2.1 Higher ideles

Let $X$ be an integral Noetherian scheme with a dimension function $d$ . Recall that a dimension function on a scheme $X$ is a set theoretic function $d:X\rightarrow \mathbb{Z}$ such that:

1. (i) for all $x\in X$ , $d(x)\geqslant 0$ ;

2. (ii) for $x,y\in X$ with $y\in \overline{\{x\}}$ of codimension one, $d(x)=d(y)+1$ , where $\overline{\{x\}}$ denotes the closure of $\{x\}$ in $X$ .

We also denote $d=d(\unicode[STIX]{x1D702})$ , where $\unicode[STIX]{x1D702}$ is the generic point of $X$ . Let $d_{m}$ be the minimal of the integers $d(x)$ for $x\in X$ . For an effective Weil divisor $D$ of $X$ , we denote $U=X\setminus D$ .

We also recall the definition of Milnor $K$ -theory.

Let ${\mathcal{P}}$ be the set of Parshin chains on the pair $(U\subset X)$ , and let ${\mathcal{Q}}$ be the set of $Q$ -chains on $(U\subset X)$ . For a Parshin chain $P=(p_{0},\ldots ,p_{d-d_{m}})\in {\mathcal{P}}$ of dimension $d$ we denote by $D(P)$ the multiplicity of the prime divisor $\overline{\{p_{d-d_{m}-1}\}}$ in $D$ .

2.2 Milnor $K$ -sheaves

Let $X$ be an integral scheme. Recall the Milnor $K$ -sheaf ${\mathcal{K}}_{\ast }^{M}$ is defined as the Nisnevich sheafification of the presheaf on affine scheme $\text{Spec}(A)$ given as follows:

where $I$ is the two-sided ideal of the tensor algebra generated by the elements $a\otimes (1-a)$ with $a,1-a\in A^{\times }$ . This sheaf is closely related to a $p$ -primary sheaf if $X$ is of characteristic $p\geqslant 0$ , so-called logarithmic de Rham–Witt sheaf $W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{r}$ on the small Nisnevich (resp. étale) site, which is a subsheaf of $W_{m}\unicode[STIX]{x1D6FA}_{X}^{r}$ (cf. [Reference IllusieIll79]) Nisnevich (resp. étale) locally generated by $d\log [x_{1}]_{m}\wedge \cdots \wedge d\log [x_{r}]_{m}$ with $x_{i}\in {\mathcal{O}}_{X}^{\times }$ for all $i$ , $d\log [x]_{m}:=d[x]_{m}/[x]_{m}$ and $[x]_{m}$ is the Teichmüller representative of $x$ in $W_{m}{\mathcal{O}}_{X}$ .

These notations can be generalized to a relative situation with respect to a divisor. Let $i:D{\hookrightarrow}X$ be an effective divisor with its complement $j:U:=X\setminus D{\hookrightarrow}X$ .

We will show relations between them in a local case, and then we may use these results in different settings. In the following, we fix the notation as follows: Let $R$ be a Henselian regular local ring of characteristic $p>0$ with the residue field $k$ . We assume that $k$ is finite. Let $D$ be an effective divisor such that $C:=$ Supp( $D$ ) is a simple normal crossing divisor on $X:=\text{Spec}(R)$ . Let $\{{D_{\unicode[STIX]{x1D706}}\}}_{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}}$ be the (regular) irreducible components of $D$ , and let $i_{\unicode[STIX]{x1D706}}:D_{\unicode[STIX]{x1D706}}{\hookrightarrow}X$ be the natural map.

