FA}_{K}^{r+1}\rightarrow \unicode[STIX]{x1D6FA}_{K}^{r+1}$ , we have

$$\begin{eqnarray}\text{Res}\circ C=C\circ \text{Res}:\unicode[STIX]{x1D6FA}_{K}^{r+1}\rightarrow \unicode[STIX]{x1D6FA}_{F}^{r}.\end{eqnarray}$$

We have $C^{b}(\mathfrak{m}_{K}^{1-a}\unicode[STIX]{x1D6FA}_{O_{K}}^{r+1}(\log ))\subset \unicode[STIX]{x1D6FA}_{O_{K}}^{r+1}(\log )$ . On $\unicode[STIX]{x1D6FA}_{O_{K}}^{r+1}(\log )$ , the residue map is the unique map which sends $\unicode[STIX]{x1D714}\wedge d\log (t)$ for any $\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}_{O_{K}}^{r}$ and any $t\in K^{\times }$ to $\text{ord}_{K}(t)\bar{\unicode[STIX]{x1D714}}$ where $\bar{\unicode[STIX]{x1D714}}$ is the image of $\unicode[STIX]{x1D714}$ in  $\unicode[STIX]{x1D6FA}_{F}^{r}$ .◻

2.1.8. Assume $[F:F^{p}]=p^{r}<\infty$ . Let $a$ be an integer such that $1\leqslant a\leqslant e_{K}$ .

Fix a ring homomorphism $\unicode[STIX]{x1D704}:F\rightarrow O_{K}/\mathfrak{m}_{K}^{a}$ which lifts the identity map of  $F$ . Fix a prime element $\unicode[STIX]{x1D70B}$ of  $K$ . Then by the isomorphism (5), we have an isomorphism

(6) $$\begin{eqnarray}\mathop{\bigoplus }_{i=0}^{a-1}\unicode[STIX]{x1D6FA}_{F}^{r}\overset{\cong }{\rightarrow }O_{K}/\mathfrak{m}_{K}^{a}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{r+1}(\log )\end{eqnarray}$$

which sends $(\unicode[STIX]{x1D714}_{i})_{0\leqslant i\leqslant a-1}$ to $\sum _{i=0}^{a-1}\unicode[STIX]{x1D70B}^{i}\unicode[STIX]{x1D704}(\unicode[STIX]{x1D714}_{i})\wedge d\log (\unicode[STIX]{x1D70B})$ .

Proof. We prove (1). If $K$ is of characteristic $p$ , this follows from 2.1.7. Using the isomorphism (5), the mixed characteristic case is reduced to the positive characteristic case.

We prove (2). The pairing $(O_{K}/\mathfrak{m}_{K}^{a}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{j}(\log ))\times (O_{K}/\mathfrak{m}_{K}^{a}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{i}(\log ))\rightarrow O_{K}/\mathfrak{m}_{K}^{a}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{r+1}(\log )\;;\;(x,y)\mapsto x\wedge y$ is a perfect duality of finitely generated free $O_{K}/\mathfrak{m}_{K}^{a}$ -modules. Hence we are reduced to the case $i=0$ and $j=r+1$ . Then by induction on $a$ , we are reduced to the statement that $F\times \unicode[STIX]{x1D6FA}_{F}^{r}\rightarrow \unicode[STIX]{x1D6FA}_{F}^{r}\;;\;(x,y)\mapsto C^{b}(xy)$ is a perfect duality of finite-dimensional vector spaces over $F$ which acts on these three spaces as follows. An element $c$ of $F$ acts on $F$ (resp. on the first $\unicode[STIX]{x1D6FA}_{F}^{r}$ , resp. on the second $\unicode[STIX]{x1D6FA}_{F}^{r}$ ) as $c^{p^{b}}$ (resp.  $c^{p^{b}}$ , resp.  $c$ ).◻

2.2.1. Let $p$ be a prime number. Let

$$\begin{eqnarray}E(T)=\mathop{\sum }_{i=0}^{p-1}\frac{T^{i}}{i!}\in \mathbb{Z}_{(p)}[T].\end{eqnarray}$$

Proof. (1) (resp. (2)) follows from the property $\exp (T_{1}+T_{2})=\exp (T_{1})\cdot \exp (T_{2})$ in $\mathbb{Q}[[T_{1},T_{2}]]$ (resp. $\exp (\sum _{i\geqslant 0}(T^{p^{i}}/p^{i}))=\prod _{i\geqslant 1,(i,p)=1}(1-T^{i})^{-\unicode[STIX]{x1D707}(i)/i}$ in $\mathbb{Q}[[T]]$ ) of the usual exponential (resp. of Artin–Hasse exponential) and the injectivity of $\mathbb{Z}_{(p)}[T_{1},T_{2}]/(T_{1},T_{2})^{p}\rightarrow \mathbb{Q}[[T_{1},T_{2}]]/(T_{1},T_{2})^{p}$ (resp. of $\mathbb{Z}_{(p)}[[T]]/(T^{p})\rightarrow \mathbb{Q}[[T]]/(T^{p})$ ).

(3) is straightforward. ◻

2.2.3. For a field $K$ , let $K_{r}^{M}(K)$ be the $r$ th Milnor $K$ -group of  $K$ . For a discrete valuation field $K$ and for $i\geqslant 1$ , we denote by $U^{i}K_{r}^{M}(K)$ the subgroup of $K_{r}^{M}(K)$ generated by elements of the form $\{u_{1},\ldots ,u_{r}\}$ where $u_{i}\in K^{\times }$ ( $1\leqslant i\leqslant r$ ) and $u_{1}\in \text{Ker}(O_{K}^{\times }\rightarrow (O_{K}/\mathfrak{m}_{K}^{i})^{\times })$ .

Proof. We prove (1). The $O_{K}$ -module $\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log )$ has the following presentation by generators and relations.

Generators. $d\log (x)$ for $x\in O_{K}\smallsetminus \{0\}$ .

Relations. $d\log (xy)=d\log (x)+d\log (y)$ for $x,y\in O_{K}\smallsetminus \{0\}$ . $x_{0}\,d\log (x_{0})=\sum _{i=1}^{n}x_{i}\,d\log (x_{i})$ if $n\geqslant 1$ , $x_{i}\in O_{K}\smallsetminus \{0\}$ , and $x_{0}=\sum _{i=1}^{n}x_{i}$ .

Let $h$ be a generator of the ideal $\mathfrak{m}_{K}^{t}$ of $O_{K}$ . By the above presentation of $\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log )$ , in the case $r=1$ , it is sufficient to prove that $\{E(hx_{0}),x_{0}\}\equiv \sum _{i=1}^{n}\{E(hx_{i}),x_{i}\}~\text{mod}~U^{pt}K_{2}^{M}(K)$ if $x_{i}\in O_{K}\smallsetminus \{0\}$ and $x_{0}=\sum _{i=1}^{n}x_{i}$ . (Here we denote the group law of $K_{2}^{M}(K)$ additively.) Since $\{E(hx_{i}),x_{i}\}=\{E(hx_{i}),hx_{i}\}-\{E(hx_{i}),h\}$ and since $\{E(hx_{0}),h\}\equiv \sum _{i=1}^{n}\{E(hx_{i}),h\}$ by (1) of 2.2.2, it is sufficient to prove that

(8) $$\begin{eqnarray}\{E(hx),hx\}\in U^{pt}K_{2}^{M}(K)\quad \text{for}~x\in O_{K}\smallsetminus \{0\}.\end{eqnarray}$$

From (2) of 2.2.2 we have, modulo $U^{pt}K_{2}^{M}(K)$ ,

$$\begin{eqnarray}\displaystyle \{E(hx),hx\} & \equiv & \displaystyle \left\{\mathop{\prod }_{i=1}^{p-1}(1-(hx)^{i})^{-\unicode[STIX]{x1D707}(i)/i},hx\right\}\nonumber\\ \displaystyle & \equiv & \displaystyle -\mathop{\sum }_{i=1}^{p-1}\unicode[STIX]{x1D707}(i)i^{-2}\{1-(hx)^{i},(hx)^{i}\}=0.\nonumber\end{eqnarray}$$

The case $r\geqslant 2$ is reduced to the case $r=1$ and to $\{E(hx),y,y\}\equiv 0~\text{mod}~U^{pt}K_{3}^{M}(K)$ for $x\in O_{K}$ and $y\in K^{\times }$ . Since $\{y,y\}=\{-1,y\}$ in $K_{2}^{M}(K)$ , it is sufficient to prove that $\{E(hx),-1\}\in U^{pt}K_{2}^{M}(K)$ . But this follows from the case $r=1$ because $d\log (-1)=0$ in $\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log )$ .

(2) is clear.

We prove (3). For $x\in O_{K}\smallsetminus \{0\}$ and $y_{1},\ldots ,y_{r-1}\in K^{\times }$ , $E$ sends $d(h\otimes x\,d\log (y_{1})\wedge \cdots \wedge d\log (y_{r-1}))$ to $\{E(hx),hx,y_{1},\ldots ,y_{r-1}\}$ . Hence (3) follows from (8).◻

2.3.1. Let $X$ be a Noetherian scheme of dimension $d<\infty$ . Let ${\mathcal{F}}$ be a sheaf of abelian groups on $X$ for Zariski topology. In 2.3.4 (resp. 2.3.5), for an abelian group  $A$ , we give an elementary understanding of a homomorphism $H^{d}(X,{\mathcal{F}})\rightarrow A$ (resp. $H_{x}^{d}(X,{\mathcal{F}})\rightarrow A$ for a closed point $x$ of  $X$ ), where $H^{d}$ is the cohomology for Zariski topology and $H_{x}^{d}$ is the cohomology with support in  $x$ .

2.3.2. Let $P(X)$ be the set of all $(d+1)$ -tuples $\mathfrak{p}=(x_{0},\ldots ,x_{d})$ of points of $X$ such that $\overline{\{x_{i}\}}\subsetneq \overline{\{x_{i+1}\}}$ for all $0\leqslant i\leqslant d-1$ .

For an integer $s$ such that $0\leqslant s\leqslant d$ , let $Q_{s}(X)$ be the set of all $d$ -tuples $\mathfrak{q}=(x_{0},\ldots ,x_{s-1},x_{s+1},\ldots ,x_{d})$ of points of $X$ such that the set $P_{\mathfrak{q}}(X):=\{(x_{i}^{\prime })_{i}\in P(X)\mid x_{i}^{\prime }=x_{i}\;\text{if}\;i\neq s\}$ is not empty.

2.3.3. For any $\mathfrak{p}=(x_{i})_{i}\in P(X)$ , we have a homomorphism $\unicode[STIX]{x1D704}_{\mathfrak{p}}:{\mathcal{F}}_{\unicode[STIX]{x1D702}(\mathfrak{p})}\rightarrow H^{d}(X,{\mathcal{F}})$ with $\unicode[STIX]{x1D702}(\mathfrak{p})=x_{d}$ defined as the composition ${\mathcal{F}}_{\unicode[STIX]{x1D702}(\mathfrak{p})}=H_{x_{d}}^{0}(X,{\mathcal{F}})\rightarrow H_{x_{d-1}}^{1}(X,{\mathcal{F}})\rightarrow \cdots \rightarrow H_{x_{0}}^{d}(X,{\mathcal{F}})\rightarrow H^{d}(X,{\mathcal{F}})$ . (See [Reference Hartshorne13, Chapter IV] for these cohomology groups with support $\{x_{i}\}$ .)

Hence for an abelian group $A$ , a homomorphism $h:H^{d}(X,{\mathcal{F}})\rightarrow A$ induces a homomorphism $h_{\mathfrak{p}}=h\circ \unicode[STIX]{x1D704}_{\mathfrak{p}}:{\mathcal{F}}_{\unicode[STIX]{x1D702}(\mathfrak{p})}\rightarrow A$ for each $\mathfrak{p}\in P(X)$ .

2.3.6. Lemmas 2.3.4 and 2.3.5 are proved together by induction on $d=\dim (X)$ as follows, by using the localization theory of cohomology with supports explained in [Reference Hartshorne13, Chapter IV].