In order to study the structure of the relative logarithmic de Rham–Witt sheaves, we introduce some notions here. We endow $\mathbb{N}^{\unicode[STIX]{x1D6EC}}$ with a semiorder by

For $\text{}\underline{n}=(n_{\unicode[STIX]{x1D706}})_{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}}\in \mathbb{N}^{\unicode[STIX]{x1D6EC}}$ let

be the associated divisor. For $\unicode[STIX]{x1D708}\in \unicode[STIX]{x1D6EC}$ we set $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}=(0,\ldots ,1,\ldots ,0)\in \mathbb{N}^{\unicode[STIX]{x1D6EC}}$ , where $1$ is on the $\unicode[STIX]{x1D708}$ th place, and we define the following Nisnevich sheaves for $r\geqslant 1$

3.1 Idele class groups

The $K$ -theoretic class group $H^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M})$ is introduced by Kato and Saito in [Reference Kato and SaitoKS86], and they also give an idelic description of the dual of this class group. In [Reference KerzKer11], we give a direct description of this class group, and prove the following theorem.

3.2 The $\ell$ -primary part

In this subsection, we study the group $H^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M})/\ell ^{m}$ , and compare it with $H^{2d}(X_{\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})$ .

The coniveau spectral sequence for an abelian étale (resp. Nisnevich) sheaf ${\mathcal{F}}$ on $X_{\acute{\text{e}}\text{t}}$ (resp. $X_{\text{Nis}}$ ) writes

where $X^{p}$ is the set of points of codimension $p$ of $X$ . Note that the degeneration of the coniveau spectral sequence due to cohomological dimension (cf. [Reference Kato and SaitoKS86, 1.2.5]) for ${\mathcal{K}}_{d,X|D}^{M}$ on $X_{\text{Nis}}$ gives rise to a short exact sequence

We now study the coniveau spectral sequence for $j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d}$ on $X_{\acute{\text{e}}\text{t}}$ .

Using the Galois symbol maps and induction on codimension, Sato constructs the localized Chern class map and proves the following theorem.

In particular, in case $i=1,r=0$ , we obtain

In summary:

3.3 The $p$ -primary part

In this subsection we want to compare the group $H^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M})/p^{m}$ with the group $H^{d}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})$ .

The coniveau spectral sequence for a $p$ -primary étale (resp. Nisnevich) sheaf ${\mathcal{F}}$ on $X_{\acute{\text{e}}\text{t}}$ (resp. $X_{\text{Nis}}$ ) writes

We know that $E_{1,\acute{\text{e}}\text{t}}^{p,q}({\mathcal{F}})=0$ if $q>1$ or $p>d$ , and $E_{1,\text{Nis}}^{p,q}({\mathcal{F}})=0$ if $q>0$ or $p>d$ .

Now, we recall the duality theorem of the relative logarithmic de Rham–Witt sheaves.

In particular, for $i=1$ and $r=0$ we get isomorphisms

and

where $A^{\vee }$ is the Pontryagin dual of a topological abelian group $A$ . These isomorphisms can be used to define a measure of ramification for étale abelian covers of $U$ whose degree divides $p^{m}$ .

The group $\unicode[STIX]{x1D70B}_{1}^{\text{ab}}(X,D)/p^{m}$ is a quotient of $\unicode[STIX]{x1D70B}_{1}^{\text{ab}}(U)/p^{m}$ , which can be thought of as classifying abelian étale coverings of $U$ whose degree divides $p^{m}$ with ramification bounded by $D$ .

3.4 Class field theory via ideles

4.1 Grothendieck’s local duality

We know that the sheaf $\unicode[STIX]{x1D6FA}_{X}^{d}$ is a dualizing sheaf of $X$ . There exists a natural homomorphism called the residue homomorphism [Reference Kunz, Cox and DickensteinKCD08, Section 5]:

By compositing with the trace map $\text{Tr}_{k/\mathbb{F}_{p}}:k\rightarrow \mathbb{F}_{p}=\mathbb{Z}/p\mathbb{Z}$ , we get the map

For any finite $A$ -module $M$ , the Yoneda pairing and the above trace map give us a canonical pairing

In particular, if $M$ is a locally free $A$ -module, we obtain the isomorphisms

where $M^{t}:=\text{Hom}_{A}(M,\unicode[STIX]{x1D6FA}_{X}^{d})$ is the dual $A$ -module, and

Note that, for a locally free $A$ -module $M$ , we have [Reference Grothendieck and HartshorneGH67]

4.2 Duality theorems

The purity result of Shiho [Reference ShihoShi07, Theorem 3.2] tells us that there exists a canonical isomorphism

Using the same method as in [Reference ZhaoZha16], we obtain a map

If we endow $H^{i}(U_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{U,\log }^{r})$ with the discrete topology and endow the inverse limit $\mathop{\varprojlim }\nolimits_{E}H_{x}^{d+1-i}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|E,\log }^{d-r})$ with the profinite topology, where $E$ runs over the set of effective divisors with support on $X\setminus U$ , then the (continuous) map $\unicode[STIX]{x1D6F7}_{m}^{i,r}$ and the trace map (4.2.1) induce a pairing of topological abelian groups:

Using Pontryagin duality, we see that $\unicode[STIX]{x1D6F7}_{m}^{i,r}$ is an isomorphism if and only if the pairing (4.2.2) is a perfect pairing of topological abelian groups for the respective $i,m,r$ .

For $r=0,i=1$ , we get

Similar to Corollary 3.3.5, the transition maps are surjective in the projective system, for our divisor $D$ we define

by Pontryagin duality, we also define

Theorem 4.2.1 gives us an isomorphism

4.3 Class field theory via ideles

For a complete regular local ring $A$ of dimension $d$ of characteristic $p>0$ , and $X,X^{\prime },U,D,D^{\prime }$ as before. An idelic description of $H_{x}^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M})$ is given by the following theorem.

In summary, the class field theory of Henselian regular local ring over $\mathbb{F}_{p}$ can be reformulated as follows:

5.1 Idele class group

We want to give an idelic description of the class group $H_{X_{s}}^{d+1}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M})$ . We use the dimension function $d(x)=\text{dim}(\overline{\{x\}})$ on $X$ .

Note that the generic fiber $X_{\unicode[STIX]{x1D702}}$ is a smooth variety over the local field $K$ . Its class field theory has been studied in several cases, for example, the case $d=1$ is well understood by the work of Bloch and Saito; see [Reference SaitoSai85, Reference HiranouchiHir16]. In [Reference ForréFor15], Forré determines the kernel of the reciprocity map in unramified $\ell$ -adic class field theory in the higher dimension case.

5.2 Kato complexes on simple normal crossing varieties

We recall notations and theorems in [Reference Jannsen and SaitoJS03]. Let $Y$ be a proper simple normal crossing variety over the finite field $k$ of dimension $d$ , and let $Y_{1},\ldots ,Y_{N}$ be its smooth irreducible components. Let

be the scheme-theoretic intersection of $Y_{i_{1}},\ldots ,Y_{i_{s}}$ , and denote

for the disjoint union of the $s$ -fold intersections of the $Y_{i}$ , for any $s>0$ . Since $Y$ is simple, all $Y^{[s]}$ are smooth of dimension $d-s+1$ . The immersions $Y_{i_{1},\ldots ,i_{s}}{\hookrightarrow}Y$ and $Y_{i_{1},\ldots ,i_{s}}{\hookrightarrow}Y_{i_{1},\ldots ,\hat{i_{v}},\ldots ,i_{s}}$ induce canonical maps

For integer $n>0,i\geqslant 0$ we define the following étale sheaves on $Y$ :

1. (i) If $p\nmid n$ , then let $\mathbb{Z}/n\mathbb{Z}(i):=\unicode[STIX]{x1D707}_{n,Y}^{\otimes i}$ be the $i$ th tensor power over $\mathbb{Z}/n\mathbb{Z}$ of the sheaf of $n$ th roots of unity.