Let $X_{i}$ be the set of all $x\in X$ such that $\dim (\overline{\{x\}})=i$ . Let $X^{i}$ be the set of all $x\in X$ such that $\dim (\text{Spec}({\mathcal{O}}_{X,x}))=i$ . By using the spectral sequence

$$\begin{eqnarray}E_{1}^{ij}=\mathop{\bigoplus }_{x\in X_{-i}}H_{x}^{i+j}(X,{\mathcal{F}})\Rightarrow H^{i+j}(X,{\mathcal{F}})\end{eqnarray}$$

and using $H_{x}^{i}(X,{\mathcal{F}})=H_{x}^{i}(\text{Spec}({\mathcal{O}}_{X,x}),{\mathcal{F}})$ , where we denote the pullback of ${\mathcal{F}}$ to $\text{Spec}({\mathcal{O}}_{X,x})$ also by ${\mathcal{F}}$ , we obtain the following (1)–(5) by induction on $\dim (X)$ .

1. (1) If $i>d$ , $H^{i}(X,{\mathcal{F}})=0$ and $H_{x}^{i}(X,{\mathcal{F}})=0$ for any $x\in X$ .

2. (2) We have an exact sequence $\bigoplus _{x\in X_{1}}H_{x}^{d-1}(X,{\mathcal{F}})\rightarrow \bigoplus _{x\in X_{0}}H_{x}^{d}(X,{\mathcal{F}})\rightarrow H^{d}(X,{\mathcal{F}})\rightarrow 0$ .

3. (3) If $x\in X^{i}$ and $i\geqslant 2$ , $H_{x}^{i}(X,{\mathcal{F}})\cong H^{i-1}(\text{Spec}({\mathcal{O}}_{X,x})\smallsetminus \{x\},{\mathcal{F}})$ and $\text{Spec}({\mathcal{O}}_{X,x})\smallsetminus \{x\}$ is of dimension $i-1$ .

4. (4) If $x\in X^{1}$ , we have an exact sequence ${\mathcal{F}}_{x}\rightarrow \bigoplus _{\unicode[STIX]{x1D702}\in \text{Spec}({\mathcal{O}}_{X,x})\smallsetminus \{x\}}{\mathcal{F}}_{\unicode[STIX]{x1D702}}\rightarrow H_{x}^{1}(X,{\mathcal{F}})\rightarrow 0$ .

5. (5) If $x\in X^{0}$ , $H_{x}^{0}(X,{\mathcal{F}})={\mathcal{F}}_{x}$ .

By (1)–(5) and by induction on $d=\dim (X)$ , we have the map $h\mapsto (h_{\mathfrak{p}})_{\mathfrak{p}}$ from the left hand side of 2.3.4 (resp. 2.3.5) to the right hand side of 2.3.4 (resp. 2.3.5), that $H^{d}(X,{\mathcal{F}})$ (resp.  $H_{x}^{d}(X,{\mathcal{F}})$ for $x\in X_{0}$ ) is generated by the images of $\unicode[STIX]{x1D704}_{\mathfrak{p}}:{\mathcal{F}}_{\unicode[STIX]{x1D702}(\mathfrak{p})}\rightarrow H^{d}(X,{\mathcal{F}})$ for $\mathfrak{p}\in P(X)$ (resp. $\unicode[STIX]{x1D704}_{x,\mathfrak{p}}:{\mathcal{F}}_{\unicode[STIX]{x1D702}(\mathfrak{p})}\rightarrow H_{x}^{d}(X,{\mathcal{F}})$ for $\mathfrak{p}\in P_{x}(X)$ ), and hence that this map $h\mapsto (h_{\mathfrak{p}})_{\mathfrak{p}}$ is injective. We give the proof of the surjectivity of this map.

Let $x\in X_{0}$ and let $(h_{\mathfrak{p}})_{\mathfrak{p}}$ be an element of the set on the right hand side of 2.3.5. If $x\in X^{i}$ with $i<d$ , $P_{x}(X)$ is empty. So we may assume $x\in X^{d}$ . Let $U=\text{Spec}({\mathcal{O}}_{X,x})\smallsetminus \{x\}$ . We have the bijection $P_{x}(X)\rightarrow P(U)\;;\mathfrak{p}=(x_{i})_{0\leqslant i\leqslant d}\mapsto \mathfrak{p}^{\prime }=(x_{i+1})_{0\leqslant i\leqslant d-1}$ . By induction on  $d$ , $(h_{\mathfrak{p}^{\prime }})_{\mathfrak{p}^{\prime }\in P(U)}$ corresponds to a homomorphism $H^{d-1}(U,{\mathcal{F}})\rightarrow A$ . If $d\geqslant 2$ , by the above (3), $(h_{\mathfrak{p}})_{\mathfrak{p}}$ corresponds to a homomorphism $h:H_{x}^{d}(X,{\mathcal{F}})\rightarrow A$ . The proof for the case $d=1$ (resp.  $d=0$ ) is similar by using the above (4) (resp. (5)).

Next let $(h_{\mathfrak{p}})_{\mathfrak{p}}$ be an element of the set on the right hand side of 2.3.4. The case $d=0$ is trivial and so assume $d\geqslant 1$ . Since $P(X)$ is the disjoint union of $P_{x}(X)$ for $x\in X_{0}$ , $(h_{\mathfrak{p}})_{\mathfrak{p}}$ induces a homomorphism $\bigoplus _{x\in X_{0}}H_{x}^{d}(X,{\mathcal{F}})\rightarrow A$ . Consider the exact sequence in the above (2). For $x\in X_{1}$ , $H_{x}^{d-1}(X,{\mathcal{F}})$ is zero unless $x\in X^{d-1}$ by the above (1), and is generated by the images of $h_{x,\mathfrak{q}}:{\mathcal{F}}_{\unicode[STIX]{x1D702}(\mathfrak{q})}\rightarrow H_{x}^{d-1}(X,{\mathcal{F}})$ if $x\in X^{d-1}$ where $\mathfrak{q}$ ranges over all elements of $P_{x}(\text{Spec}({\mathcal{O}}_{X,x}))$ . By this fact and by the fact $Q_{0}(X)$ is the disjoint union of $P_{x}(\text{Spec}({\mathcal{O}}_{X,x}))$ for $x\in X_{1}\cap X^{d-1}$ , the homomorphism $\bigoplus _{x}H_{x}^{d}(X,{\mathcal{F}})\rightarrow A$ kills the image of $\bigoplus _{x\in X_{1}}H_{x}^{d-1}(X,{\mathcal{F}})$ by the property (i) with $s=0$ of $(h_{\mathfrak{p}})_{\mathfrak{p}}$ in 2.3.4. Hence $(h_{\mathfrak{p}})_{\mathfrak{p}}$ corresponds to a homomorphism $H^{d}(X,{\mathcal{F}})\rightarrow A$ .

2.4.1. Recall that the notion $d$ -dimensional local field is as follows. A 0-dimensional local field is a finite field. For $d\geqslant 1$ , a $d$ -dimensional local field is a complete discrete valuation field whose residue field is a $d-1$ -dimensional local field.

2.4.2. The following remark is used in 2.4.8 later.

To give a $d$ -dimensional local field $K$ is equivalent to giving a valuation ring $V$ having the following properties (i)–(iii). (i) The residue field of $V$ is a finite field. (ii) The value group of $V$ is isomorphic to $\mathbb{Z}^{d}$ with the lexicographic order. (iii) If $P_{0}\supsetneq \cdots \supsetneq P_{d}=(0)$ are all prime ideals of  $V$ , for each $1\leqslant i\leqslant d$ , the local ring of $V/P_{i}$ at the prime ideal $P_{i-1}/P_{i}$ (which is a discrete valuation ring) is complete.

In fact, $K$ is obtained from $V$ as the field of fractions of  $V$ . $V=V_{K}$ is a subring of $K$ defined by induction on $d$ as follows. If $d=0$ , then $V_{K}=K$ . If $d\geqslant 1$ , $V_{K}$ is the subring of the discrete valuation ring $O_{K}$ consisting of all elements whose images in the residue field $F$ of $K$ belong to  $V_{F}$ .

2.4.3. By higher dimensional local field theory (see [Reference Kato17, Reference Parshin31]), for a $d$ -dimensional local field  $K$ , we have a canonical homomorphism $K_{d}^{M}(K)\rightarrow \text{Gal}(K^{\text{ab}}/K)$ called the reciprocity map, where $K^{\text{ab}}$ is the maximal abelian extension of  $K$ .

2.4.4. For $n\geqslant 0$ , define the category ${\mathcal{F}}_{n}$ inductively as follows. ${\mathcal{F}}_{0}$ is the category of finite sets. For $n\geqslant 1$ , ${\mathcal{F}}_{n}=\text{ind}(\text{pro}({\mathcal{F}}_{n-1}))$ where $\text{pro}(\;)$ is the category of pro-objects and $\text{ind}(\;)$ is the category of ind-objects. Let ${\mathcal{F}}_{\infty }=\bigcup _{n}{\mathcal{F}}_{n}$ .

2.4.5. For a $d$ -dimensional local field $K$ , we can regard $K$ and $K^{\times }$ as objects of ${\mathcal{F}}_{\infty }$ (actually of ${\mathcal{F}}_{d}$ ) canonically [Reference Kato22, Introduction]. A homomorphism $K_{d}^{M}(K)\rightarrow \mathbb{Q}/\mathbb{Z}$ is said to be continuous if the composition $K^{\times }\times \cdots \times K^{\times }\;(d~\text{times})\overset{\{\;\}}{\rightarrow }K_{d}^{M}(K)\rightarrow \mathbb{Q}/\mathbb{Z}$ is a morphism of  ${\mathcal{F}}_{\infty }$ .

A main result of higher dimensional local class field theory formulated in [Reference Kato22, Theorem 2] is the following. Via the reciprocity map in 2.4.3, $H^{1}(K,\mathbb{Q}/\mathbb{Z})=\text{Hom}_{\text{cont}}(\text{Gal}(K^{\text{ab}}/K),\mathbb{Q}/\mathbb{Z})$ is isomorphic to the group of continuous homomorphisms $K_{d}^{M}(K)\rightarrow \mathbb{Q}/\mathbb{Z}$ of finite orders.

2.4.6. Let $X$ be an integral scheme of finite type over $\mathbb{Z}$ of dimension $d$ , and let $K$ be the function field of  $X$ . By a place of $K$ along $X$ , we mean a subring $v$ of $K$ satisfying the following conditions (i)–(iii).

1. (i) $K$ is the field of fractions of $v$ .

2. (ii) $v$ is a valuation ring and the value group is isomorphic to $\mathbb{Z}^{d}$ with the lexicographic order.

3. (iii) There is an element $\mathfrak{p}=(x_{i})_{0\leqslant i\leqslant d}\in P(X)$ such that if $P_{0}\supsetneq \cdots \supsetneq P_{d}$ denotes the set of all prime ideals of  $v$ , then for $0\leqslant i\leqslant d$ , the local ring of $v$ at $P_{i}$ (which is a valuation ring) dominates  $x_{i}$ .

Let $Pl(X)$ be the set of all places of $K$ along $X$ . We have a canonical map $Pl(X)\rightarrow P(X)$ which sends $v\in Pl(X)$ to $\mathfrak{p}$ in (iii).

For $v\in Pl(X)$ , we will define a $d$ -dimensional local field $K_{v}\supset K$ called the local field of $K$ at  $v$ . The class field theory of $K$ is related to the local class field theory of $K_{v}$ as is explained below.

2.4.7. To obtain local fields $K_{v}$ of $K$ for $v\in Pl(X)$ and their henselian versions $K_{v}^{h}$ , we use the following iterated completion and iterated henselization.

For $0\leqslant i\leqslant d$ and for a sequence $(x_{0},\ldots ,x_{i})$ of points of $X$ such that $\overline{\{x_{0}\}}\subset \cdots \subset \overline{\{x_{i}\}}$ and such that $\dim (\overline{\{x_{j}\}})=j$ for $0\leqslant j\leqslant i$ , we define the rings $\hat{{\mathcal{O}}}_{X,x_{0},\ldots ,x_{i}}$ and ${\mathcal{O}}_{X,x_{0},\ldots ,x_{i}}^{h}$ over ${\mathcal{O}}_{X,x_{i}}$ as follows, inductively.

In the case $i=0$ , $\hat{{\mathcal{O}}}_{X,x_{0}}$ (resp. ${\mathcal{O}}_{X,x_{0}}^{h}$ ) is the completion (resp. henselization) of ${\mathcal{O}}_{X,x_{0}}$ .

For $i\geqslant 1$ , let

$$\begin{eqnarray}\hat{{\mathcal{O}}}_{X,x_{0},\ldots ,x_{i}}=\mathop{\prod }_{P}(\hat{{\mathcal{O}}}_{X,x_{0},\ldots ,x_{i-1}})_{P}^{\wedge },\quad {\mathcal{O}}_{X,x_{0},\ldots ,x_{i}}^{h}=\mathop{\prod }_{P}({\mathcal{O}}_{X,x_{0},\ldots ,x_{i-1}}^{h})_{P}^{h}\end{eqnarray}$$

where $P$ ranges over all prime ideals of $\hat{{\mathcal{O}}}_{X,x_{0},\ldots ,x_{i-1}}$ (resp. ${\mathcal{O}}_{X,x_{0},\ldots ,x_{i-1}}^{h}$ ) lying over the prime ideal of ${\mathcal{O}}_{X,x_{i-1}}$ corresponding to $x_{i}$ , and $(\;)_{P}^{\wedge }$ (resp. $(\;)_{P}^{h}$ ) denotes the completion (resp. henselization) of the local ring $(\;)_{P}$ of the ring $(\;)$ at  $P$ . By induction on $i$ , we see that the set of such $P$ is a nonempty finite set and that $\hat{{\mathcal{O}}}_{X,x_{0},\ldots ,x_{i}}$ and ${\mathcal{O}}_{X,x_{0},\ldots ,x_{i}}^{h}$ are finite products of complete (resp. henselian) local integral domains of dimension $d-i$ .

For $\mathfrak{p}=(x_{i})_{i}\in P(X)$ , let

$$\begin{eqnarray}K_{\mathfrak{p}}=\hat{{\mathcal{O}}}_{X,x_{0},\ldots ,x_{d}},\quad K_{\mathfrak{p}}^{h}={\mathcal{O}}_{X,x_{0},\ldots ,x_{d}}^{h}.\end{eqnarray}$$

Then $K_{\mathfrak{p}}$ and $K_{\mathfrak{p}}^{h}$ are finite products of fields.

2.4.8. Let $X$ and $K$ be as above. The following statements are proved in 2.4.9 by induction on  $d$ .

1. (1) Let $\mathfrak{p}\in P(X)$ , and let $Pl(\mathfrak{p})=Pl(X,\mathfrak{p})$ be the inverse image of $\mathfrak{p}$ in $Pl(X)$ . Then each field factor of $K_{\mathfrak{p}}$ has canonically a structure of a $d$ -dimensional local field. We have a bijection $\text{Spec}(K_{\mathfrak{p}})\rightarrow Pl(\mathfrak{p})$ which sends the point of $\text{Spec}(K_{\mathfrak{p}})$ corresponding to a field factor $L$ of $K_{\mathfrak{p}}$ to the valuation ring $V_{L}\cap K$ , where $V_{L}\subset L$ is the valuation ring of rank $d$ associated to $L$ by 2.4.2. In particular, the set $Pl(\mathfrak{p})$ is a nonempty finite set.

2. (2) Let $\mathfrak{p}\in P(X)$ . Then the map $\text{Spec}(K_{\mathfrak{p}})\rightarrow \text{Spec}(K_{\mathfrak{p}}^{h})$ induced by the inclusion map $K_{\mathfrak{p}}^{h}\rightarrow K_{\mathfrak{p}}$ is bijective.

3. (3) Let $X^{\prime }\rightarrow X$ be a finite surjective morphism of integral schemes of finite type over $\mathbb{Z}$ and let $K^{\prime }$ be the function field of  $X^{\prime }$ . Then we have canonical isomorphisms

   $$\begin{eqnarray}K^{\prime }\otimes _{K}K_{v}=\mathop{\prod }_{v^{\prime }|v}K_{v^{\prime }}^{\prime },\quad K^{\prime }\otimes _{K}K_{v}^{h}=\mathop{\prod }_{v^{\prime }}(K^{\prime })_{v^{\prime }}^{h}\end{eqnarray}$$
   where $v^{\prime }|v$ means that $v^{\prime }$ ranges over all elements of $Pl(X^{\prime })$ such that $v^{\prime }\cap K=v$ .

2.4.9. We prove statements in 2.4.8 by induction on $d$ .

We prove (1). Let $\mathfrak{p}=(x_{i})_{i}\in P(X)$ , and let $Y\subset X$ be the closure of $x_{d-1}$ in $X$ with the reduced scheme structure. Let $\mathfrak{q}=(x_{0},\ldots ,x_{d-1})\in P(Y)$ . Let  $F$ be the residue field of $x_{d-1}$ , that is, the function field of  $Y$ . Let $A$ be the normalization of ${\mathcal{O}}_{X,x_{d-1}}$ and let $\unicode[STIX]{x1D6E5}$ be the set of all maximal ideals of  $A$ . Then $\unicode[STIX]{x1D6E5}$ is finite. For $z\in \unicode[STIX]{x1D6E5}$ , let $\unicode[STIX]{x1D705}(z)$ be the residue field of $z$ which is a finite extension of  $F$ , let $Y(z)$ be the integral closure of $Y$ in $\unicode[STIX]{x1D705}(z)$ and let $Pl(Y(z),\mathfrak{q})\subset Pl(Y(z))$ be the inverse image of $\mathfrak{q}$ under the map $Pl(Y(z))\rightarrow P(Y)$ . Then we have a bijection $\coprod _{z\in \unicode[STIX]{x1D6E5}}Pl(Y(z),\mathfrak{q})\rightarrow Pl(\mathfrak{p})$ which sends $w\in Pl(Y(z),\mathfrak{q})$ with $z\in \unicode[STIX]{x1D6E5}$ to the valuation ring consisting of all elements of the local ring of $A$ at $z$ whose residue classes in $\unicode[STIX]{x1D705}(z)$ belong to  $w$ . The ring $A$ is Noetherian normal one-dimensional semilocal integral domain, and hence is a principal ideal domain. Let $t$ be a generator of the intersection of all maximal ideals of  $A$ . Let $B=\hat{{\mathcal{O}}}_{X,x_{0},\ldots ,x_{d-1}}$ and let $C=A\otimes _{{\mathcal{O}}_{X,x_{d-1}}}B$ . Then $C$ is a finite product of one-dimensional local integral domains, $C/tC=(\prod _{z\in \unicode[STIX]{x1D6E5}}\unicode[STIX]{x1D705}(z))\otimes _{F}B/\mathfrak{m}B$ where $\mathfrak{m}$ is the maximal ideal of ${\mathcal{O}}_{X,x_{d-1}}$ , and $B/\mathfrak{m}B=\hat{{\mathcal{O}}}_{Y,x_{0},\ldots ,x_{d-1}}=\prod _{\unicode[STIX]{x1D706}\in Pl(\mathfrak{q})}F_{\unicode[STIX]{x1D706}}$ where the last $=$ is by the statement 1 for dimension $d-1$ . Hence $\unicode[STIX]{x1D705}(z)\otimes _{F}B/\mathfrak{m}B=\unicode[STIX]{x1D705}(z)\otimes _{F}\prod _{\unicode[STIX]{x1D706}\in Pl(\mathfrak{q})}F_{\unicode[STIX]{x1D706}}=\prod _{w\in Pl(Y(z),\mathfrak{q})}\unicode[STIX]{x1D705}(z)_{w}$ by the statement (3) for dimension $d-1$ . This shows that $\text{Spec}(K_{\mathfrak{p}})$ is identified with $\coprod _{z\in \unicode[STIX]{x1D6E5}}Pl(Y(z),\mathfrak{q})$ . Hence $\text{Spec}(K_{\mathfrak{p}})$ is identified with $Pl(\mathfrak{p})$ , and for $v\in Pl(\mathfrak{p})$ corresponding to $w\in Pl(Y(z),\mathfrak{q})$ , $K_{v}$ is the field of fractions of a complete discrete valuation ring whose residue field is $\unicode[STIX]{x1D705}(z)_{w}$ . This proves (1).

We prove (2). Replacing the iterated completion in the above arguments by the iterated henselization, and by using the induction on  $d$ , we obtain similarly a bijection between $\text{Spec}(K_{\mathfrak{p}}^{h})$ and $Pl(\mathfrak{p})$ . This shows that the map $\text{Spec}(K_{\mathfrak{p}})\rightarrow \text{Spec}(K_{\mathfrak{p}}^{h})$ is bijective.

We prove (3) for $K_{v}$ . Let $f$ be the morphism $X^{\prime }\rightarrow X$ . By induction on $i$ , we have that $(f_{\ast }{\mathcal{O}}_{X^{\prime }})_{x_{i}}\otimes _{{\mathcal{O}}_{X,x_{d-1}}}\hat{{\mathcal{O}}}_{X,x_{0},\ldots ,x_{i}}\overset{\cong }{\rightarrow }\prod _{x_{0}^{\prime },\ldots ,x_{i}^{\prime }}\hat{{\mathcal{O}}}_{X^{\prime },x_{0}^{\prime },\ldots ,x_{i}^{\prime }}$ where $(x_{0}^{\prime },\ldots ,x_{i}^{\prime })$ ranges over all sequences of points of $X^{\prime }$ such that $x_{j}^{\prime }$ lies over $x_{j}$ for $0\leqslant j\leqslant i$ and such that $\overline{\{x_{0}^{\prime }\}}\subset \cdots \subset \overline{\{x_{i}^{\prime }\}}$ . The case $i=d$ gives an isomorphism $K^{\prime }\otimes _{K}K_{\mathfrak{p}}\overset{\cong }{\rightarrow }\prod _{\mathfrak{p}^{\prime }}K_{\mathfrak{p}^{\prime }}^{\prime }$ where $\mathfrak{p}^{\prime }$ ranges over all elements of $P(X^{\prime })$ lying over $\mathfrak{p}$ . By looking at the $v$ -factor of this isomorphism, we have (3) for $K_{v}$ .

The proof of (3) for $K_{v}^{h}$ is similar to that for $K_{v}$ .

2.4.10. In 2.4.10–2.4.12, let $X$ be a proper normal integral scheme over $\mathbb{Z}$ and let $K$ be the function field of  $X$ . We assume $X(\mathbb{R})=\emptyset$ .

The following is what we use in this paper from the higher dimensional global class field theory.

We have a unique continuous homomorphism

(9) $$\begin{eqnarray}\mathop{\varprojlim }_{I}H^{d}(X,K_{d}^{M}({\mathcal{O}}_{X},I))\rightarrow \text{Gal}(K^{ab}/K)\end{eqnarray}$$

called the reciprocity map, where $I$ ranges over all nonzero coherent ideals of ${\mathcal{O}}_{X}$ , $K_{d}^{M}({\mathcal{O}}_{X},I)$ denotes the kernel of $K_{d}^{M}({\mathcal{O}}_{X})\rightarrow K_{d}^{M}({\mathcal{O}}_{X}/I)$ with $K_{d}^{M}$ the sheaf of $d$ th Milnor $K$ -groups, the cohomology groups are the Zariski cohomology groups, and the left hand side is regarded as a topological space for the projective limit of the discrete topologies of $H^{d}(X,K_{d}^{M}({\mathcal{O}}_{X},I))$ , which is characterized by the following property. For $\unicode[STIX]{x1D712}\in H^{1}(K,\mathbb{Q}/\mathbb{Z})=\text{Hom}_{\text{cont}}(\text{Gal}(K^{\text{ab}}/K),\mathbb{Q}/\mathbb{Z})$ , the homomorphism

$$\begin{eqnarray}H^{d}(X,K_{d}^{M}({\mathcal{O}}_{X},I))\rightarrow \mathbb{Q}/\mathbb{Z}\end{eqnarray}$$

induced by $\unicode[STIX]{x1D712}$ for some $I$ corresponds to $(h_{\mathfrak{p}})_{\mathfrak{p}\in P(X)}$ via 2.3.4 where $h_{\mathfrak{p}}:K_{d}^{M}(K)\rightarrow \mathbb{Q}/\mathbb{Z}$ is the composition

$$\begin{eqnarray}K_{d}^{M}(K)\rightarrow \bigoplus _{v\in Pl(\mathfrak{p})}K_{d}^{M}(K_{v})\overset{\unicode[STIX]{x1D712}}{\rightarrow }\mathbb{Q}/\mathbb{Z}\end{eqnarray}$$

where $Pl(\mathfrak{p})$ is as in 2.4.8, the first arrow is the diagonal map, and the map $K_{d}^{M}(K_{v})\rightarrow \mathbb{Q}/\mathbb{Z}$ is induced by the image of $\unicode[STIX]{x1D712}$ in $H^{1}(K_{v},\mathbb{Q}/\mathbb{Z})$ and by the $d$ -dimensional local class field theory of  $K_{v}$ .

2.4.11. In [Reference Kato and Saito23], a continuous map

(10) $$\begin{eqnarray}\mathop{\varprojlim }_{I}H^{d}(X_{\text{Nis}},K_{d}^{M}({\mathcal{O}}_{X},I))\rightarrow \text{Gal}(K^{ab}/K)\end{eqnarray}$$

called the reciprocity map is defined, where the cohomology groups are Nisnevich cohomology, instead of Zariski cohomology, and the left hand side is endowed with the topology of the projective limit of the discrete sets.

The map (9) is induced from the map (10) and the canonical map $H^{d}(X,K_{d}^{M}({\mathcal{O}}_{X},I))\rightarrow H^{d}(X_{\text{Nis}},K_{d}^{M}({\mathcal{O}}_{X},I))$ .

In fact, by the definition of the reciprocity map (10) in [Reference Kato and Saito23], for $\unicode[STIX]{x1D712}\in H^{1}(K,\mathbb{Q}/\mathbb{Z})$ , the induced homomorphism (9) corresponds to $(h_{\mathfrak{p}})_{\mathfrak{p}\in P(X)}$ where $h_{\mathfrak{p}}:K_{d}^{M}(K)\rightarrow \mathbb{Q}/\mathbb{Z}$ is the composition $K_{d}^{M}(K)\rightarrow \bigoplus _{v\in Pl(\mathfrak{p})}K_{d}^{M}(K_{v}^{h})$ $\overset{\unicode[STIX]{x1D712}}{\rightarrow }\mathbb{Q}/\mathbb{Z}$ with $K_{d}^{M}(K_{v}^{h})\rightarrow \mathbb{Q}/\mathbb{Z}$ a homomorphism defined in [Reference Kato and Saito23]. This last homomorphism coincides with the composition $K_{d}^{M}(K_{v}^{h})\rightarrow K_{d}^{M}(K_{v})\rightarrow \mathbb{Q}/\mathbb{Z}$ where the last map is that in 2.4.10.

2.4.12. The main result of [Reference Kato and Saito23] concerning the class field theory of $K$ is as follows. (In [Reference Kato and Saito23], we do not need to assume $X(\mathbb{R})=\emptyset$ , but if we do not assuming $X(\mathbb{R})=\emptyset$ , the left hand side of (10) should be modified by adding archimedean objects to have results below.)

In the case $K$ is of characteristic 0, the map (10) is an isomorphism of topological groups.

In the case $K$ is of characteristic $p>0$ , the map (10) induces an isomorphism of topological groups from the left hand side of (10) to the fiber product of $\text{Gal}(K^{\text{ab}}/K)\rightarrow \text{Gal}(\mathbb{F}_{p}^{\text{ab}}/\mathbb{F}_{p})\leftarrow \mathbb{Z}$ , where $\mathbb{Z}$ is discrete and the right arrow sends $1\in \mathbb{Z}$ to the Frobenius $\mathbb{F}_{p}^{\text{ab}}\rightarrow \mathbb{F}_{p}^{\text{ab}}\;;\;x\mapsto x^{p}$ .

By Raskind [Reference Raskind32] and Kerz and Saito [Reference Kerz and Saito26], these results on the class field theory of $K$ hold also for Zariski cohomology $\mathop{\varprojlim }\nolimits_{I}H^{d}(X,K_{d}^{m}({\mathcal{O}}_{X},I))$ (replacing Nisnevich cohomology) if $K$ is of characteristic $\neq 2$ .

2.4.13. There is another formulation of higher dimensional class field theory due to Wiesend [Reference Wiesend34] which was studied more in [Reference Kerz and Saito26]. But we do not use it in this paper.

3.1.1. Let $K$ be a complete discrete valuation field with residue field $F$ , and assume $F$ is of characteristic $p>0$ . Let $n\geqslant 1$ and let $m=\max (n-e_{K},[n/p])$ .

We define a homomorphism

$$\begin{eqnarray}\text{Rsw}:F_{n}H^{1}(K,\mathbb{Q}/\mathbb{Z})\rightarrow \mathfrak{m}_{K}^{-n}/\mathfrak{m}_{K}^{-m}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log )\end{eqnarray}$$

which we call the refined Swan conductor modulo $p$ .

3.1.2. The homomorphism $\text{Rsw}$ in 3.1.1 is characterized by the following properties (i) and (ii).

1. (i) $\text{Rsw}$ is compatible with any homomorphisms of cdvf. That is, the following diagram is commutative for an extension of complete discrete valuation fields $K^{\prime }/K$ , where $n^{\prime }=e(K^{\prime }/K)n$ with $e(K^{\prime }/K)$ the ramification index of the extension $K^{\prime }/K$ and $m^{\prime }=\max (n^{\prime }-e_{K^{\prime }},[n^{\prime }/p])$ .

   $$\begin{eqnarray}\begin{array}{@{}ccc@{}}F_{n}H^{1}(K,\mathbb{Q}/\mathbb{Z}) & \rightarrow & \mathfrak{m}_{K}^{-n}/\mathfrak{m}_{K}^{-m}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log )\\ \downarrow & & \downarrow \\ F_{n^{\prime }}H^{1}(K^{\prime },\mathbb{Q}/\mathbb{Z}) & \rightarrow & \mathfrak{m}_{K^{\prime }}^{-n^{\prime }}/\mathfrak{m}_{K^{\prime }}^{-m^{\prime }}\otimes _{O_{K^{\prime }}}\unicode[STIX]{x1D6FA}_{O_{K^{\prime }}}^{1}(\log )\end{array}\end{eqnarray}$$
   (The fact $F_{n}H^{1}(K,\mathbb{Q}_{p}/\mathbb{Z}_{p})$ is sent to $F_{n^{\prime }}H^{1}(K^{\prime },\mathbb{Q}_{p}/\mathbb{Z}_{p})$ is proved in [Reference Kato20].)

2. (ii) If $F$ is an $r$ -dimensional local field ( $r\geqslant 0$ ), Rsw is characterized by the property

   $$\begin{eqnarray}\unicode[STIX]{x1D712}(E(\unicode[STIX]{x1D6FC}))=\text{Res}_{F}(R_{b}(\unicode[STIX]{x1D6FC}\wedge \text{Rsw}(\unicode[STIX]{x1D712})))\end{eqnarray}$$
   for $\unicode[STIX]{x1D712}\in F_{n}H^{1}(K,\mathbb{Q}_{p}/\mathbb{Z}_{p})$ , $\unicode[STIX]{x1D6FC}\in \mathfrak{m}_{K}^{m+1}/\mathfrak{m}_{K}^{n+1}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{r}(\log )$ . Here $E(\unicode[STIX]{x1D6FC})$ denotes the image of $\unicode[STIX]{x1D6FC}$ under the truncated exponential map
   $$\begin{eqnarray}\mathfrak{m}_{K}^{m+1}/\mathfrak{m}_{K}^{n+1}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{r}(\log )\rightarrow U^{m+1}K_{r+1}^{M}(K)/U^{n+1}K_{r+1}^{M}(K),\end{eqnarray}$$
   $\unicode[STIX]{x1D712}(E(\unicode[STIX]{x1D6FC}))$ denotes the image of $E(\unicode[STIX]{x1D6FC})$ under the composition $K_{r+1}^{M}(K)\rightarrow \text{Gal}(K^{\text{ab}}/K)\overset{\unicode[STIX]{x1D712}}{\rightarrow }\mathbb{Q}_{p}/\mathbb{Z}_{p}$ of the reciprocity map of the $r+1$ -dimensional local field $K$ and  $\unicode[STIX]{x1D712}$ , which kills $U^{n+1}K_{r+1}^{M}(K)$ by the condition $\text{Sw}(\unicode[STIX]{x1D712})\leqslant n$ , and
   $$\begin{eqnarray}\text{Res}_{F}:\unicode[STIX]{x1D6FA}_{F}^{r}\rightarrow \mathbb{F}_{p}\end{eqnarray}$$
   is the residue map (3.3.2). Note that $\unicode[STIX]{x1D6FC}\wedge \text{Rsw}(\unicode[STIX]{x1D712})\in \mathfrak{m}_{K}^{m+1-n}/\mathfrak{m}_{K}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{r+1}(\log )$ and $n-m\leqslant e_{K}$ and hence its $R_{b}$ is defined in  $\unicode[STIX]{x1D6FA}_{F}^{r}$ .

3.1.3. If $\text{char}(K)=p>0$ ,

$$\begin{eqnarray}\text{Rsw}:F_{n}H^{1}(K,\mathbb{Q}/\mathbb{Z})\rightarrow \mathfrak{m}_{K}^{-n}/\mathfrak{m}_{K}^{-[n/p]}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log )\end{eqnarray}$$

is the homomorphism of Matsuda [Reference Matsuda30, Remark 3.2.2] and Borger [Reference Borger6, 3.6] defined by using Artin–Schreier–Witt theory. It was studied in [Reference Leal29, Section 2]. See Sections 3.2 and 3.5.9.

In the mixed characteristic case, we will construct $\text{Rsw}$ using higher dimensional local class field theory (Section 3.3) and also higher dimensional global class field theory (Section 3.4).

3.1.4. It may seem strange that we use higher dimensional global class field theory for the local subject $\text{Rsw}$ . Our idea is that since any field is a union of finitely generated fields over a prime field and since each finitely generated field over a prime field has class field theory, any subject about one-dimensional Galois representations of any field can be studied by using higher dimensional global class field theory. Our method in 3.5.6 to define $\text{Rsw}$ actually follows this idea.

A purely local method for the local subject $\text{Rsw}$ exists also in the mixed characteristic case as in 3.1.5.

3.1.5. In the case $F$ is perfect, our homomorphism Rsw can be defined also by using the local class field theory of Serre [Reference Serre33] and Hazewinkel [Reference Hazewinkel14].

In the case $[F:F^{p}]=p^{r}<\infty$ , Rsw can be also defined by using the duality theory [Reference Kato and Suzuki25] which is a generalization of the local class field theory of Serre [Reference Serre33] and Hazewinkel [Reference Hazewinkel14] to the case $[F:F^{p}]=p^{r}$ .

These things will be explained elsewhere.

3.1.6. The authors wonder whether Rsw in general can be obtained by the reduction to the perfect residue field case using the local class field theory of Serre [Reference Serre33] and Hazewinkel [Reference Hazewinkel14] and the work [Reference Borger5] of Borger.

3.1.7. The authors wonder whether our refined Swan conductor mod $p$ can be obtained from the relation of $p$ -adic étale cohomology and de Rham–Witt complex in [Reference Geisser and Hesselholt11] or from the relation of $p$ -adic étale cohomology and Hochschild homology in [Reference Bhatt, Morrow and Scholze4].

3.2.1. Let $K$ be a complete discrete valuation field of characteristic $p>0$ . We briefly review the definition of

$$\begin{eqnarray}\text{Rsw}:F_{n}H^{1}(K,\mathbb{Q}/\mathbb{Z})\rightarrow \mathfrak{m}_{K}^{-n}/\mathfrak{m}_{K}^{-[n/p]}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log ).\end{eqnarray}$$

For details, see [Reference Borger6, Reference Illusie16, Reference Matsuda30].

In 3.5.9, we will show that this Rsw has the properties (i) and (ii) in 3.1.2.

3.2.2. From Artin–Schreier–Witt theory, there are isomorphisms

(11) $$\begin{eqnarray}W_{s}(K)/(\unicode[STIX]{x1D719}-1)W_{s}(K)\simeq H^{1}(K,\mathbb{Z}/p^{s}\mathbb{Z}),\end{eqnarray}$$

where $W_{s}(K)$ denotes the ring of Witt vectors of length $s$ , and $\unicode[STIX]{x1D719}$ the endomorphism of Frobenius.

3.2.3. For a Witt vector $a=(a_{s-1},\ldots ,a_{0})\in W_{s}(K)$ , let $\text{ord}_{K}(a)=\min _{i}\{p^{i}\text{ord}_{K}(a_{i})\}$ . In [Reference Brylinski7], Brylinski defined an increasing filtration of $W_{s}(K)$ as

$$\begin{eqnarray}F_{n}W_{s}(K)=\{a\in W_{s}(K)\mid \text{ord}_{K}(a)\geqslant -n\}\end{eqnarray}$$

for $n\in \mathbb{Z}_{{\geqslant}0}$ . The filtration $F_{n}H^{1}(K,\mathbb{Z}/p^{s}\mathbb{Z})$ defined in [Reference Kato20] is the image of $F_{n}W_{s}(K)$ under (11).

3.2.4. The homomorphism $-d:W_{s}(K)\rightarrow \unicode[STIX]{x1D6FA}_{K}^{1}$ given by $a=(a_{s-1},\ldots ,a_{0})\mapsto -\sum _{i}a_{i}^{p^{i}-1}\,da_{i}$ produces a map

$$\begin{eqnarray}F_{n}W_{s}(K)/F_{[n/p]}W_{s}(K)\rightarrow \mathfrak{m}_{K}^{-n}/\mathfrak{m}_{K}^{-[n/p]}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log ),\end{eqnarray}$$

which factors through

$$\begin{eqnarray}F_{n}H^{1}(K,\mathbb{Z}/p^{s}\mathbb{Z})/F_{[n/p]}H^{1}(K,\mathbb{Z}/p^{s}\mathbb{Z})\rightarrow \mathfrak{m}_{K}^{-n}/\mathfrak{m}_{K}^{-[n/p]}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log ),\end{eqnarray}$$

inducing

$$\begin{eqnarray}\text{Rsw}:F_{n}H^{1}(K,\mathbb{Q}/\mathbb{Z})\rightarrow \mathfrak{m}_{K}^{-n}/\mathfrak{m}_{K}^{-[n/p]}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log ).\end{eqnarray}$$

Here the minus sign of the definition of $-d$ may seem strange, but we put it to have the compatibility with the refined Swan conductor mod $\mathfrak{m}_{K}$ in [Reference Kato20]. The minus sign naturally appears in the argument in 3.5.11.

3.3.1. Let $K$ be a complete discrete valuation field with residue field  $F$ . In the case $F$ is an $r$ -dimensional local field, we define Rsw by using higher dimensional local class field theory as below.

We will use the continuity of $K_{r+1}^{M}(K)\rightarrow \mathbb{Q}/\mathbb{Z}$ induced by the reciprocity map of the $r+1$ -dimensional local field $K$ and by $\unicode[STIX]{x1D712}\in H^{1}(K,\mathbb{Q}/\mathbb{Z})$ and we will use also the self-duality of the additive group  $F$ . Here the continuity is the one defined in [Reference Kato22], and the duality is also treated by using such continuity. (In the case $r=1$ , this duality is the usual self-duality of the locally compact abelian group  $F$ .)

3.3.2. Let $F$ be an $r$ -dimensional local field of characteristic $p>0$ . We have the residue map

$$\begin{eqnarray}\text{Res}_{F}:\unicode[STIX]{x1D6FA}_{F}^{r}\rightarrow \mathbb{F}_{p}\end{eqnarray}$$

defined as follows. (The following definition and properties of $\text{Res}_{F}$ are contained in [Reference Kato17, Chapter 2, Lemmas 12 and 14].) Let $F_{r}=F$ and for $1\leqslant i\leqslant r$ , define the field $F_{i-1}$ to be the residue field of $F_{i}$ by downward induction on  $i$ . Then $\text{Res}_{F}$ is the composition

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{F}^{r}=\unicode[STIX]{x1D6FA}_{F_{r}}^{r}\overset{\text{Res}}{\rightarrow }\unicode[STIX]{x1D6FA}_{F_{r-1}}^{r-1}\rightarrow \cdots \rightarrow \unicode[STIX]{x1D6FA}_{F_{1}}^{1}\overset{\text{Res}}{\rightarrow }\unicode[STIX]{x1D6FA}_{F_{0}}^{0}=F_{0}\overset{\text{trace}}{\rightarrow }\mathbb{F}_{p}\end{eqnarray}$$

where each $\text{Res}:\unicode[STIX]{x1D6FA}_{F_{i}}^{i}\rightarrow \unicode[STIX]{x1D6FA}_{F_{i-1}}^{i-1}$ is the residue map in 2.1.7 defined by taking $F_{i-1}\rightarrow O_{F_{i}}$ and a prime element $\unicode[STIX]{x1D70B}_{i}$ of the discrete valuation field $F_{i}$ as in 2.1.7. This composition $\text{Res}_{F}$ is independent of the choices of $F_{i-1}\rightarrow O_{F_{i}}$ and  $\unicode[STIX]{x1D70B}_{i}$ .

We have

(12) $$\begin{eqnarray}\text{Res}_{F}\circ C=\text{Res}_{F},\end{eqnarray}$$

where $C$ is the Cartier operator. For a finite extension $F^{\prime }$ of  $F$ , we have

(13) $$\begin{eqnarray}\text{Res}_{F}\circ \text{Tr}_{F^{\prime }/F}=\text{Res}_{F^{\prime }},\end{eqnarray}$$

where $\text{Tr}_{F^{\prime }/F}$ is the trace map $\unicode[STIX]{x1D6FA}_{F^{\prime }}^{r}\rightarrow \unicode[STIX]{x1D6FA}_{F}^{r}$ . (This trace map for differential forms is defined in [Reference Kato19] as

$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6FA}_{F^{\prime }}^{r}\cong U^{1}K_{r+1}^{M}(F^{\prime }((T)))/U^{2}K_{r+1}^{M}(F^{\prime }((T))) & \displaystyle \nonumber\\ \displaystyle & \overset{\text{norm}}{\rightarrow }U^{1}K_{r+1}^{M}(F((T)))/U^{2}K_{r+1}^{M}(F((T)))\cong \unicode[STIX]{x1D6FA}_{F}^{r}. & \displaystyle \nonumber\end{eqnarray}$$

The trace map for differential forms in a more general setting is defined in [Reference Garel10]. The formula (13) contains the formula (12) because for a field $k$ of characteristic $p$ such that $[k:k^{p}]=p^{r}$ , the Cartier operator $\unicode[STIX]{x1D6FA}_{k}^{r}\rightarrow \unicode[STIX]{x1D6FA}_{k}^{r}$ is the trace map of the homomorphism $k\rightarrow k\;;\;x\mapsto x^{p}$ .)

3.3.3. Let $F$ be as in 3.3.2 and let $V$ be a finite-dimensional vector space over  $F$ . Then $V$ is canonically regarded as an object of the category ${\mathcal{F}}_{r,\text{ab}}$ of abelian group objects of the category ${\mathcal{F}}_{r}$  (2.4.4).

Proof. This follows from [Reference Kato22, Proposition 3]. ◻

3.3.5. Let $K$ be a complete discrete valuation field with residue field $F$ and assume that $F$ is an $r$ -dimensional local field of characteristic $p>0$ . (Hence $K$ is an $r+1$ -dimensional local field.)

Let $\unicode[STIX]{x1D712}\in F_{n}H^{1}(K,\mathbb{Q}_{p}/\mathbb{Z}_{p})$ . Then by [Reference Kato and Saito23], $\unicode[STIX]{x1D712}$ induces a continuous homomorphism $K_{r+1}^{M}(K)/U^{n+1}K_{r+1}^{M}(K)\rightarrow \mathbb{Q}_{p}/\mathbb{Z}_{p}$ , where the continuity is that of [Reference Kato22]. Let $m=\max (n-e_{K},[n/p])$ . Then via

$$\begin{eqnarray}E:\mathfrak{m}_{K}^{m+1}/\mathfrak{m}_{K}^{n+1}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{r}(\log )\rightarrow K_{r+1}^{M}(K)/U^{n+1}K_{r+1}^{M}(K)\end{eqnarray}$$

(2.2.4(1)), we obtain a continuous homomorphism $u_{\unicode[STIX]{x1D712}}:\mathfrak{m}_{K}^{m+1}/\mathfrak{m}_{K}^{n+1}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{r}(\log )\rightarrow \mathbb{F}_{p}$ .

Take an integer $b\geqslant 0$ such that $p^{b}\geqslant n-m$ . Then

$$\begin{eqnarray}(\mathfrak{m}_{K}^{-n}/\mathfrak{m}_{K}^{-m}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log ))\times (\mathfrak{m}_{K}^{m+1}/\mathfrak{m}_{K}^{n+1}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{r}(\log ))\rightarrow \unicode[STIX]{x1D6FA}_{F}^{r};\end{eqnarray}$$

$(x,y)\mapsto R_{b}(x\wedge y)$ is a perfect duality of finite-dimensional $F$ -vector spaces where $F$ acts on the two $O_{K}/\mathfrak{m}_{K}^{n-m}$ -modules on the left hand side via $\unicode[STIX]{x1D704}_{b}$ ((2) of 2.1.9). Hence by 3.3.4, $u_{\unicode[STIX]{x1D712}}$ corresponds to an element $\text{Rsw}_{K}(\unicode[STIX]{x1D712})\in \mathfrak{m}_{K}^{-n}/\mathfrak{m}_{K}^{-m}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log )$ . This element is independent of the choice of  $b$ .

3.3.7. We prove the case (i) of 3.3.6. We have a commutative diagram

$$\begin{eqnarray}\begin{array}{@{}ccc@{}}K_{d}^{M}(K^{\prime }) & \rightarrow & \text{Gal}((K^{\prime })^{\text{ab}}/K^{\prime })\\ \downarrow & & \downarrow \\ K_{d}^{M}(K) & \rightarrow & \text{Gal}(K^{\text{ab}}/K)\end{array}\end{eqnarray}$$

where the horizontal arrows are the reciprocity maps, the left vertical arrow is the norm map and the right vertical arrow is the natural one [Reference Kato17, Section 3.2, Corollary 1].

Let $R_{K}:\mathfrak{m}_{K}^{1-n+m}/\mathfrak{m}_{K}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{d}(\log )\rightarrow \mathbb{F}_{p}$ be the composite map $\text{Res}_{F}\circ R_{b}$ ( $p^{b}\geqslant n-m$ ) which is independent of  $b$ , and let $R_{K^{\prime }}:\mathfrak{m}_{K^{\prime }}^{1-n^{\prime }+m^{\prime }}/\mathfrak{m}_{K^{\prime }}\otimes _{O_{K^{\prime }}}\unicode[STIX]{x1D6FA}_{O_{K^{\prime }}}^{d}(\log )\rightarrow \mathbb{F}_{p}$ ( $n^{\prime }=e(K^{\prime }/K)n$ , $m^{\prime }=\max (n^{\prime }-e_{K^{\prime }},[n^{\prime }/p])$ ) be the corresponding map for  $K^{\prime }$ .

Let $\unicode[STIX]{x1D712}\in F_{n}H^{1}(K,\mathbb{Q}_{p}/\mathbb{Z}_{p})$ , let $\unicode[STIX]{x1D712}_{K^{\prime }}\in F_{n^{\prime }}H^{1}(K^{\prime },\mathbb{Q}_{p}/\mathbb{Z}_{p})$ be the image of  $\unicode[STIX]{x1D712}$ , and let $\text{Rsw}(\unicode[STIX]{x1D712})_{K^{\prime }}\in \mathfrak{m}_{K^{\prime }}^{-n^{\prime }}/\mathfrak{m}_{K^{\prime }}^{-m^{\prime }}\otimes _{O_{K^{\prime }}}\unicode[STIX]{x1D6FA}_{O_{K^{\prime }}}^{1}(\log )$ be the image of $\text{Rsw}(\unicode[STIX]{x1D712})=\text{Rsw}_{K}(\unicode[STIX]{x1D712})\in \mathfrak{m}_{K}^{-n}/\mathfrak{m}_{K}^{-m}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{1}(\log )$ . Since any finite extension of $K$ is a successive extension of a tame extension and extensions of degree  $p$ , it is sufficient to prove $\text{Rsw}(\unicode[STIX]{x1D712}_{K^{\prime }})=\text{Rsw}(\unicode[STIX]{x1D712})_{K^{\prime }}$ in the case $K^{\prime }/K$ is tame and in the case $[K^{\prime }:K]=p$ . It is sufficient to prove in these cases that

(14) $$\begin{eqnarray}R_{K^{\prime }}(\text{Rsw}(\unicode[STIX]{x1D712}_{K^{\prime }})\wedge \unicode[STIX]{x1D6FC})=R_{K^{\prime }}(\text{Rsw}(\unicode[STIX]{x1D712})_{K^{\prime }}\wedge \unicode[STIX]{x1D6FC})\end{eqnarray}$$

for any $\unicode[STIX]{x1D6FC}\in \mathfrak{m}_{K^{\prime }}^{m^{\prime }+1}/\mathfrak{m}_{K^{\prime }}^{n^{\prime }+1}\otimes _{O_{K^{\prime }}}\unicode[STIX]{x1D6FA}_{O_{K^{\prime }}}^{d-1}(\log )$ . In these cases, the last group is generated additively by elements of the following three forms.

1. (a) $f\unicode[STIX]{x1D714}$ where $f\in \mathfrak{m}_{K^{\prime }}^{m^{\prime }+1}$ and $\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}_{O_{K}}^{d-1}(\log )$ .

2. (b) $\unicode[STIX]{x1D714}\wedge df$ where $f$ is as in (a) and $\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}_{O_{K}}^{d-2}(\log )$ .

3. (c) $\unicode[STIX]{x1D714}\wedge d\log (f)$ where $\unicode[STIX]{x1D714}\in \mathfrak{m}_{K}^{m+1}/\mathfrak{m}_{K}^{n+1}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{d-2}(\log )$ and $f\in (K^{\prime })^{\times }$ .

(This generation is deduced from 2.1.3. If $K^{\prime }/K$ is tamely ramified, $\mathfrak{m}_{K^{\prime }}^{m^{\prime }+1}/\mathfrak{m}_{K^{\prime }}^{n^{\prime }+1}\otimes _{O_{K^{\prime }}}\unicode[STIX]{x1D6FA}_{O_{K^{\prime }}}^{d-1}(\log )$ is additively generated by elements of the form (a). If the ramification index of $K^{\prime }/K$ is one and if the residue field of $K^{\prime }$ is a purely inseparable extension of $K$ of degree $p$ generated by the residue class of an element $g$ of $O_{K^{\prime }}$ , and if $\unicode[STIX]{x1D70B}$ denotes a prime element of  $K$ , it is additively generated by elements of the form (a), elements of the form (b) with $f=\unicode[STIX]{x1D70B}^{j}g^{j}$ ( $m+1\leqslant i\leqslant n$ , $1\leqslant j\leqslant p-1$ ), and elements of the form (c) with $f=g$ . If the residue field of $K^{\prime }$ coincides with that of $K$ and $[K^{\prime }:K]=p$ , and if $\unicode[STIX]{x1D70B}^{\prime }$ denotes a prime element of  $K^{\prime }$ , it is additively generated by elements of the form (a), elements of the form (b) with $f=(\unicode[STIX]{x1D70B}^{\prime })^{i}$ ( $m^{\prime }+1\leqslant i\leqslant n^{\prime }$ ), and elements of the form (c) with $f=\unicode[STIX]{x1D70B}^{\prime }$ .)

Assume $\unicode[STIX]{x1D6FC}$ is an element as in (a) (resp. (b), resp. (c)), and let $\unicode[STIX]{x1D6FD}\in \mathfrak{m}_{K}^{m+1}/\mathfrak{m}_{K}^{n+1}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{d-1}(\log )$ be $\text{Tr}_{K^{\prime }/K}(f)\unicode[STIX]{x1D714}$ (resp. $\unicode[STIX]{x1D714}\wedge d(\text{Tr}_{K^{\prime }/K}(f))$ , resp. $\unicode[STIX]{x1D714}\wedge d\log (N(f))$ where $N$ is the norm map $(K^{\prime })^{\times }\rightarrow K^{\times }$ ). Then $R_{K^{\prime }}(\text{Rsw}(\unicode[STIX]{x1D712}_{K^{\prime }})\wedge \unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D712}_{K^{\prime }}(E(\unicode[STIX]{x1D6FC}))=\unicode[STIX]{x1D712}(N(E(\unicode[STIX]{x1D6FC})))$ (here $E$ is as in 3.3.5 and $N$ is the norm map)  $=\unicode[STIX]{x1D712}(E(\unicode[STIX]{x1D6FD}))=R_{K}(\text{Rsw}(\unicode[STIX]{x1D712})\wedge \unicode[STIX]{x1D6FD})$ . Hence, for the proof of (14), it is sufficient to prove

1. (A) $\!R_{K^{\prime }}(f\unicode[STIX]{x1D714})=R_{K}(\text{Tr}_{K^{\prime }/K}(f)\unicode[STIX]{x1D714})$ for $f\in \mathfrak{m}_{K^{\prime }}^{1-n^{\prime }+m^{\prime }}$ and $\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}_{O_{K}}^{d}(\log )$ ;

2. (B) $\!R_{K^{\prime }}(\unicode[STIX]{x1D714}\wedge df)=R_{K}(\unicode[STIX]{x1D714}\wedge d(\text{Tr}_{K^{\prime }/K}(f)))$ for $f$ as in (A) and $\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}_{O_{K}}^{d-1}(\log )$ ;

3. (C) $\!R_{K^{\prime }}(\unicode[STIX]{x1D714}\wedge d\log (f))=R_{K}(\unicode[STIX]{x1D714}\wedge d\log (N(f)))$ for $\unicode[STIX]{x1D714}\in \mathfrak{m}_{K}^{1-n+m}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{d-1}(\log )$ and for $f\in (K^{\prime })^{\times }$ .

If $K$ is of characteristic $p$ , $\mathfrak{m}_{K}^{1-n+m}/\mathfrak{m}_{K}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{d}(\log )$ is a subquotient of $\unicode[STIX]{x1D6FA}_{K}^{d}$ and $R_{K}$ coincides with the map induced from $\text{Res}_{K}:\unicode[STIX]{x1D6FA}_{K}^{d}\rightarrow \mathbb{F}_{p}$ on the subquotient, and the same thing holds for  $K^{\prime }$ . Hence (A), (B), (C) follow from $\text{Res}_{K^{\prime }}=\text{Res}_{K}\circ \text{Tr}_{K^{\prime }/K}$  (3.3.2).

Assume $K$ is of characteristic 0. Let $L=F((T))$ . Then via the isomorphism $F[T]/(T^{n-m})\overset{\cong }{\rightarrow }O_{K}/\mathfrak{m}_{K}^{n-m}$ of rings with log structures ((5) with $a=n-m$ ), $\mathfrak{m}_{K}^{1-n+m}/\mathfrak{m}_{K}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{d}(\log )$ is a subquotient of $\unicode[STIX]{x1D6FA}_{L}^{d}$ and $R_{K}$ coincides with the map induced from $\text{Res}_{L}:\unicode[STIX]{x1D6FA}_{L}^{d}\rightarrow \mathbb{F}_{p}$ on the subquotient. As is easily seen, there is an extension $L^{\prime }$ of $L$ of degree $[K^{\prime }:K]$ and an isomorphism $O_{K^{\prime }}/\mathfrak{m}_{K}^{n-m}O_{K^{\prime }}\cong O_{L^{\prime }}/\mathfrak{m}_{L}^{n-m}O_{L^{\prime }}$ of rings with log structures extending the isomorphism $O_{K}/\mathfrak{m}_{K}^{n-m}\cong O_{L}/\mathfrak{m}_{L}^{n-m}$ of rings with log structures, and $R_{K^{\prime }}:\mathfrak{m}_{K^{\prime }}^{1-n^{\prime }+m^{\prime }}/\mathfrak{m}_{K^{\prime }}\otimes _{O_{K^{\prime }}}\unicode[STIX]{x1D6FA}_{O_{K^{\prime }}}^{d}(\log )\rightarrow \mathbb{F}_{p}$ is identified with the map induced from $\text{Res}_{L^{\prime }}:\unicode[STIX]{x1D6FA}_{L^{\prime }}^{d}\rightarrow \mathbb{F}_{p}$ on the subquotient. Hence (A), (B), (C) follow from $\text{Res}_{L^{\prime }}=\text{Res}_{L}\circ \text{Tr}_{L^{\prime }/L}$  (3.3.2).

3.3.8. We prove the case (ii) of 3.3.6. By [Reference Kato19, Theorem 2], we have a commutative diagram

$$\begin{eqnarray}\begin{array}{@{}ccc@{}}\hat{K}_{d+1}^{M}(K^{\prime }) & \rightarrow & \text{Gal}((K^{\prime })^{\text{ab}}/K^{\prime })\\ \downarrow & & \downarrow \\ \hat{K}_{d}^{M}(K) & \rightarrow & \text{Gal}(K^{\text{ab}}/K)\end{array}\end{eqnarray}$$

where $\hat{K}_{d}^{M}(K)=\mathop{\varprojlim }\nolimits_{i}K_{d}^{M}(K)/U^{i}K_{d}^{M}(K)$ , $\hat{K}_{d+1}^{M}(K^{\prime })$ is defined similarly, the horizontal arrows are the reciprocity maps, the left vertical arrow is minus the residue homomorphism in [Reference Kato19] and the right vertical arrow is the natural one.

Let $R_{K}:\mathfrak{m}_{K}^{1-n+m}/\mathfrak{m}_{K}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{d}(\log )\rightarrow \mathbb{F}_{p}$ and $R_{K^{\prime }}:\mathfrak{m}_{K^{\prime }}^{1-n+m}/\mathfrak{m}_{K^{\prime }}\otimes _{O_{K^{\prime }}}\unicode[STIX]{x1D6FA}_{O_{K^{\prime }}}^{d+1}(\log )\rightarrow \mathbb{F}_{p}$ be the maps defined as in 3.3.7.

Let $\unicode[STIX]{x1D712}\in F_{n}H^{1}(K,\mathbb{Q}_{p}/\mathbb{Z}_{p})$ . We prove $\text{Rsw}(\unicode[STIX]{x1D712}_{K^{\prime }})=\text{Rsw}(\unicode[STIX]{x1D712})_{K^{\prime }}$ . For this, it is sufficient to prove that $R_{K^{\prime }}(\text{Rsw}(\unicode[STIX]{x1D712}_{K^{\prime }})\wedge \unicode[STIX]{x1D6FC})=R_{K^{\prime }}(\text{Rsw}(\unicode[STIX]{x1D712})_{K^{\prime }}\wedge \unicode[STIX]{x1D6FC})$ for any $\unicode[STIX]{x1D6FC}\in \mathfrak{m}_{K^{\prime }}^{m+1}/\mathfrak{m}_{K^{\prime }}^{n+1}\otimes _{O_{K^{\prime }}}\unicode[STIX]{x1D6FA}_{O_{K^{\prime }}}^{d}(\log )$ . By 2.1.3, the last group is generated additively by elements as in the following (i)–(iii).

1. (a) Elements in the image of

   $$\begin{eqnarray}\displaystyle & & \displaystyle (\mathfrak{m}_{K^{\prime }}^{m+1}/\mathfrak{m}_{K^{\prime }}^{n+1}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{d}(\log ))\nonumber\\ \displaystyle & & \displaystyle \quad \oplus \,(\mathfrak{m}_{K}^{m+1}/\mathfrak{m}_{K}^{n+1}\otimes _{O_{K}}O_{K}[[T]]\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{d-1}(\log )\wedge dT).\nonumber\end{eqnarray}$$

2. (b) $\unicode[STIX]{x1D714}\otimes d\log (T)$ with $\unicode[STIX]{x1D714}\in \mathfrak{m}_{K}^{m+1}/\mathfrak{m}_{K}^{n+1}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{d-1}(\log )$ .

3. (c) $T^{-i}\unicode[STIX]{x1D714}\otimes d\log (T)$ with $\unicode[STIX]{x1D714}\in \mathfrak{m}_{K}^{m+1}/\mathfrak{m}_{K}^{n+1}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{d-1}(\log )$ and with $i\geqslant 1$ .

Assume $\unicode[STIX]{x1D6FC}$ is an element as in (a) (resp. (b), resp. (c)). Let $\text{Res}_{K^{\prime }/K}:\hat{K}_{d+1}^{M}(K^{\prime })\rightarrow \hat{K}_{d}^{M}(K)$ be the residue map. Then $R_{K^{\prime }}(\text{Rsw}(\unicode[STIX]{x1D712}_{K^{\prime }})\wedge \unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D712}_{K^{\prime }}(E(\unicode[STIX]{x1D6FC}))=-\unicode[STIX]{x1D712}(\text{Res}_{K^{\prime }/K}(E(\unicode[STIX]{x1D6FC})))$ where $E$ is as in 3.3.5. By (ii) of 2.2.2 and by the definition of $\text{Res}_{K^{\prime }/K}$ in [Reference Kato19], we have $\unicode[STIX]{x1D712}(\text{Res}_{K^{\prime }/K}(E(\unicode[STIX]{x1D6FC})))=\unicode[STIX]{x1D712}(E(\unicode[STIX]{x1D6FD}))=R_{K}(\text{Rsw}(\unicode[STIX]{x1D712})\wedge \unicode[STIX]{x1D6FD})$ where $\unicode[STIX]{x1D6FD}=0$ in cases (a) and (c) and $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D714}$ in the case (b). Hence it is sufficient to prove

(15) $$\begin{eqnarray}\displaystyle & & \displaystyle R_{K^{\prime }}\left(\mathop{\sum }_{i\gg -\infty }T^{i}\unicode[STIX]{x1D714}_{i}\wedge d\log (T)\right)\nonumber\\ \displaystyle & & \displaystyle \quad =-R_{K}(\unicode[STIX]{x1D714}_{0})\quad (\unicode[STIX]{x1D714}_{i}\in \mathfrak{m}_{K}^{1-n+m}/\mathfrak{m}_{K}\otimes _{O_{K}}\unicode[STIX]{x1D6FA}_{O_{K}}^{d}(\log )).\end{eqnarray}$$

Take a lifting $\unicode[STIX]{x1D704}:F\rightarrow O_{K}/\mathfrak{m}_{K}^{n-m}$ and extend it to $\unicode[STIX]{x1D704}:F((T))\rightarrow O_{K^{\prime }}/\mathfrak{m}_{K^{\prime }}^{n-m}$ ; $\sum _{i\gg -\infty }a_{i}T^{i}\mapsto \sum _{i\gg -\infty }\unicode[STIX]{x1D704}(a_{i})T^{i}$ . Let $\unicode[STIX]{x1D70B}$ be a prime element of  $K$ . Write

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{i}=\mathop{\sum }_{j=0}^{n-m-1}\unicode[STIX]{x1D70B}^{-j}\unicode[STIX]{x1D704}(\unicode[STIX]{x1D714}_{ij})\wedge d\log (\unicode[STIX]{x1D70B})\quad (\unicode[STIX]{x1D714}_{ij}\in \unicode[STIX]{x1D6FA}_{F}^{d-1}).\end{eqnarray}$$

We have

$$\begin{eqnarray}\displaystyle & & \displaystyle R_{K^{\prime }}\left(\mathop{\sum }_{i\gg -\infty }T^{i}\unicode[STIX]{x1D714}_{i}\wedge d\log (T)\right)\nonumber\\ \displaystyle & & \displaystyle \quad =R_{K^{\prime }}\left(\unicode[STIX]{x1D704}\left(\mathop{\sum }_{i,j}T^{i}\unicode[STIX]{x1D714}_{ij}\right)\unicode[STIX]{x1D70B}^{-j}\,d\log (\unicode[STIX]{x1D70B})\wedge d\log (T)\right)\nonumber\\ \displaystyle & & \displaystyle \quad =-\text{Res}_{F((T))}\left(\mathop{\sum }_{i}T^{i}\unicode[STIX]{x1D714}_{i0}\wedge d\log (T)\right)=-\text{Res}_{F}(\unicode[STIX]{x1D714}_{00})=-R_{K}(\unicode[STIX]{x1D714}_{0}).\nonumber\end{eqnarray}$$

This proves (15).

3.4.1. Let $X$ be a proper normal integral scheme over $\mathbb{Z}$ with function field $J$ and let $\unicode[STIX]{x1D708}$ be a point of $X$ of codimension one whose residue field is of characteristic $p>0$ . We assume $X(\mathbb{R})=\emptyset$ .

Let $n\geqslant 1$ and let $F_{\unicode[STIX]{x1D708},n}H^{1}(J,\mathbb{Q}_{p}/\mathbb{Z}_{p})\subset H^{1}(J,\mathbb{Q}_{p}/\mathbb{Z}_{p})$ be the inverse image of $F_{n}H^{1}({\hat{J}}_{\unicode[STIX]{x1D708}},\mathbb{Q}_{p}/\mathbb{Z}_{p})\subset H^{1}({\hat{J}}_{\unicode[STIX]{x1D708}},\mathbb{Q}_{p}/\mathbb{Z}_{p})$ where ${\hat{J}}_{\unicode[STIX]{x1D708}}$ denotes the field of fractions of the completion of the discrete valuation ring $O_{X,\unicode[STIX]{x1D708}}$ . Let $\mathfrak{m}_{\unicode[STIX]{x1D708}}$ be the maximal ideal of ${\mathcal{O}}_{X,\unicode[STIX]{x1D708}}$ . Let $m=\max (n-e_{{\hat{J}}_{\unicode[STIX]{x1D708}}},[n/p])$ .

We define a canonical homomorphism

$$\begin{eqnarray}\text{Rsw}_{X,\unicode[STIX]{x1D708}}:F_{\unicode[STIX]{x1D708},n}H^{1}(J,\mathbb{Q}_{p}/\mathbb{Z}_{p})\rightarrow \mathfrak{m}_{\unicode[STIX]{x1D708}}^{-n}/\mathfrak{m}_{\unicode[STIX]{x1D708}}^{-m}\otimes _{O_{X,\unicode[STIX]{x1D708}}}\unicode[STIX]{x1D6FA}_{X,\unicode[STIX]{x1D708}}^{1}(\log )\end{eqnarray}$$

by using the higher dimensional global class field theory in [Reference Kato and Saito23].

3.4.2. By the class field theory of $X$ [Reference Kato and Saito23], we have a canonical continuous homomorphism

$$\begin{eqnarray}C_{X}:=\mathop{\varprojlim }_{I}H^{d}(X,K_{d}^{M}({\mathcal{O}}_{X},I))\rightarrow \text{Gal}(J^{\text{ab}}/J)\end{eqnarray}$$

where $d=\dim (X)$ , and hence $\unicode[STIX]{x1D712}\in H^{1}(J,\mathbb{Q}_{p}/\mathbb{Z}_{p})$ induces a homomorphism $C_{X}\rightarrow \mathbb{Q}_{p}/\mathbb{Z}_{p}$ which factors through $H^{d}(X,K_{d}^{M}({\mathcal{O}}_{X},I))$ for some $I$  (2.4.10).

3.4.3. Let $Y$ be the closure of $\unicode[STIX]{x1D708}$ in $X$ . We will identify an element $\mathfrak{p}=(y_{i})_{i}\in P(Y)$ with the element $(x_{i})_{i}\in P(X)$ where $x_{i}=y_{i}$ for $0\leqslant i\leqslant d-1$ and $x_{d}$ is the generic point of  $X$ .

Let $\unicode[STIX]{x1D712}\in F_{\unicode[STIX]{x1D708},n}H^{1}(J,\mathbb{Q}_{p}/Z_{p})$ . For $\mathfrak{p}\in P(X)$ , let $h_{\mathfrak{p}}:K_{d}^{M}(J)\rightarrow \mathbb{Q}_{p}/\mathbb{Z}_{p}$ be the homomorphism induced by $\unicode[STIX]{x1D712}:C_{X}\rightarrow \mathbb{Q}_{p}/\mathbb{Z}_{p}$  (2.3.3). Then if $\mathfrak{p}\in P(Y)$ , $h_{\mathfrak{p}}$ kills $U^{n+1}K_{d}^{M}(J)$ where $U^{\bullet }$ is defined with respect to the discrete valuation ring ${\mathcal{O}}_{X,\unicode[STIX]{x1D708}}$ . For $\mathfrak{p}\in P(Y)$ , let $s_{\mathfrak{p}}:\mathfrak{m}_{\unicode[STIX]{x1D708}}^{m+1}/\mathfrak{m}_{\unicode[STIX]{x1D708}}^{n+1}\otimes _{{\mathcal{O}}_{X,\unicode[STIX]{x1D708}}}\unicode[STIX]{x1D6FA}_{X,\unicode[STIX]{x1D708}}^{d-1}(\log )\rightarrow \mathbb{F}_{p}$ be the homomorphism induced by $h_{\mathfrak{p}}$ and the truncated exponential map $E$ of 3.3.5.

Proof. Let $U$ be a regular dense open subset of $X$ such that $D:=U\cap Y$ is also a regular dense open subset of $Y$ and that $\unicode[STIX]{x1D712}$ is unramified on $U\smallsetminus D$ . Let ${\mathcal{F}}_{U}$ be the coherent ${\mathcal{O}}_{U}$ -module ${\mathcal{O}}_{U}(-(m+1)D)/{\mathcal{O}}_{U}(-(n+1)D)\otimes _{{\mathcal{O}}_{U}}\unicode[STIX]{x1D6FA}_{U}^{d-1}(\log D)$ . Note that ${\mathcal{F}}_{U,\unicode[STIX]{x1D708}}$ equals ${\mathcal{F}}_{\unicode[STIX]{x1D708}}:=\mathfrak{m}_{\unicode[STIX]{x1D708}}^{m+1}/\mathfrak{m}_{\unicode[STIX]{x1D708}}^{n+1}\otimes _{{\mathcal{O}}_{X,\unicode[STIX]{x1D708}}}\unicode[STIX]{x1D6FA}_{X,\unicode[STIX]{x1D708}}^{d-1}(\log )$ . We show that it is sufficient to prove the following (i) and (ii). Let $\unicode[STIX]{x1D709}\in Y$ be a point of codimension one.

1. (i) If $\unicode[STIX]{x1D709}\in U$ , then for any $\mathfrak{p}=(y_{i})_{i}\in P(Y)$ such that $y_{d-2}=\unicode[STIX]{x1D709}$ , $s_{\mathfrak{p}}$ kills the image of ${\mathcal{F}}_{U,\unicode[STIX]{x1D709}}\rightarrow {\mathcal{F}}_{U,\unicode[STIX]{x1D708}}$ .

   (Note that there are only finitely many $\unicode[STIX]{x1D709}\in Y$ of codimension one such that $\unicode[STIX]{x1D709}\notin U$ .)

2. (ii) There is a finitely generated ${\mathcal{O}}_{X,\unicode[STIX]{x1D709}}$ -submodule ${\mathcal{F}}_{\unicode[STIX]{x1D709}}$ of ${\mathcal{F}}_{\unicode[STIX]{x1D708}}$ which generates ${\mathcal{F}}_{\unicode[STIX]{x1D708}}$ over ${\mathcal{O}}_{X,\unicode[STIX]{x1D708}}$ having the following property: For any $\mathfrak{p}=(y_{i})_{i}\in P(Y)$ such that $y_{d-2}=\unicode[STIX]{x1D709}$ , $s_{\mathfrak{p}}$ kills  ${\mathcal{F}}_{\unicode[STIX]{x1D709}}$ .

Let $j:U\rightarrow X$ be the open immersion and ${\mathcal{F}}$ be a coherent ${\mathcal{O}}_{X}$ -module such that ${\mathcal{F}}|_{U}={\mathcal{F}}_{U}$ and ${\mathcal{F}}_{\unicode[STIX]{x1D709}}$ is given in (ii) for each codimension one point $\unicode[STIX]{x1D709}$ of $Y$ not contained in  $U$ . Then, $(s_{\mathfrak{p}})_{\mathfrak{p}\in P(Y)}$ satisfies the conditions (i) and (ii) in Lemma 2.3.4 for $H^{d-1}(Y,{\mathcal{F}})$ since the kernel of ${\mathcal{F}}_{\unicode[STIX]{x1D708}}=H_{\unicode[STIX]{x1D708}}^{0}(Y,{\mathcal{F}})\rightarrow H_{\unicode[STIX]{x1D709}}^{1}(Y,{\mathcal{F}})$ is ${\mathcal{F}}_{U,\unicode[STIX]{x1D709}}$ (resp.  ${\mathcal{F}}_{\unicode[STIX]{x1D709}}$ ) for codimension one point $\unicode[STIX]{x1D709}$ in $U$ (resp. not contained in  $U$ ).

Note that for any $\mathfrak{p}=(y_{i})_{i}\in P(Y)$ , we have $\sum _{\mathfrak{p}^{\prime }\in R(\mathfrak{p})}h_{\mathfrak{p}^{\prime }}=0$ on $K_{d}^{M}(J)$ where $R(\mathfrak{p})$ denotes the set of $\mathfrak{p}^{\prime }=(x_{i})_{i}\in P(X)$ such that $x_{i}=y_{i}$ for $0\leqslant i\leqslant d-2$ .

We prove (i). Let $\unicode[STIX]{x1D70B}$ be an element of ${\mathcal{O}}_{X,\unicode[STIX]{x1D709}}$ which defines $Y$ at  $\unicode[STIX]{x1D709}$ . For $\mathfrak{p}$ as in (i) and for $g\in {\mathcal{O}}_{X,\unicode[STIX]{x1D709}}$ and $u_{1},\ldots ,u_{d-1}\in {\mathcal{O}}_{X,\unicode[STIX]{x1D709}}^{\times }\cdot \unicode[STIX]{x1D70B}^{\mathbb{Z}}$ , we have $s_{\mathfrak{p}}(\unicode[STIX]{x1D70B}^{m+1}g\,d\log (u_{1})\wedge \cdots \wedge d\log (u_{d-1}))=h_{\mathfrak{p}}(\{E(\unicode[STIX]{x1D70B}^{m+1}g),u_{1},\ldots ,u_{d-1}\})$ . This element $\{E(\unicode[STIX]{x1D70B}^{m+1}g),u_{1},\ldots ,u_{d-1}\}\in K_{d}^{M}(J)/U^{n+1}K_{d}^{M}(J)$ belongs to the subgroup of $K_{d}^{M}(J)/U^{n+1}K_{d}^{M}(J)$ generated by all elements of the form $\{u_{1},\ldots ,u_{d}\}\in K_{d}^{M}(J)$ such that $u_{1},\ldots ,u_{d}\in {\mathcal{O}}_{X,\unicode[STIX]{x1D709}}^{\times }\cdot \unicode[STIX]{x1D70B}^{\mathbb{Z}}$ . For any $\mathfrak{p}^{\prime }=(x_{i})_{i}\in R(\mathfrak{p})\smallsetminus \{\mathfrak{p}\}$ , $h_{\mathfrak{p}^{\prime }}:K_{d}^{M}(J)\rightarrow \mathbb{Q}_{p}/\mathbb{Z}_{p}$ kills such $\{u_{1},\ldots ,u_{d}\}$ because $\unicode[STIX]{x1D712}$ is unramified at $x_{d-1}$ and hence $h_{\mathfrak{p}^{\prime }}$ factors through the boundary map $K_{d}^{M}(J)\rightarrow K_{d-1}^{M}(\unicode[STIX]{x1D705}(x_{d-1}))$ ( $\unicode[STIX]{x1D705}$ denotes the residue field) which kills $\{u_{1},\ldots ,u_{d}\}$ . Hence $h_{\mathfrak{p}}(\{u_{1},\ldots ,u_{d}\})=0$ .

We prove (ii). Take an element $\unicode[STIX]{x1D70B}$ of ${\mathcal{O}}_{X,\unicode[STIX]{x1D709}}$ which is a prime element of the discrete valuation ring ${\mathcal{O}}_{X,\unicode[STIX]{x1D708}}$ . Take an element $f\in {\mathcal{O}}_{X,\unicode[STIX]{x1D709}}$ having the following properties (a1) and (a2). (a1)  $f$ is a unit in ${\mathcal{O}}_{X,\unicode[STIX]{x1D708}}$ . (a2)  $\text{ord}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70B}^{m+1}f)>\text{Sw}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D712})$ for any point $\unicode[STIX]{x1D707}$ of $X$ of codimension one such that $\unicode[STIX]{x1D709}\in \overline{\{\unicode[STIX]{x1D707}\}}$ and $\unicode[STIX]{x1D707}\neq \unicode[STIX]{x1D708}$ and such that either $\unicode[STIX]{x1D712}$ ramifies at $\unicode[STIX]{x1D707}$ or $\text{ord}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70B})>0$ . Let ${\mathcal{F}}_{\unicode[STIX]{x1D709}}$ be the ${\mathcal{O}}_{X,\unicode[STIX]{x1D709}}$ -submodule of ${\mathcal{F}}_{\unicode[STIX]{x1D708}}$ generated by the images of $\unicode[STIX]{x1D70B}^{m+1}f\otimes \unicode[STIX]{x1D6FA}_{X,\unicode[STIX]{x1D709}}^{d-1}$ and $\unicode[STIX]{x1D70B}^{m+1}f\otimes \unicode[STIX]{x1D6FA}_{X,\unicode[STIX]{x1D709}}^{d-2}\wedge d\log (\unicode[STIX]{x1D70B})$ . We prove that $s_{\mathfrak{p}}$ kills ${\mathcal{F}}_{\unicode[STIX]{x1D709}}$ . For $g\in {\mathcal{O}}_{X,\unicode[STIX]{x1D709}}$ and $u_{1},\ldots ,u_{d-1}\in {\mathcal{O}}_{X,\unicode[STIX]{x1D709}}^{\times }\cdot \unicode[STIX]{x1D70B}^{\mathbb{Z}}$ , $s_{\mathfrak{p}}(\unicode[STIX]{x1D70B}^{m+1}f\otimes g\,d\log (u_{1})\wedge \cdots \wedge d\log (u_{d-1}))=h_{\mathfrak{p}}(\unicode[STIX]{x1D6FC})$ where $\unicode[STIX]{x1D6FC}=\{E(\unicode[STIX]{x1D70B}^{m+1}fg),u_{1},\ldots ,u_{d-1}\}\in K_{d}^{M}(J)$ . To prove that $h_{\mathfrak{p}}(\unicode[STIX]{x1D6FC})=0$ , it is enough to prove that $h_{\mathfrak{p}^{\prime }}(\unicode[STIX]{x1D6FC})=0$ for any $\mathfrak{p}^{\prime }=(x_{i})_{i}\in R(\mathfrak{p})\smallsetminus \{\mathfrak{p}\}$ . Let $\unicode[STIX]{x1D707}=x_{d-1}$ . Assume first that either $\unicode[STIX]{x1D712}$ ramifies at $\unicode[STIX]{x1D707}$ or $\text{ord}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70B})>0$ . Then $\unicode[STIX]{x1D6FC}\in U_{\unicode[STIX]{x1D707}}^{s+1}K_{d}^{M}(J)$ where $s=\text{Sw}_{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D712})$ and $U_{\unicode[STIX]{x1D707}}^{\bullet }$ is the filtration defined for the discrete valuation ring ${\mathcal{O}}_{X,\unicode[STIX]{x1D707}}$ . Hence $h_{\mathfrak{p}^{\prime }}$ kills $\unicode[STIX]{x1D6FC}$ . Assume next $\unicode[STIX]{x1D712}$ is unramified at $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D70B}$ is a unit at $\unicode[STIX]{x1D707}$ . Then $h_{\mathfrak{p}^{\prime }}$ factors through the boundary map $K_{d}^{M}(J)\rightarrow K_{d-1}^{M}(\unicode[STIX]{x1D705}(\unicode[STIX]{x1D707}))$ which kills  $\unicode[STIX]{x1D6FC}$ .◻

3.4.5. By Serre–Grothendieck duality of the cohomology of coherent sheaves [Reference Hartshorne13], we have a canonical isomorphism

$$\begin{eqnarray}\mathop{\varinjlim }_{{\mathcal{F}}}\text{Hom}(H^{d-1}(Y,{\mathcal{F}}),\mathbb{F}_{p})\cong \mathfrak{m}_{\unicode[STIX]{x1D708}}^{-n}/\mathfrak{m}_{\unicode[STIX]{x1D708}}^{-m}\otimes _{O_{X,\unicode[STIX]{x1D708}}}\unicode[STIX]{x1D6FA}_{X,\unicode[STIX]{x1D708}}^{1}(\log )\end{eqnarray}$$

where ${\mathcal{F}}$ ranges over coherent ${\mathcal{O}}_{X}$ -modules ${\mathcal{F}}$ as in (i) in 3.4.4. (The inductive system is given by making ${\mathcal{F}}$ smaller and smaller. If ${\mathcal{F}}$ and ${\mathcal{F}}^{\prime }$ are as in (i) in 3.4.4 and ${\mathcal{F}}^{\prime }\subset {\mathcal{F}}$ , the canonical map $H^{d-1}(Y,{\mathcal{F}}^{\prime })\rightarrow H^{d-1}(Y,{\mathcal{F}})$ is a surjective map of finite abelian groups.) By 3.4.4, $\unicode[STIX]{x1D712}\in F_{\unicode[STIX]{x1D708},n}H^{1}(J,\mathbb{Q}_{p}/\mathbb{Z}_{p})$ gives an element of the left hand side of this isomorphism, and hence gives an element of $\mathfrak{m}_{\unicode[STIX]{x1D708}}^{-n}/\mathfrak{m}_{\unicode[STIX]{x1D708}}^{-m}\otimes _{O_{X,\unicode[STIX]{x1D708}}}\unicode[STIX]{x1D6FA}_{X,\unicode[STIX]{x1D708}}^{1}(\log )$ . This is our $\text{Rsw}_{X,\unicode[STIX]{x1D708}}(\unicode[STIX]{x1D712})$ .

Proof. If $K$ is of characteristic $p$ , take a proper normal integral scheme $Y$ over $\mathbb{F}_{p}$ with function field  $F$ , let $X=\mathbf{P}_{Y}^{1}$ , and let $\unicode[STIX]{x1D708}$ be the generic point of the image of any section $Y\rightarrow \mathbf{P}_{Y}^{1}$ . Next assume $K$ is of characteristic $0$ . Let $(t_{i})_{1\leqslant i\leqslant r}$ be a transcendental basis of $F$ over $\mathbb{F}_{p}$ and let $(T_{i})_{1\leqslant i\leqslant r}$ be its lifting to  $O_{K}$ . Let $J_{0}=\mathbb{Q}(T_{1},\ldots ,T_{r})$ . Then the algebraic closure $J_{\infty }$ of $J_{0}$ in $K$ is a henselian discrete valuation field whose completion is  $K$ , and hence $H^{1}(J_{\infty },\mathbb{Q}_{p}/\mathbb{Z}_{p})\overset{\cong }{\rightarrow }H^{1}(K,\mathbb{Q}_{p}/\mathbb{Z}_{p})$ . Hence $\unicode[STIX]{x1D712}$ comes from $H^{1}(J,\mathbb{Q}_{p}/\mathbb{Z}_{p})$ for some finite extension $J$ of $J_{0}$ in $J_{\infty }$ such that $J$ is dense in  $K$ . Let $X$ be the integral closure of $\mathbf{P}_{\mathbb{Z}}^{r}\supset \text{Spec}(\mathbb{Z}[T_{1},\ldots ,T_{r}])$ in  $J$ . By replacing $J$ by a bigger  $J$ , we have $X(\mathbb{R})=\emptyset$ . (Indeed, $K$ contains a purely imaginary algebraic number  $\unicode[STIX]{x1D6FD}$ , for example, a square root of $1-p$ if $p\neq 2$ , and a square root of $-7=1-8$ in the case $p=2$ . By replacing $J$ by $J(\unicode[STIX]{x1D6FD})\subset K$ , we have $\unicode[STIX]{x1D6FD}\in J$ . For such  $J$ , $X(\mathbb{R})=\emptyset$ .) Let $\unicode[STIX]{x1D708}$ be the image of the closed point of $\text{Spec}(O_{K})$ under $\text{Spec}(O_{K})\rightarrow X$ .◻