2. (ii) If $n=mp^{r},r\geqslant 0$ with $p\nmid m$ , then let

   $$\begin{eqnarray}\mathbb{Z}/n\mathbb{Z}(i):=\unicode[STIX]{x1D708}_{r,Y}^{i}[-i]\oplus \unicode[STIX]{x1D707}_{m,Y}^{\otimes i}\end{eqnarray}$$
   where $\unicode[STIX]{x1D708}_{r,Y}^{i}(U):=\ker (\unicode[STIX]{x2202}:\bigoplus _{x\in U^{0}}W_{r}\unicode[STIX]{x1D6FA}_{x,\log }^{i}\rightarrow \bigoplus _{x\in U^{1}}W_{r}\unicode[STIX]{x1D6FA}_{x,\log }^{i-1})$ for $U\subset Y$ open. Note that $\unicode[STIX]{x1D708}_{r,Y}^{d}=W_{r}\unicode[STIX]{x1D6FA}_{Y,\log }^{d}$ if $Y$ is smooth [Reference SatoSat07, 1.3.2].

The Kato complex $C^{1,0}(Y,\mathbb{Z}/n\mathbb{Z}(d))$ is defined to be the complex:

where $\mathbb{Z}/n\mathbb{Z}(i)$ is defined as above for the residue field of $Y$ at $y$ , and put the term $\bigoplus _{y\in Y^{a}}$ in degree $a-d$ as an object in derived category. Similarly, for each $s$ , on $Y^{[s]}$ we define the complex $C^{1,0}(Y^{[s]},\mathbb{Z}/n\mathbb{Z}(d-s+1))$ , and moreover we define the complex $C(Y^{\bullet },\mathbb{Z}/n\mathbb{Z})$ as

where $\unicode[STIX]{x1D70B}_{0}(Z)$ is the set of connected components of a scheme $Z$ , the last term of this complex is placed in degree $0$ , and the differential $d_{s}$ is $\sum _{\unicode[STIX]{x1D708}=1}^{s+1}(-1)^{\unicode[STIX]{x1D708}+1}(\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})_{\ast }$ .

5.3 The $\ell$ -primary part

Let $\ell$ be a prime number and $\ell \neq p$ . The cup product induces the following morphism

As $\unicode[STIX]{x1D707}_{\ell ^{m},U}^{\otimes d+1}=\jmath ^{\ast }\unicode[STIX]{x1D707}_{\ell ^{m},X}^{\otimes d+1}$ the adjoint pair $(\jmath _{!},\jmath ^{\ast })$ gives an isomorphism

Using the adjoint pair $(i_{\ast },Ri^{!})$ and these two maps above, we obtain a pairing on $X_{\acute{\text{e}}\text{t}}$ :

Therefore a pairing of cohomology groups:

We have the following duality theorem; see [Reference GeisserGei10, Theorem 7.5].

For $r=0,i=1$ , we obtain

and by Pontryagin duality

For any abelian sheaf ${\mathcal{F}}$ on $X_{\text{Nis}}$ or $X_{\acute{\text{e}}\text{t}}$ , we have the following two coniveau spectral sequences:

By purity the complex $E_{1,\acute{\text{e}}\text{t}}^{\bullet ,d+2}(\unicode[STIX]{x1D707}_{\ell ^{m},X}^{\otimes d+1})$ is isomorphic to the complex Kato complex $C^{1,0}(X_{s},\mathbb{Z}/\ell ^{m}\mathbb{Z}(d))$ from Section 5.2 (up to a shift), that is, to

where we set the last term in degree $0$ as an object in the derived category.

In summary, combining Theorems 5.3.4 and 5.1.2 with the identification (5.3.3), we reformulate the $\ell$ -primary part of class field theory in this setting as follows.

In the case that $\text{Supp}(D)=X_{s}$ , we have the following diagram:

where the last row is the exact sequence (5.3.4), the morphism $\unicode[STIX]{x1D70C}_{X_{\unicode[STIX]{x1D702}}}$ is the reciprocity map of variety over the local field $K$ (cf. [Reference Kato and SaitoKS83]), and the map $\unicode[STIX]{x1D719}$ is induced by the connection map $H^{d}(X_{\unicode[STIX]{x1D702}},{\mathcal{K}}_{d+1,X_{\unicode[STIX]{x1D702}}}^{M})\rightarrow H_{X_{s}}^{d+1}(X_{\text{Nis}},{\mathcal{K}}_{d+1,X|D}^{M})$ .

In the remainder of this subsection, we explain why our new approach recovers the known result for varieties over local fields (cf. [Reference ForréFor15]) in the good reduction case.

To prove this theorem, we may further assume that $D=X_{s}$ , since the multiplicity of $D$ has no contribution to $\widehat{SK}_{1}(X_{\unicode[STIX]{x1D702}})/\ell ^{m}$ . To simplify our notations, we denote $\unicode[STIX]{x1D6EC}(i)_{Y}:=\mathbb{Z}/\ell ^{m}\mathbb{Z}\otimes \mathbb{Z}(i)_{Y}$ for a scheme $Y$ and $i\in \mathbb{Z}$ , where $\mathbb{Z}(i)$ is Bloch’s cycle complex on the small Nisnevich site (cf. [Reference GeisserGei04]).

We can define the restriction map $r_{i}:\unicode[STIX]{x1D6EC}(i)_{X}\rightarrow i_{\ast }\unicode[STIX]{x1D6EC}(i)_{X_{s}}$ as the composition

where the middle arrow is given by multiplication by $\unicode[STIX]{x1D70B}$ , and the last arrow is the localization map.

Let

be the homotopy fiber of $r_{i}$ . By rigidity [Reference GeisserGei04, Theorem 1.2(3)] we get an isomorphism $j_{!}\unicode[STIX]{x1D6EC}(i)_{X_{\unicode[STIX]{x1D702}}}\cong \unicode[STIX]{x1D6EC}(i)_{X|X_{s}}$ . Notice that we also have an analogous isomorphism $j_{!}{\mathcal{K}}_{i,X_{\unicode[STIX]{x1D702}}}^{M}/\ell ^{m}\cong {\mathcal{K}}_{i,X|X_{s}}^{M}/\ell ^{m}$ . So we conclude:

Note that Proposition 5.3.8 implies that the canonical map

is an isomorphism.

To finish the proof of Theorem 5.3.7, we also need the following result:

5.4 The $p$ -primary part: equicharacteristic

Due to the lack of ramified duality in the mixed characteristic case for $p$ -primary sheaves, we only treat the case that $R=\mathbb{F}_{q}[[t]]$ in this subsection and assume $X_{s}$ is reduced. In [Reference ZhaoZha16], we proved the following duality theorem for the relative logarithmic de Rham–Witt sheaves in this setting.

For $r=0,i=1$ , we get

Similar to Corollary 3.3.5, the transition maps are surjective in the projective limit, for our divisor $D$ we define

by Pontryagin duality, we also define

Therefore Theorem 5.4.1 gives us an isomorphism

As before we want to compare the group $H_{X_{s}}^{d+1}(X_{\text{Nis}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d+1})$ with $H_{X_{s}}^{d+1}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d+1})$ , by using the coniveau spectral sequence.

For any abelian sheaf ${\mathcal{F}}$ on $X_{\text{Nis}}$ or $X_{\acute{\text{e}}\text{t}}$ , we have the following two coniveau spectral sequences:

By purity [Reference ShihoShi07] the complex $E_{1,\acute{\text{e}}\text{t}}^{\bullet ,1}(W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d+1})$ is isomorphic to the Kato complex $C^{1,0}(X_{s},\mathbb{Z}/p^{m}\mathbb{Z}(d))$ (up to a shift), that is, to

where $\mathbb{Z}/p^{m}\mathbb{Z}(d)=\unicode[STIX]{x1D708}_{m,X_{s}}^{d}[-d]$ and where the last term is placed in degree $0$ .

The $p$ -primary part of class field theory in this setting can be reformulated as follows: