1 Introduction
In higher dimensional class field theory one tries to describe the abelian fundamental group of a scheme
$X$
of arithmetic interest in terms of idelic or cycle theoretic data on
$X$
. More precisely, assume that
$X$
is regular and connected and fix a modulus data, that is, an effective divisor
$D$
on
$X$
. Let
$\unicode[STIX]{x1D70B}_{1}^{\text{ab}}(X,D)$
be the abelian fundamental group classifying étale coverings with ramification bounded by
$D$
. One defines an idele class group
$C(X,D)$
which is a quotient of the idele group
$$\begin{eqnarray}I(U\subset X):=\bigoplus _{P\in {\mathcal{P}}}K_{d(P)}^{M}(k(P))\end{eqnarray}$$
by a modulus subgroup depending on
$D$
and certain reciprocity relations. Here
$P\in {\mathcal{P}}$
runs through some set of chains of prime ideals and
$k(P)$
is a generalized form of Henselian local residue field at the chain
$P$
; see Section 2.1 and [Reference KerzKer11].
One then constructs a residue map
$$\begin{eqnarray}\unicode[STIX]{x1D70C}:C(X,D)\rightarrow \unicode[STIX]{x1D70B}_{1}^{\text{ab}}(X,D)\end{eqnarray}$$
which we show to be an isomorphism after tensoring with
$\mathbb{Z}/n\mathbb{Z}$
(
$n>0$
) in the following situations:
(i)
$X$
is a smooth proper variety over a finite field, recovering (with simpler proof) the main result of [Reference Kato and SaitoKS86] for varieties; see Section 3.(ii)
$X$
is an (equal characteristic) complete regular local ring with finite residue field, recovering in case
$\dim (X)=2$
results of [Reference SaitoSai87], recovering in case
$n$
is invertible on
$X$
results of [Reference SatoSat09] and completing our understanding in case
$X$
is of equal characteristic
$p$
and
$n$
is a power of
$p$
; see Section 4.(iii)
$X$
is a smooth proper scheme over an (equal characteristic) complete discrete valuation ring with finite residue field, recovering results of Bloch and Saito, see [Reference SaitoSai85], for
$\dim (X)=2$
and results of [Reference ForréFor15] for
$n$
invertible on
$X$
and completing our understanding in case
$X$
is of characteristic
$p$
and
$n$
is a power of
$p$
; see Section 5.
Here is an outline of our universal strategy to all three cases of the reciprocity isomorphism
$\unicode[STIX]{x1D70C}$
in higher dimensional class field theory listed above:
Step 1: Show that
$C(X,D)$
is isomorphic to a Nisnevich cohomology group of relative Milnor
$K$
-sheaf
${\mathcal{K}}_{X,D}^{M}$
, for example, in case (i) above one has an isomorphism
$$\begin{eqnarray}C(X,D)\cong H^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M}),\end{eqnarray}$$
where
$d=\dim (X)$
.
Step 2: Show that the Nisnevich cohomology of the relative Milnor
$K$
-sheaf with finite coefficients is isomorphic to a certain analogous étale cohomology group, for example, in case (i) and for
$n=p^{m}$
a power of the characteristic
$p$
of the base field one has an isomorphism
$$\begin{eqnarray}H^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M}/n)\cong H^{d}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})\end{eqnarray}$$
where
$W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d}$
is a relative de Rham–Witt sheaf. This isomorphism is established by comparing coniveau spectral sequences and observing that based on cohomological dimension arguments there is just one additional potentially nonvanishing row in the spectral sequence in the étale situation, which however disappears at the end by known cases of the Kato conjecture.
Step 3: Arithmetic duality tells us that the étale cohomology group from Step 2 is isomorphic to an abelian étale fundamental group, for example, in the special case as in Step 2 the profinite group
$\lim _{D}H^{d}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})$
, where
$D$
runs through all effective divisors with a fixed support
$X\setminus U$
, is Pontryagin dual to the (discrete) cohomology group
$H^{1}(U_{\acute{\text{e}}\text{t}},\mathbb{Z}/n\mathbb{Z})$
.
2 Higher ideles and Milnor
$K$
-sheaves
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:
(i) for all
$x\in X$
,
$d(x)\geqslant 0$
;(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
$U=X\setminus D$
.
Definition 2.1.1.
(i) A chain on
$X$
is a sequence of points
$P=(p_{0},p_{1},\ldots ,p_{s})$
of
$X$
such that $$\begin{eqnarray}\overline{\{p_{0}\}}\subset \overline{\{p_{1}\}}\subset \cdots \subset \overline{\{p_{s}\}}.\end{eqnarray}$$
(ii) A Parshin chain on
$X$
is a chain
$P=(p_{0},p_{1},\ldots ,p_{s})$
on
$X$
such that
$d(p_{i})=i+d_{m}$
, for
$0\leqslant i\leqslant s$
.(iii) A Parshin chain on the pair
$(U\subset X)$
is a Parshin chain
$P=(p_{0},p_{1},\ldots ,p_{s})$
on
$X$
such that
$p_{i}\in D$
for
$0\leqslant i<s$
and such that
$p_{s}\in U$
.(iv) The dimension
$d(P)$
of a chain
$P=(p_{0},p_{1},\ldots ,p_{s})$
is defined to be
$d(p_{s})$
.(v) A
$Q$
-chain on
$(U\subset X)$
is defined as a chain
$P=(p_{0},\ldots ,p_{s-2},p_{s})$
on
$X$
for
$1\leqslant s\leqslant d$
, such that
$d(p_{i})=i+d_{m}$
for
$i\in \{0,1,\ldots ,s-2,s\}$
,
$p_{i}\in D$
for
$0\leqslant i\leqslant s-2$
and
$p_{s}\in U$
.
We also recall the definition of Milnor
$K$
-theory.
Definition 2.1.2.
(i) For a commutative unital ring
$R$
, the Milnor
$K$
-ring
$K_{\bullet }^{M}(R)$
of
$R$
is the graded ring
$T(R^{\times })/I$
, where
$I$
is the ideal of the tensor algebra
$T(R^{\times })$
over
$R^{\times }$
generated by elements
$a\otimes (1-a)$
with
$a,1-a\in R^{\times }$
. The image of
$a_{1}\otimes \cdots \otimes a_{r}$
in
$K_{r}^{M}(R)$
is denoted by
$\{a_{1},\ldots ,a_{r}\}$
.(ii) If
$R$
is a discrete valuation ring with quotient field
$K$
and maximal ideal
$\mathfrak{m}\subset R$
we define
$K_{r}^{M}(K,n)\subset K_{r}^{M}(K)$
be the subgroup generated by
$\{1+\mathfrak{m}^{n},K^{\times },\ldots ,K^{\times }\}$
for an integer
$n\geqslant 0$
.
Definition 2.1.3. Let
$P=(p_{0},\ldots ,p_{s})$
be a chain on
$X$
.
(i) We define the ring
${\mathcal{O}}_{X,P}^{h}$
, which is a finite product of Henselian local rings, as follows: If
$s=0$
set
${\mathcal{O}}_{X,P}^{h}={\mathcal{O}}_{X,p_{0}}^{h}$
. If
$s>0$
assume that
${\mathcal{O}}_{X,P^{\prime }}^{h}$
has been defined for chains of the form
$P^{\prime }=(p_{0},\ldots ,p_{s-1})$
. Denote
$R={\mathcal{O}}_{X,P^{\prime }}^{h}$
, let
$T$
be the finite set of prime ideals of
$R$
lying over
$p_{s}$
. Then we define $$\begin{eqnarray}{\mathcal{O}}_{X,P}^{h}:=\mathop{\prod }_{\mathfrak{p}\in T}R_{\mathfrak{p}}^{h}.\end{eqnarray}$$
(ii) For a chain
$P=(p_{0},\ldots ,p_{s})$
on
$X$
we let
$k(P)$
be the finite product of the residue fields of
${\mathcal{O}}_{X,P}^{h}$
. If
$s\geqslant 1$
each of these residue fields has a natural discrete valuation such that the product of their rings of integers is equal to the normalization of
${\mathcal{O}}_{X,P^{\prime }}^{h}/p_{s}$
, where
$P^{\prime }=(p_{0},\ldots ,p_{s-1})$
.
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
$D$
.
Definition 2.1.4.
(i) The idele class group of
$(U\subset X)$
is defined as and endow this group with the topology generated by the open subgroups$$\begin{eqnarray}I(U\subset X):=\bigoplus _{P\in {\mathcal{P}}}K_{d(P)}^{M}(k(P)),\end{eqnarray}$$
where$$\begin{eqnarray}\bigoplus _{\substack{ P\in {\mathcal{P}} \\ d(P)=d}}K_{d}^{M}(k(P),D(P))\subset I(U\subset X),\end{eqnarray}$$
$D$
runs through all effective Weil divisors with support
$X\setminus U$
.(ii) The idele group of
$X$
relative to the fixed effective divisor
$D$
with complement
$U$
is defined as $$\begin{eqnarray}I(X,D):=\text{Coker}\biggl(\bigoplus _{\substack{ P\in {\mathcal{P}} \\ d(P)=d}}K_{d}^{M}(k(P),D(P))\rightarrow I(U\subset X)\biggr).\end{eqnarray}$$
(iii) The idele class group
$C(U\subset X)$
is where$$\begin{eqnarray}C(U\subset X):=\text{Coker}\biggl(\bigoplus _{P\in {\mathcal{Q}}}K_{d(P)}^{M}(k(P))\xrightarrow[{}]{Q}I(U\subset X)\biggr),\end{eqnarray}$$
$Q$
is defined to be the sum of all
$Q^{P^{\prime }\rightarrow P}$
for
$P^{\prime }=(p_{0},\ldots ,p_{s-2},p)\in {\mathcal{Q}}$
and
$P=(p_{0},\ldots ,p_{s-2},p_{s-1},p_{s})\in {\mathcal{P}}$
:– if
$p_{s-1}\in D$
, then
$Q^{P^{\prime }\rightarrow P}$
is the natural map
$K_{d(P^{\prime })}^{M}(k(P^{\prime }))\rightarrow K_{d(P)}^{M}(k(P))$
induced on Milnor
$K$
-groups by the ring homomorphism
$k(P^{\prime })\rightarrow k(P)$
;– if
$p_{s-1}\in U$
, then
$Q^{P^{\prime }\rightarrow P}$
is the residue symbol
$K_{d(P^{\prime })}^{M}(k(P^{\prime }))\rightarrow K_{d(P^{\prime \prime })}^{M}(k(P^{\prime \prime }))$
where
$P^{\prime \prime }=(p_{0},\ldots ,p_{s-1})$
.
(iv) The idele class group
$C(X,D)$
of
$X$
relative to the effective divisor
$D$
is defined as $$\begin{eqnarray}C(X,D):=\text{Coker}\biggl(\bigoplus _{P\in {\mathcal{Q}}}K_{d(P)}^{M}(k(P))\xrightarrow[{}]{Q}I(X,D)\biggr).\end{eqnarray}$$
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:
$$\begin{eqnarray}A\mapsto K_{\bullet }^{M}(A)=\bigoplus _{i\in \mathbb{N}}\underbrace{(A^{\times }\otimes _{\mathbb{ Z}}\cdots \otimes _{\mathbb{Z}}A^{\times })}_{i\;\text{times}}/I,\end{eqnarray}$$
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
$j:U:=X\setminus D{\hookrightarrow}X$
.
Definition 2.2.1. Let
$r\in \mathbb{N}$
. We define:
(i) [Reference Rülling and SaitoRS18, Definition 2.4] the relative Milnor
$K$
-sheaf
${\mathcal{K}}_{r,X|D}^{M}$
on the small Nisnevich (resp. étale) site is defined to be the subsheaf of
$j_{\ast }{\mathcal{K}}_{r,U}^{M}$
Nisnevich (resp. étale) locally generated by
$\{x_{1},\ldots ,x_{r}\}$
with
$x_{1}\in \ker ({\mathcal{O}}_{X}^{\times }\rightarrow {\mathcal{O}}_{D}^{\times })$
and
$x_{i}\in {\mathcal{O}}_{U}^{\times }$
for all
$i$
. Note that if
$X$
is a regular scheme over a field, then
${\mathcal{K}}_{r,X|D}^{M}\subset {\mathcal{K}}_{r,X}^{M}$
by the known Gersten conjecture [Reference KerzKer09] (see also [Reference Rülling and SaitoRS18, Corollary 2.9]).(ii) [Reference Jannsen, Saito and ZhaoJSZ18, Definition 1.1.1] in the case that
$X$
is of characteristic
$p\geqslant 0$
, the relative logarithmic de Rham–Witt sheaf
$W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{r}$
on the small Nisnevich (resp. étale) site is the subsheaf of
$j_{\ast }W_{m}\unicode[STIX]{x1D6FA}_{U,\log }^{r}$
Nisnevich (resp. étale) locally generated by
$d\log [x_{1}]_{m}\wedge \cdots \wedge d\log [x_{r}]_{m}$
with
$x_{1}\in \ker ({\mathcal{O}}_{X}^{\times }\rightarrow {\mathcal{O}}_{D}^{\times })$
and
$x_{i}\in {\mathcal{O}}_{U}^{\times }$
for all
$i$
. Similar to the relative Milnor
$K$
-group, we also have
$W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{r}\subset W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{r}$
in the case that
$X$
is a regular scheme.
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.
Theorem 2.2.2. The
$d\log$
map induces an isomorphism of Nisnevich sheaves on
$X_{\text{Nis}}$
$$\begin{eqnarray}\displaystyle d\log [-]:{\mathcal{K}}_{r,X|D}^{M}/(p^{m}{\mathcal{K}}_{r,X}^{M}\cap {\mathcal{K}}_{r,X|D}^{M}) & \xrightarrow[{}]{\cong } & \displaystyle W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{r}\nonumber\\ \displaystyle \{x_{1},\ldots ,x_{r}\} & \mapsto & \displaystyle d\log [x_{1}]_{m}\wedge \cdots \wedge d\log [x_{r}]_{m}.\nonumber\end{eqnarray}$$
Proof. The assertion follows directly by the following commutative diagram
where the right vertical map is an isomorphism by Bloch–Gabber–Kato theorem [Reference Bloch and KatoBK86] and Gersten resolutions of
$\unicode[STIX]{x1D716}_{\ast }{\mathcal{K}}_{r,X}^{M}$
and
$\unicode[STIX]{x1D716}_{\ast }W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{r}$
from [Reference KerzKer09, Reference Gros and SuwaGS88]; here
$\unicode[STIX]{x1D716}:X_{\text{Nis}}\rightarrow X_{\text{Zar}}$
is the canonical 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
$$\begin{eqnarray}\text{}\underline{n}:=(n_{\unicode[STIX]{x1D706}})_{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}}\geqslant \text{}\underline{n^{\prime }}:=(n_{\unicode[STIX]{x1D706}}^{\prime })_{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}}\quad \text{if }n_{\unicode[STIX]{x1D706}}\geqslant n_{\unicode[STIX]{x1D706}}^{\prime }\text{for all }\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}.\end{eqnarray}$$
For
$\text{}\underline{n}=(n_{\unicode[STIX]{x1D706}})_{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}}\in \mathbb{N}^{\unicode[STIX]{x1D6EC}}$
let
$$\begin{eqnarray}D_{\text{}\underline{n}}=\mathop{\sum }_{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}}n_{\unicode[STIX]{x1D706}}D_{\unicode[STIX]{x1D706}}\end{eqnarray}$$
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
$r\geqslant 1$
$$\begin{eqnarray}\displaystyle & \displaystyle \text{gr}^{\text{}\underline{n},\unicode[STIX]{x1D708}}{\mathcal{K}}_{r,X}^{M}:={\mathcal{K}}_{r,X|D_{\text{}\underline{n}}}^{M}/{\mathcal{K}}_{r,X|D_{\text{}\underline{n}+\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}}}^{M}\!; & \displaystyle \nonumber\\ \displaystyle & \displaystyle \text{gr}^{\text{}\underline{n},\unicode[STIX]{x1D708}}W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{r}:=W_{m}\unicode[STIX]{x1D6FA}_{X|D_{\text{}\underline{n}},\log }^{r}/W_{m}\unicode[STIX]{x1D6FA}_{X|D_{\text{}\underline{n}+\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}},\log }^{r}. & \displaystyle \nonumber\end{eqnarray}$$
Proposition 2.2.3. [Reference Rülling and SaitoRS18, Proposition 2.10]
Let
$\text{}\underline{n}=(n_{\unicode[STIX]{x1D706}})_{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}}\in \mathbb{N}^{\unicode[STIX]{x1D6EC}}$
, and let
$\unicode[STIX]{x1D708}\in \unicode[STIX]{x1D6EC},r\geqslant 1$
. Assume
$n_{\unicode[STIX]{x1D708}}=0$
and set
$$\begin{eqnarray}D_{\unicode[STIX]{x1D708},\text{}\underline{n}}:=\mathop{\sum }_{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}\setminus \{\unicode[STIX]{x1D708}\}}n_{\unicode[STIX]{x1D706}}(D_{\unicode[STIX]{x1D706}}\cap D_{\unicode[STIX]{x1D708}}).\end{eqnarray}$$
Then there is a natural isomorphism of Nisnevich sheaves
$$\begin{eqnarray}\text{gr}^{\text{}\underline{n},\unicode[STIX]{x1D708}}{\mathcal{K}}_{r,X}^{M}\xrightarrow[{}]{\cong }i_{\unicode[STIX]{x1D708},\ast }{\mathcal{K}}_{r,D_{\unicode[STIX]{x1D708}}|D_{\unicode[STIX]{x1D708},\text{}\underline{n}}}^{M}.\end{eqnarray}$$
Proof. The argument in [Reference Rülling and SaitoRS18] works verbatim for our case. ◻
Theorem 2.2.4. If
$D$
is reduced, then
$d\log$
induces an isomorphism of Nisnevich sheaves
$$\begin{eqnarray}\displaystyle d\log [-]:{\mathcal{K}}_{r,X|D}^{M}/p^{m} & \xrightarrow[{}]{\cong } & \displaystyle W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{r}\nonumber\\ \displaystyle \{x_{1},\ldots ,x_{r}\} & \mapsto & \displaystyle d\log [x_{1}]_{m}\wedge \cdots \wedge d\log [x_{r}]_{m}.\nonumber\end{eqnarray}$$
Proof. By the commutative diagram
it is enough to show that
${\mathcal{K}}_{r,X|D}^{M}/p^{m}{\hookrightarrow}{\mathcal{K}}_{r,X}^{M}/p^{m}$
. On the other hand, we have the following commutative diagram:
Combining the fact [Reference Geisser and LevineGL00, Theorem 8.1] and the Gersten resolution [Reference KerzKer09], we know that
${\mathcal{K}}_{r,X}^{M}$
is
$p$
-torsion free. Therefore the middle vertical map is injective, so is the first vertical map. By the snake lemme, it is sufficient to check that the third vertical map
$p^{m}:{\mathcal{K}}_{r,X}^{M}/{\mathcal{K}}_{r,X|D}^{M}\rightarrow {\mathcal{K}}_{r,X}^{M}/{\mathcal{K}}_{r,X|D}^{M}$
is injective. This follows from the above Proposition 2.2.3, by noting that
${\mathcal{K}}_{r,X}^{M}/{\mathcal{K}}_{r,X|D}^{M}$
is a successive extension of sheaves
$\text{gr}^{\text{}\underline{n},\unicode[STIX]{x1D708}}{\mathcal{K}}_{r,X}^{M}$
and the map
$p^{m}:i_{\unicode[STIX]{x1D708},\ast }{\mathcal{K}}_{r,D_{\unicode[STIX]{x1D708}}|D_{\unicode[STIX]{x1D708},\text{}\underline{n}}}^{M}\rightarrow i_{\unicode[STIX]{x1D708},\ast }{\mathcal{K}}_{r,D_{\unicode[STIX]{x1D708}}|D_{\unicode[STIX]{x1D708},\text{}\underline{n}}}^{M}$
is injective (similar to the injectivity of the first vertical map in the above diagram). We remark that the assumption in Proposition 2.2.3 is satisfied, since
$D$
is reduced.◻
Proposition 2.2.5. [Reference Jannsen, Saito and ZhaoJSZ18, Proposition 1.1.9]
Let
$X,D$
be as above. Then we have:
(i)
$W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d}=W_{m}\unicode[STIX]{x1D6FA}_{X|D_{\text{red}},\log }^{d}$
;(ii) for
$\text{}\underline{n}\geqslant \text{}\underline{1}$
, the quotient
$\text{gr}^{\text{}\underline{n},\unicode[STIX]{x1D708}}W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{r}$
is a coherent
${\mathcal{O}}_{D_{\unicode[STIX]{x1D708}}}^{p^{e}}$
-module, for some
$e\gg 0$
.
Proof. In the case that
$d=1$
(i.e.,
$R$
is a discrete valuation ring), the assertions have been given in [Reference Bloch and KatoBK86, (4.7), (4.8)]. For general
$d$
, in [Reference Jannsen, Saito and ZhaoJSZ18], the graded pieces have been studied in the case that
$R$
is the Henselization of a local ring of a smooth scheme over
$k$
. But note that the argument also works in our setting. We only need to show (i). By Theorem 2.2.4, we see that, for
$\text{}\underline{n}<\text{}\underline{1}$
,
$$\begin{eqnarray}\text{gr}^{\text{}\underline{n},\unicode[STIX]{x1D708}}{\mathcal{K}}_{d,X}^{M}/p^{m}\cong i_{\unicode[STIX]{x1D708},\ast }{\mathcal{K}}_{d,D_{\unicode[STIX]{x1D708}}|D_{\unicode[STIX]{x1D708},\text{}\underline{n}}}^{M}/p^{m}=i_{\unicode[STIX]{x1D708},\ast }W_{m}\unicode[STIX]{x1D6FA}_{D_{\unicode[STIX]{x1D708}}|D_{\unicode[STIX]{x1D708},\text{}\underline{n}},\log }^{d}=0,\end{eqnarray}$$
where the vanishing is by dimension. ◻
3 Class field theory for proper varieties over finite fields
In this section we reprove the main results of the class field theory of smooth proper varieties over finite fields with ramification along divisors
$D$
, which originally are due to Kato and Saito [Reference Kato and SaitoKS86].
Let
$X$
be a smooth proper variety of dimension
$d$
over a finite field
$k$
, let
$D$
be an effective divisor such that
$C:=$
Supp(
$D$
) is a simple normal crossing divisor on
$X$
, and let
$j:U:=X-C{\hookrightarrow}X$
be the complement of
$C$
. Let
$\{{D_{\unicode[STIX]{x1D706}}\}}_{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}}$
be the (smooth) irreducible components of
$D$
, and let
$i_{\unicode[STIX]{x1D706}}:D_{\unicode[STIX]{x1D706}}{\hookrightarrow}X$
be the natural map. We use the dimension function
$d(x)=\text{dim}(\overline{\{x\}})$
for
$x\in X$
. We also denote by
$X_{r}:=\{x\in X|~d(x)=r\}$
the set of points of dimension
$r$
of
$X$
and
$X^{r}:=X_{d-r}$
the set of points of codimension
$r$
of
$X$
.
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.
Theorem 3.1.1. [Reference KerzKer11, Theorem 8.4]
There exists a unique isomorphism
$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{X,D}:C(X,D)\cong H^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M})\end{eqnarray}$$
such that the following triangle commutes
where
$\imath$
is the obvious map, and
$\imath _{\text{Nis}}$
is the map from [Reference Kato and SaitoKS86, Theorem 2.5].
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
$$\begin{eqnarray}\displaystyle & \displaystyle E_{1,\acute{\text{e}}\text{t}}^{p,q}({\mathcal{F}}):=\bigoplus _{x\in X^{p}}H_{x}^{p+q}(X_{\acute{\text{e}}\text{t}},{\mathcal{F}})\Longrightarrow H^{p+q}(X_{\acute{\text{e}}\text{t}},{\mathcal{F}}) & \displaystyle \nonumber\\ \displaystyle & \displaystyle E_{1,\text{Nis}}^{p,q}({\mathcal{F}}):=\bigoplus _{x\in X^{p}}H_{x}^{p+q}(X_{\text{Nis}},{\mathcal{F}})\Longrightarrow H^{p+q}(X_{\text{Nis}},{\mathcal{F}}), & \displaystyle \nonumber\end{eqnarray}$$
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
$$\begin{eqnarray}\displaystyle \bigoplus _{x\in X^{d-1}}H_{x}^{d-1}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M}) & \rightarrow & \displaystyle \bigoplus _{x\in X^{d}}H_{x}^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M})\nonumber\\ \displaystyle & \rightarrow & \displaystyle H^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M})\rightarrow 0.\end{eqnarray}$$
We now study the coniveau spectral sequence for
$j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d}$
on
$X_{\acute{\text{e}}\text{t}}$
.
Proposition 3.2.1. Let
$X$
be a smooth (not necessarily proper) variety over a finite field of dimension
$d$
. For any
$x\in X^{a}$
, we have
$$\begin{eqnarray}H_{x}^{a+d+1}(X_{\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})=H_{x}^{a+d+1}(X_{\acute{\text{e}}\text{t}},\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d}),\end{eqnarray}$$
that is,
$E_{1,\acute{\text{e}}\text{t}}^{\bullet ,d+1}(j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})=E_{1,\acute{\text{e}}\text{t}}^{\bullet ,d+1}(\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})$
. In particular, we have
$E_{2,\acute{\text{e}}\text{t}}^{d-2,d+1}(j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})=E_{2,\acute{\text{e}}\text{t}}^{d-1,d+1}(j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})=0$
.
Proof. We prove the first claim by induction on the codimension
$a$
. For
$x\in X^{a}$
, we denote by
$X_{x}=\text{Spec}({\mathcal{O}}_{X,x}^{h})$
the Henselization of
$X$
at
$x$
, and
$Y_{x}=X_{x}\setminus \{x\}$
. If
$a=1$
, then any divisor of
$X_{x}$
must have support in the closed point
$\{x\}$
. Therefore
$$\begin{eqnarray}j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d}|_{Y_{x}}=\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d}|_{Y_{x}}\end{eqnarray}$$
by the definition of
$j_{!}$
. Using the localization exact sequences twice, we obtain
$$\begin{eqnarray}\displaystyle H_{x}^{d+2}(X_{\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d}) & \cong & \displaystyle H^{d+1}(Y_{x,\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})=H^{d+1}(Y_{x,\acute{\text{e}}\text{t}},\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})\nonumber\\ \displaystyle & \cong & \displaystyle H_{x}^{d+2}(X_{\acute{\text{e}}\text{t}},\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d}),\nonumber\end{eqnarray}$$
where the first isomorphism is due to
$j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d}|_{x}=0$
, and the second isomorphism is by the vanishing
$H^{d+2}(X_{x,\acute{\text{e}}\text{t}},\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})\cong H^{d+2}(x_{\acute{\text{e}}\text{t}},\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})=0=H^{d+1}(x_{\acute{\text{e}}\text{t}},\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})\cong H^{d+1}(X_{x,\acute{\text{e}}\text{t}},\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})$
, where we use the fact that
$\text{cd}_{\ell }(x)\leqslant d+1-\text{codim}_{X}(x)$
(cf. [Reference SatoSat09, Lemma 4.2(1)]).
For general codimension
$a>1$
, the coniveau spectral sequence on
$Y_{x}$
and cohomological vanishing give us an exact sequence
$$\begin{eqnarray}\displaystyle \bigoplus _{y\in Y_{x}^{a-2}}H_{y}^{a+d-1}(Y_{x,\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d}) & \rightarrow & \displaystyle \bigoplus _{y\in Y_{x}^{a-1}}H_{y}^{a+d}(Y_{x,\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})\nonumber\\ \displaystyle & \rightarrow & \displaystyle H^{a+d}(Y_{x,\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})\rightarrow 0.\end{eqnarray}$$
On the other hand, the localization exact sequence for
$j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d}$
on
$X_{x}$
tells us
$$\begin{eqnarray}H^{a+d}(Y_{x,\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})\cong H_{x}^{a+d+1}(X_{x,\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d}),\end{eqnarray}$$
Indeed due to
$\text{cd}_{\ell }(x)\leqslant d+1-\text{codim}_{X}(x)$
we have
$$\begin{eqnarray}H^{a+d}(X_{x,\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})=0=H^{a+d+1}(X_{x,\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d}).\end{eqnarray}$$
Combining these facts, we get the following diagram with exact rows
The first two vertical maps are isomorphisms by induction. Hence the third vertical arrow is also an isomorphism. Thanks to [Reference Jannsen, Saito and SatoJSS14, Theorem 3.5.1], we see that the complex
$E_{1,\acute{\text{e}}\text{t}}^{\bullet ,d+1}(\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})$
is the Kato complex of
$\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d}$
(cf. [Reference Kerz and SaitoKS12, (0.2)]) up to a sign. By the known Kato conjecture on vanishing of cohomology groups of this complex at places
$d-1$
and
$d-2$
(cf. [Reference Kerz and SaitoKS12, Theorem 8.1]) we obtain the second part of Proposition 3.2.1.◻
Corollary 3.2.2. We have the following exact sequence
$$\begin{eqnarray}\bigoplus _{x\in X^{d-1}}H_{x}^{2d-1}(X_{\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})\rightarrow \bigoplus _{x\in X^{d}}H_{x}^{2d}(X_{\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})\rightarrow H^{2d}(X_{\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})\rightarrow 0.\end{eqnarray}$$
Proof. By the above proposition, we have
$E_{2,\acute{\text{e}}\text{t}}^{d,d}(j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})=H^{2d}(X_{\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})$
.◻
Using the Galois symbol maps and induction on codimension, Sato constructs the localized Chern class map and proves the following theorem.
Theorem 3.2.3. [Reference SatoSat09, Theorem 1.2 and Section 3]
For any
$x\in X^{a}$
, there exists a canonical surjective map
$$\begin{eqnarray}\text{cl}_{X,D,x,\ell ^{m}}^{d,\text{loc}}:H_{x}^{a}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M})/\ell ^{m}{\twoheadrightarrow}H_{x}^{d+a}(X_{\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d}),\end{eqnarray}$$
which is called the localized Chern class map. Moreover, if
$x\in X^{d}$
, the localized Chern class map
$$\begin{eqnarray}\text{cl}_{X,D,x,\ell ^{m}}^{d,\text{loc}}:H_{x}^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M})/\ell ^{m}\xrightarrow[{}]{\cong }H_{x}^{2d}(X_{\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})\end{eqnarray}$$
is bijective.
Corollary 3.2.4. There is a canonical isomorphism
$$\begin{eqnarray}H^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M})/\ell ^{m}\cong H^{2d}(X_{\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d}).\end{eqnarray}$$
Proof. We have the following commutative diagram with exact rows:
where the first exact row follows from the exact sequence (3.2.1) by tensoring with
$\mathbb{Z}/\ell ^{m}\mathbb{Z}$
, the second is Corollary 3.2.2. By Theorem 3.2.3 the first vertical arrow is surjective and the second is bijective. Then the assertion follows from an easy diagram chasing.◻
Theorem 3.2.5. [Reference SaitoSai89, Lemma 2.9]
There is a perfect pairing of finite
$\mathbb{Z}/\ell ^{m}\mathbb{Z}$
-modules
$$\begin{eqnarray}H^{i}(U_{\acute{\text{e}}\text{t}},\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes r})\times H^{2d+1-i}(X_{\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d-r})\rightarrow H^{2d+1}(X_{\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})\xrightarrow[{}]{\cong }\mathbb{Z}/\ell ^{m}\mathbb{Z}.\end{eqnarray}$$
In particular, in case
$i=1,r=0$
, we obtain
$$\begin{eqnarray}H^{d}(X_{\acute{\text{e}}\text{t}},j_{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d})/\ell ^{m}\cong \unicode[STIX]{x1D70B}_{1}^{\text{ab}}(U)/\ell ^{m}.\end{eqnarray}$$
In summary:
Corollary 3.2.6. We obtain canonical isomorphisms
$$\begin{eqnarray}C(X,D)/\ell ^{m}\stackrel{\unicode[STIX]{x1D70C}_{X,D}}{\cong }H^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M})/\ell ^{m}\cong \unicode[STIX]{x1D70B}_{1}^{\text{ab}}(U)/\ell ^{m}.\end{eqnarray}$$
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
$$\begin{eqnarray}\displaystyle & \displaystyle E_{1,\acute{\text{e}}\text{t}}^{p,q}({\mathcal{F}}):=\bigoplus _{x\in X^{p}}H_{x}^{p+q}(X_{\acute{\text{e}}\text{t}},{\mathcal{F}})\Longrightarrow H^{p+q}(X_{\acute{\text{e}}\text{t}},{\mathcal{F}}) & \displaystyle \nonumber\\ \displaystyle & \displaystyle E_{1,\text{Nis}}^{p,q}({\mathcal{F}}):=\bigoplus _{x\in X^{p}}H_{x}^{p+q}(X_{\text{Nis}},{\mathcal{F}})\Longrightarrow H^{p+q}(X_{\text{Nis}},{\mathcal{F}}). & \displaystyle \nonumber\end{eqnarray}$$
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
$p>d$
.
Theorem 3.3.1. The canonical map
$$\begin{eqnarray}H^{d}(X_{\text{Nis}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})\xrightarrow[{}]{\cong }H^{d}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})\end{eqnarray}$$
is an isomorphism.
Proof. By the coniveau spectral sequences, it follows from the following two propositions. ◻
Proposition 3.3.2. Let
$X$
be a smooth (not necessarily proper) variety over a finite field of dimension
$d$
. The map
$E_{1,\acute{\text{e}}\text{t}}^{\bullet ,1}(W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})\xrightarrow[{}]{\cong }E_{1,\acute{\text{e}}\text{t}}^{\bullet ,1}(W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d})$
is an isomorphism of complexes. Therefore we have
$E_{2,\acute{\text{e}}\text{t}}^{d-1,1}(W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})=E_{2,\acute{\text{e}}\text{t}}^{d-2,1}(W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})=0$
.
Proof. For
$x\in X^{a}$
, we denote by
$X_{x}:=\text{Spec}({\mathcal{O}}_{X,x}^{h})$
the Henselization of
$X$
at
$x$
, and
$Y_{x}:=X_{x}\setminus \{x\}$
. We want to prove that
$$\begin{eqnarray}H_{x}^{a+1}(X,W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})\cong H_{x}^{a+1}(X,W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d}).\end{eqnarray}$$
If
$a=1$
, then any divisor of
$X_{x}$
must have support in the closed point
$\{x\}$
. Therefore, we have
$$\begin{eqnarray}W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d}|_{Y_{x}}=W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d}|_{Y_{x}}\end{eqnarray}$$
by the definition of
$W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d}$
. Using the localization exact sequences twice, we obtain
We claim that the first vertical arrow is surjective: Indeed, we have the exact sequence
$$\begin{eqnarray}\displaystyle H^{1}(X_{x,\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d}) & \rightarrow & \displaystyle H^{1}(X_{x,\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d})\nonumber\\ \displaystyle & \rightarrow & \displaystyle H^{1}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d}/W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d}),\nonumber\end{eqnarray}$$
where
$H^{1}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d}/W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})=0$
since this sheaf is a successive extension of coherent sheaves by Proposition 2.2.5. We conclude that the third vertical map in the previous commutative diagram is an isomorphism.
For general codimension
$a>1$
, we prove this by induction. The coniveau spectral sequence on
$Y_{x}$
gives us the exact sequence
$$\begin{eqnarray}\displaystyle \bigoplus _{y\in Y_{x}^{a-2}}H_{y}^{a-1}(Y_{x,\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d}) & \rightarrow & \displaystyle \bigoplus _{y\in Y_{x}^{a-1}}H_{y}^{a}(Y_{x,\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})\nonumber\\ \displaystyle & \rightarrow & \displaystyle H^{a}(Y_{x,\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})\rightarrow 0.\end{eqnarray}$$
On the other hand, the localization exact sequence for
$W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d}$
on
$X_{x}$
tells us
$$\begin{eqnarray}H^{a}(Y_{x,\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})\cong H_{x}^{a+1}(X_{x,\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d}),\end{eqnarray}$$
since we know that
$H^{a+1}(X_{x,\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})\cong H^{a+1}(x_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})=0$
and similarly
$H^{a}(X_{x,\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})\cong H^{a}(x_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})=0$
. Combining these facts, we get the following diagram with exact rows:
The first two vertical maps are isomorphisms by induction. Hence the third vertical arrow is also an isomorphism. Thanks to [Reference Jannsen, Saito and SatoJSS14, Theorem 4.11.1], we see that the complex
$E_{1,\acute{\text{e}}\text{t}}^{\bullet ,1}(W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d})$
is the Kato complex of
$W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d}$
(cf. [Reference Kerz and SaitoKS12, (0.2)]) up to a sign. By the known Kato conjecture on vanishing of the cohomology groups of this complex at places
$d-1$
and
$d-2$
(cf. [Reference Jannsen and SaitoJS03]), we obtain the second part of Proposition 3.3.2.◻
Proposition 3.3.3. Let
$X$
be a smooth (not necessarily proper) over a finite field
$k$
of dimension
$d$
. For any
$x\in X^{a}$
, the canonical map
$$\begin{eqnarray}H_{x}^{a}(X_{\text{Nis}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})\rightarrow H_{x}^{a}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})\end{eqnarray}$$
is an isomorphism.
That is, there is a natural isomorphism of complexes
$$\begin{eqnarray}E_{1,\text{Nis}}^{\bullet ,0}(W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})\xrightarrow[{}]{\cong }E_{1,\acute{\text{e}}\text{t}}^{\bullet ,0}(W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})\end{eqnarray}$$
Proof. To prove this, we use Proposition 2.2.5(ii). We reduced to the case that
$D$
is reduced, since the quotient
$W_{m}\unicode[STIX]{x1D6FA}_{X|D}^{d}/W_{m}\unicode[STIX]{x1D6FA}_{X|D_{\text{red}}}^{d}$
on
$X_{\text{Nis}}$
is a successive extension of coherent sheaves, for which the étale and Nisnevich cohomology groups are the same. By Proposition 2.2.5(i), it is equivalent to show that the canonical map
$$\begin{eqnarray}H_{x}^{a}(X_{\text{Nis}},W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d})\xrightarrow[{}]{\cong }H_{x}^{a}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d})\end{eqnarray}$$
is an isomorphism. This is true since both are isomorphic to
$K_{d-a}^{M}(k(x))/p^{m}=W_{m}\unicode[STIX]{x1D6FA}_{x,\log }^{d-a}$
by purity [Reference MilneMil86, Proposition 2.1] and the known Gersten conjecture [Reference Gros and SuwaGS88].◻
Corollary 3.3.4. There is a canonical isomorphism
$$\begin{eqnarray}H^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M})/p^{m}\cong H^{d}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d}).\end{eqnarray}$$
Proof. First we have
$$\begin{eqnarray}\displaystyle H^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M})/p^{m} & \cong & \displaystyle H^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M}/p^{m})\nonumber\\ \displaystyle & \cong & \displaystyle H^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M}/p^{m}{\mathcal{K}}_{d,X}^{M}\cap {\mathcal{K}}_{d,X|D}^{M}),\nonumber\end{eqnarray}$$
where the first isomorphism is due to the fact that the Nisnevich cohomological dimension of
$X$
is
$d$
, and the second follows from the observation that the support of
$p^{m}{\mathcal{K}}_{d,X}^{M}\cap {\mathcal{K}}_{d,X|D}^{M}/p^{m}{\mathcal{K}}_{d,X|D}^{M}$
is contained in
$D$
, which is of dimension
$d-1$
.
By Theorems 2.2.2 and 3.3.1, hence we have
$$\begin{eqnarray}H^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M})/p^{m}\cong H^{d}(X_{\text{Nis}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})\cong H^{d}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d}).\hspace{-11.99998pt}\square\end{eqnarray}$$
Corollary 3.3.5. Let
$D_{1},D_{2}$
be two effective divisors on
$X$
whose supports are simple normal crossing divisors. Assume
$D_{1}\geqslant D_{2}$
. Then the canonical map
$$\begin{eqnarray}H^{d}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D_{1},\log }^{d})\rightarrow H^{d}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D_{2},\log }^{d})\end{eqnarray}$$
is surjective.
Proof. Note that we have the following exact sequence on
$X_{\text{Nis}}$
$$\begin{eqnarray}0\rightarrow {\mathcal{K}}_{d,X|D_{1}}^{M}\rightarrow {\mathcal{K}}_{d,X|D_{2}}^{M}\rightarrow {\mathcal{K}}_{d,X|D_{2}}^{M}/{\mathcal{K}}_{d,X|D_{1}}^{M}\rightarrow 0,\end{eqnarray}$$
but the Nisnevich sheaf
${\mathcal{K}}_{d,X|D_{2}}^{M}/{\mathcal{K}}_{d,X|D_{1}}^{M}$
is supported in
$D_{2}$
, which is of dimension
$d-1$
. Hence the associated long exact sequence implies that
$$\begin{eqnarray}H^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D_{1}}^{M})\rightarrow H^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D_{2}}^{M})\end{eqnarray}$$
is surjective. Therefore the claim follows from Corollary 3.3.4. ◻
Now, we recall the duality theorem of the relative logarithmic de Rham–Witt sheaves.
Theorem 3.3.6. [Reference Jannsen, Saito and ZhaoJSZ18, Theorem 4.1.4]
Let
$X,U,D$
be as before. For
$i\in \mathbb{N},r\in \mathbb{N}$
, there are natural perfect pairings of topological groups
$$\begin{eqnarray}\displaystyle & & \displaystyle H^{i}(U_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{U,\log }^{r})\times \underset{\substack{ E \\ \text{Supp}(E)\subset X\setminus U}}{\varprojlim }H^{d+1-i}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|E,\log }^{d-r})\nonumber\\ \displaystyle & & \displaystyle \quad \rightarrow H^{d+1}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d})\xrightarrow[{}]{\text{Tr}}\mathbb{Z}/p^{m}\mathbb{Z},\nonumber\end{eqnarray}$$
where the first group is endowed with discrete topology, the second is endowed with profinite topology, and the limit with respect to all effective divisor
$E$
with
$\text{Supp}(E)\subset X\setminus U$
.
In particular, for
$i=1$
and
$r=0$
we get isomorphisms
$$\begin{eqnarray}\underset{E}{\varprojlim }H^{d}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|E,\log }^{d})\xrightarrow[{}]{\cong }H^{1}(U_{\acute{\text{e}}\text{t}},\mathbb{Z}/p^{m}\mathbb{Z})^{\vee }\cong \unicode[STIX]{x1D70B}_{1}^{ab}(U)/p^{m},\end{eqnarray}$$
and
$$\begin{eqnarray}H^{1}(U_{\acute{\text{e}}\text{t}},\mathbb{Z}/p^{m}\mathbb{Z})\xrightarrow[{}]{\cong }\underset{E}{\varinjlim }H^{d}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|E,\log }^{d})^{\vee },\end{eqnarray}$$
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
$p^{m}$
.
Definition 3.3.7. For our divisor
$D$
, we define
$$\begin{eqnarray}\text{Fil}_{D}H^{1}(U_{\acute{\text{e}}\text{t}},\mathbb{Z}/p^{m}\mathbb{Z}):=H^{d}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})^{\vee }.\end{eqnarray}$$
Dually we define
$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{1}^{\text{ab}}(X,D)/p^{m}:=\text{Hom}(\text{Fil}_{D}H^{1}(U_{\acute{\text{e}}\text{t}},\mathbb{Z}/p^{m}\mathbb{Z}),\mathbb{Z}/p^{m}\mathbb{Z}).\end{eqnarray}$$
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
$D$
.
Corollary 3.3.8. We have canonical isomorphisms
$$\begin{eqnarray}C(X,D)/p^{m}\cong H^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M})/p^{m}\xrightarrow[{}]{\cong }\unicode[STIX]{x1D70B}_{1}^{\text{ab}}(X,D)/p^{m}.\end{eqnarray}$$
3.4 Class field theory via ideles
Theorem 3.4.1. (Logarithmic version of wildly ramified class field theory)
For any integer
$n$
, there exists a canonical isomorphism
$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{X,D,n}:C(X,D)/n\xrightarrow[{}]{\cong }\unicode[STIX]{x1D70B}_{1}^{\text{ab}}(X,D)/n,\end{eqnarray}$$
such that the following triangle commutes
where the right diagonal map
$\unicode[STIX]{x1D70C}_{U}$
sends
$1$
at the point
$x$
to the Frobenius
$\text{Frob}_{x}$
. In particular,
$\unicode[STIX]{x1D70C}_{X,D,n}$
induces an isomorphism
$$\begin{eqnarray}\underset{D,n}{\varprojlim }C(X,D)/n\cong \unicode[STIX]{x1D70B}_{1}^{\text{ab}}(U).\end{eqnarray}$$
Proof. For
$n=p^{m}$
, this follows from Corollary 3.3.8 and Theorem 3.1.1 directly. For
$n$
prime to
$p$
, this is Corollary 3.2.6.◻
Remark 3.4.2. The wildly ramified class field theory in [Reference Kerz and SaitoKS16], where we work with the relative Chow group of zero cycles instead of the idelic class group, comprises Theorem 3.4.1.
4 Class field theory for complete local rings over
$\mathbb{F}_{q}$
Let
$(A,\mathfrak{m})$
be a complete regular local ring of dimension
$d$
and of characteristic
$p>0$
, and let
$k:=A/\mathfrak{m}$
be the residue field. We assume that
$k$
is finite. We denote
$X=\text{Spec}(A),x=\mathfrak{m}\in X$
. Let
$D$
be an effective divisor with
$\text{Supp}(D)$
is a simple normal crossing divisor, let
$U=X\setminus D$
be its complement. Set
$X^{\prime }=X\setminus \{x\},D^{\prime }=D\setminus \{x\}$
. We use the dimension function on
$X$
(hence also induces one on
$X^{\prime }$
) by
$d(x)=\text{dim}(\overline{\{x\}})$
.
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]:
$$\begin{eqnarray}\text{res}:H_{x}^{d}(X,\unicode[STIX]{x1D6FA}_{X}^{d})\rightarrow k.\end{eqnarray}$$
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
$$\begin{eqnarray}\text{Tr}_{k/\mathbb{F}_{p}}\circ \text{res}:H_{x}^{d}(X,\unicode[STIX]{x1D6FA}_{X}^{d})\rightarrow \mathbb{Z}/p\mathbb{Z}.\end{eqnarray}$$
For any finite
$A$
-module
$M$
, the Yoneda pairing and the above trace map give us a canonical pairing
$$\begin{eqnarray}H_{x}^{i}(X,M)\times \text{Ext}_{X}^{d-i}(M,\unicode[STIX]{x1D6FA}_{X}^{d})\rightarrow \mathbb{Z}/p\mathbb{Z}.\end{eqnarray}$$
Theorem 4.1.1. (Grothendieck local duality [Reference Grothendieck and HartshorneGH67])
For each integer
$i\geqslant 0$
, the pairing (4.1.1) induces the isomorphisms
$$\begin{eqnarray}\displaystyle & \displaystyle \text{Ext}_{A}^{d-i}(M,\unicode[STIX]{x1D6FA}_{X}^{d})\cong \text{Hom}_{\mathbb{Z}/p\mathbb{Z}}(H_{x}^{i}(X,M),\mathbb{Z}/p\mathbb{Z}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle H_{x}^{i}(X,M)\cong \text{Hom}_{\text{cont}}(\text{Ext}_{A}^{d-i}(M,\unicode[STIX]{x1D6FA}_{X}^{d}),\mathbb{Z}/p\mathbb{Z}), & \displaystyle \nonumber\end{eqnarray}$$
where
$\text{Hom}_{\text{cont}}$
denotes the set of continuous homomorphisms with respect to
$\mathfrak{m}$
-adic topology on Ext group.
In particular, if
$M$
is a locally free
$A$
-module, we obtain the isomorphisms
$$\begin{eqnarray}H^{d-i}(M^{t})\cong \text{Hom}_{\mathbb{ Z}/p\mathbb{Z}}(H_{x}^{i}(X,M),\mathbb{Z}/p\mathbb{Z}),\end{eqnarray}$$
where
$M^{t}:=\text{Hom}_{A}(M,\unicode[STIX]{x1D6FA}_{X}^{d})$
is the dual
$A$
-module, and
$$\begin{eqnarray}H_{x}^{i}(X,M)\cong \text{Hom}_{\text{cont}}(H^{d-i}(M^{t}),\mathbb{Z}/p\mathbb{Z}).\end{eqnarray}$$
Note that, for a locally free
$A$
-module
$M$
, we have [Reference Grothendieck and HartshorneGH67]
$$\begin{eqnarray}H_{x}^{i}(X,M)=0\quad \text{if }i\neq d.\end{eqnarray}$$
4.2 Duality theorems
The purity result of Shiho [Reference ShihoShi07, Theorem 3.2] tells us that there exists a canonical isomorphism
$$\begin{eqnarray}\text{Tr}:H_{x}^{d+1}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d})\xrightarrow[{}]{\cong }H^{1}(x,\mathbb{Z}/p^{m}\mathbb{Z})\cong \mathbb{Z}/p^{m}\mathbb{Z}.\end{eqnarray}$$
Using the same method as in [Reference ZhaoZha16], we obtain a map
$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6F7}_{m}^{i,r}:H^{i}(U_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{U,\log }^{r})\rightarrow \underset{E}{\varinjlim }\text{Hom}_{\mathbb{Z}/p^{n}\mathbb{Z}}\nonumber\\ \displaystyle & & \displaystyle \quad \times \,(H_{x}^{d+1-i}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|E,\log }^{d-r}),H_{x}^{d+1}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d})).\nonumber\end{eqnarray}$$
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:
$$\begin{eqnarray}H^{i}(U_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{U,\log }^{r})\times \underset{E}{\varprojlim }H_{x}^{d+1-i}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|E,\log }^{d-r})\rightarrow \mathbb{Z}/p^{m}\mathbb{Z}.\end{eqnarray}$$
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
$i,m,r$
.
Theorem 4.2.1. For any integers
$r\geqslant 0,m\geqslant 1$
, the maps
$\unicode[STIX]{x1D6F7}_{m}^{i,r}$
are isomorphisms.
Proof. We are reduced to the case
$m=1$
by induction on
$m$
and the following two exact sequences on the small étale site
$$\begin{eqnarray}0\rightarrow W_{m-1}\unicode[STIX]{x1D6FA}_{U,\log }^{r}\xrightarrow[{}]{\cdot p}W_{m}\unicode[STIX]{x1D6FA}_{U,\log }^{r}\xrightarrow[{}]{R}\unicode[STIX]{x1D6FA}_{U,\log }^{r}\rightarrow 0\end{eqnarray}$$
and
$$\begin{eqnarray}0\rightarrow W_{m-1}\unicode[STIX]{x1D6FA}_{X|[E/p],\log }^{d-r}\xrightarrow[{}]{\cdot p}W_{m}\unicode[STIX]{x1D6FA}_{X|E,\log }^{d-r}\xrightarrow[{}]{R}\unicode[STIX]{x1D6FA}_{X|E,\log }^{d-r}\rightarrow 0,\end{eqnarray}$$
where
$[E/p]=\sum _{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}}[n_{\unicode[STIX]{x1D706}}/p]D_{\unicode[STIX]{x1D706}}$
if
$D=\sum _{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}}n_{\unicode[STIX]{x1D706}}D_{\unicode[STIX]{x1D706}}$
; here
$[n/p]=\text{min}\{n^{\prime }\in \mathbb{Z}|pn^{\prime }\geqslant n\}$
, and the exactness of the second complex follows from [Reference Jannsen, Saito and ZhaoJSZ18, Theorem 1.1.6].
Using the relation between logarithmic forms and differential forms ([Reference IllusieIll79, 0, Corollary 2.1.18] and [Reference Jannsen, Saito and ZhaoJSZ18, Theorem 1.2.1]), we see that the assertion for
$i\neq 0,1$
follows from the vanishing (4.1.4) directly. We have the following diagram with exact rows
where
$A^{\ast }:=\text{Hom}_{\mathbb{Z}/p\mathbb{Z}}(A,\mathbb{Z}/p\mathbb{Z})$
for an abelian group
$A$
,
$$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{X|E}^{d-r}:=\unicode[STIX]{x1D6FA}_{X}^{d-r}(\log E_{\text{red}})\otimes {\mathcal{O}}_{X}(-E),\end{eqnarray}$$
and
$d\unicode[STIX]{x1D6FA}_{X|E}^{d-r-1}:=\text{Image}(d:\unicode[STIX]{x1D6FA}_{X|E}^{d-r-1}\rightarrow \unicode[STIX]{x1D6FA}_{X}^{d-r})$
, and
$Z\unicode[STIX]{x1D6FA}_{U}^{r}:=\text{Ker}(d:\unicode[STIX]{x1D6FA}_{U}^{r}\rightarrow \unicode[STIX]{x1D6FA}_{U}^{r+1})$
.
The proof is same as the proof in [Reference Jannsen, Saito and ZhaoJSZ18, Reference ZhaoZha16], we quickly recall the argument: since
$j:U\rightarrow X$
is affine, we may rewrite
$H^{0}(U,\unicode[STIX]{x1D6FA}_{U}^{r})$
as
$\mathop{\varinjlim }\nolimits_{E}H^{0}(X,\unicode[STIX]{x1D6FA}_{X}^{i}(\log E_{\text{red}})\otimes {\mathcal{O}}_{X}(E))$
. Then we use Theorem 4.1.1 for sheaves
$\unicode[STIX]{x1D6FA}_{X}^{i}(\log E_{\text{red}})(-E)$
to conclude that the second and the third vertical arrows are isomorphisms. Hence the assertion follows.◻
For
$r=0,i=1$
, we get
$$\begin{eqnarray}H^{1}(U_{\acute{\text{e}}\text{t}},\mathbb{Z}/p^{m}\mathbb{Z})\cong \underset{E}{\varinjlim }\text{Hom}(H_{x}^{d}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|E,\log }^{d}),\mathbb{Z}/p^{m}\mathbb{Z}).\end{eqnarray}$$
Similar to Corollary 3.3.5, the transition maps are surjective in the projective system, for our divisor
$D$
we define
$$\begin{eqnarray}\text{Fil}_{D}H^{1}(U_{\acute{\text{e}}\text{t}},\mathbb{Z}/p^{m}\mathbb{Z}):=\text{Hom}(H_{x}^{d}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d}),\mathbb{Z}/p^{m}\mathbb{Z});\end{eqnarray}$$
by Pontryagin duality, we also define
$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{1}^{\text{ab}}(X,D)/p^{m}:=\text{Hom}(\text{Fil}_{D}H^{1}(U_{\acute{\text{e}}\text{t}},\mathbb{Z}/p^{m}\mathbb{Z}),\mathbb{Z}/p^{m}\mathbb{Z}).\end{eqnarray}$$
Theorem 4.2.1 gives us an isomorphism
$$\begin{eqnarray}H_{x}^{d}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})\xrightarrow[{}]{\cong }\unicode[STIX]{x1D70B}_{1}^{\text{ab}}(X,D)/p^{m}.\end{eqnarray}$$
Proposition 4.2.2. We have
$$\begin{eqnarray}H_{x}^{d}(X_{\text{Nis}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d})\cong H_{x}^{d}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d}).\end{eqnarray}$$
Proof. This is similar to the argument in the proof of Proposition 3.3.3. Only the last step, to claim
$$\begin{eqnarray}H_{x}^{a}(X_{\text{Nis}},W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d})\xrightarrow[{}]{\cong }H_{x}^{a}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d})\end{eqnarray}$$
is an isomorphism, uses different results. In this case, it is an isomorphism since both are isomorphic to
$K_{d-a}^{M}(k(x))/p^{m}=W_{m}\unicode[STIX]{x1D6FA}_{x,\log }^{d-a}$
by purity [Reference ShihoShi07, Theorem 3.2] and the known Gersten conjecture [Reference KerzKer09].◻
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.
Theorem 4.3.1. [Reference KerzKer11, Theorem 8.2]
There exists an isomorphism
$$\begin{eqnarray}C(X^{\prime },D^{\prime })\cong H_{x}^{d}(X_{\text{Nis}},{\mathcal{K}}_{d,X|D}^{M}).\end{eqnarray}$$
In summary, the class field theory of Henselian regular local ring over
$\mathbb{F}_{p}$
can be reformulated as follows:
Corollary 4.3.2. There is a canonical isomorphism
$$\begin{eqnarray}C(X^{\prime },D^{\prime })/p^{m}\xrightarrow[{}]{\cong }\unicode[STIX]{x1D70B}_{1}^{\text{ab}}(X,D)/p^{m}.\end{eqnarray}$$
Remark 4.3.3. The case
$d=2$
has been studied in [Reference SaitoSai87]. The case
$d=3$
has been investigated in [Reference MatsumiMat02] using a slightly different class group. The
$\ell$
-primary analog has been studied by Sato in [Reference SatoSat09].
5 Class field theory for schemes over discrete valuation rings
Let
$R$
be a Henselian discrete valuation ring with fraction field
$K$
, and let
$k$
be its residue field of characteristic
$p>0$
which we assume to be finite. We fix an uniformizer
$\unicode[STIX]{x1D70B}$
of
$R$
. We use the notation as in the following diagram:
where
$f$
is a flat projective of fiber dimension
$d$
. We assume that
$X$
is a regular scheme with smooth generic fiber
$X_{\unicode[STIX]{x1D702}}$
such that the reduced special fiber
$X_{s,\text{red}}$
is a simple normal crossing divisor. Let
$\jmath :U{\hookrightarrow}X$
be an open subscheme contained in the generic fiber such that
$X\setminus U$
is the support of a simple normal crossing divisor
$D$
.
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
$X$
.
Definition 5.1.1.
(i) A
$Q^{o}$
-chain on
$(U\subset X)$
is a
$Q$
-chain on$$\begin{eqnarray}P=(p_{0},\ldots ,p_{s-2},p_{s})\end{eqnarray}$$
$(U\subset X)$
such that
$s\geqslant 2$
. We denote the set of
$Q^{o}$
-chain on
$(U\subset X)$
by
${\mathcal{Q}}^{o}$
.(ii) The idele class group
$C(U\subset X;X_{s})$
is $$\begin{eqnarray}\displaystyle & & \displaystyle C(U\subset X;X_{s})\nonumber\\ \displaystyle & & \displaystyle \quad :=\text{Coker}\biggl(\bigoplus _{P\in {\mathcal{Q}}^{o}}K_{d(P)}^{M}(k(P))\oplus \bigoplus _{y\in U_{\unicode[STIX]{x1D702}}^{d-1}}K_{2}^{M}(k(y))\xrightarrow[{}]{Q}I(U\subset X)\biggr);\nonumber\end{eqnarray}$$
(iii) The idele class group
$C(X,D;X_{s})$
of
$X$
relative to the effective divisor
$D$
is defined as $$\begin{eqnarray}\displaystyle & & \displaystyle C(X,D;X_{s})\nonumber\\ \displaystyle & & \displaystyle \quad :=\text{Coker}\biggl(\bigoplus _{P\in {\mathcal{Q}}^{o}}K_{d(P)}^{M}(k(P))\oplus \bigoplus _{y\in U_{\unicode[STIX]{x1D702}}^{d-1}}K_{2}^{M}(k(y))\xrightarrow[{}]{Q}I(X,D)\biggr).\nonumber\end{eqnarray}$$
Theorem 5.1.2.
(i) There exists a canonical isomorphism
$$\begin{eqnarray}C(X,D;X_{s})\cong H_{X_{s}}^{d+1}(X_{\text{Nis}},{\mathcal{K}}_{d+1,X|D}^{M}).\end{eqnarray}$$
(ii)
$H^{d+1}(X_{\text{Nis}},{\mathcal{K}}_{d+1,X|D}^{M})=0.$
Proof. Let
${\mathcal{F}}$
be the Nisnevich sheaf
${\mathcal{K}}_{d+1,X|D}^{M}$
. We start with part (i). We have seen that the degeneration of the coniveau spectral sequence
$$\begin{eqnarray}E_{1,\text{Nis}}^{p,q}({\mathcal{F}}):=\bigoplus _{x\in X^{p}}H_{x}^{p+q}(X_{\text{Nis}},{\mathcal{F}})\Longrightarrow H^{p+q}(X_{\text{Nis}},{\mathcal{F}})\end{eqnarray}$$
implies
$$\begin{eqnarray}H_{X_{s}}^{d+1}(X_{\text{Nis}},{\mathcal{F}})=\text{Coker}\biggl(\bigoplus _{x\in X_{1}\cap X_{s}}H_{x}^{d}(X_{\text{Nis}},{\mathcal{F}})\rightarrow \bigoplus _{x\in X_{0}}H_{x}^{d+1}(X_{\text{Nis}},{\mathcal{F}})\biggr).\end{eqnarray}$$
By definition and [Reference KerzKer11, Theorem 8.2] we obtain an isomorphism
$$\begin{eqnarray}C(X,D;X_{s})\cong \text{Coker}\biggl(\bigoplus _{y\in U_{\unicode[STIX]{x1D702}}^{d-1}}K_{2}^{M}(k(y))\rightarrow \bigoplus _{x\in X_{0}}H_{x}^{d+1}(X_{\text{Nis}},{\mathcal{F}})\biggr).\end{eqnarray}$$
It is sufficient to observe that the canonical map
$$\begin{eqnarray}\bigoplus _{y\in U_{\unicode[STIX]{x1D702}}^{d-1}}K_{2}^{M}(k(y))\rightarrow \bigoplus _{x\in X_{1}\cap X_{s}}H_{x}^{d}(X_{\text{Nis}},{\mathcal{F}})\end{eqnarray}$$
is surjective; see [Reference KerzKer11, Section 6]. This finishes the proof of part (i).
For part (ii) we use the isomorphism
$$\begin{eqnarray}H^{d+1}(X_{\text{Nis}},{\mathcal{F}})=\text{Coker}\biggl(\bigoplus _{x\in X_{1}}H_{x}^{d}(X_{\text{Nis}},{\mathcal{F}})\rightarrow \bigoplus _{x\in X_{0}}H_{x}^{d+1}(X_{\text{Nis}},{\mathcal{F}})\biggr)\end{eqnarray}$$
and the surjectivity of
$$\begin{eqnarray}\bigoplus _{x\in X_{1}\cap X_{\unicode[STIX]{x1D702}}}K_{1}^{M}(k(x))\rightarrow \bigoplus _{x\in X_{0}}H_{x}^{d+1}(X_{\text{Nis}},{\mathcal{F}});\end{eqnarray}$$
see [Reference KerzKer11, Section 6]. ◻
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.
Definition 5.1.3. Assume
$\text{Supp}(D)\supset X_{s}$
, we denote
$D_{\unicode[STIX]{x1D702}}=D\times _{X}X_{\unicode[STIX]{x1D702}}$
, and define
$$\begin{eqnarray}\widehat{SK}_{1}(U):=\underset{D}{\varprojlim }C(X,D;X_{s})=\underset{E}{\varprojlim }H_{X_{s}}^{d+1}(X_{\text{Nis}},{\mathcal{K}}_{d+1,X|E}^{M}),\end{eqnarray}$$
where the limit is over all effective divisors
$E$
with support
$X\setminus U$
.
$$\begin{eqnarray}SK_{1}(X_{\unicode[STIX]{x1D702}},D_{\unicode[STIX]{x1D702}}):=H^{d}(X_{\unicode[STIX]{x1D702},\text{Nis}},{\mathcal{K}}_{d+1,X|D}^{M}).\end{eqnarray}$$
Remark 5.1.4.
(i) We have seen that, by the degeneration of the coniveau spectral sequence, the group
$SK_{1}(X_{\unicode[STIX]{x1D702}},D_{\unicode[STIX]{x1D702}})=H^{d}(X_{\unicode[STIX]{x1D702},\text{Nis}},{\mathcal{K}}_{d+1,X|D}^{M})$
is isomorphic to (5.1.3)Using the methods from [Reference KerzKer11] it is easy to write down an idelic description of this group, for example, if$$\begin{eqnarray}\displaystyle & & \displaystyle \text{coker}\biggl(\bigoplus _{y\in (X_{\unicode[STIX]{x1D702}})_{1}}H_{y}^{d-1}(X_{\unicode[STIX]{x1D702},\text{Nis}},{\mathcal{K}}_{d+1,X|D}^{M})\nonumber\\ \displaystyle & & \displaystyle \quad \xrightarrow[{}]{\unicode[STIX]{x2202}}\bigoplus _{x\in (X_{\unicode[STIX]{x1D702}})_{0}}H_{x}^{d}(X_{\unicode[STIX]{x1D702},\text{Nis}},{\mathcal{K}}_{d+1,X|D}^{M})\biggr).\end{eqnarray}$$
$D_{\unicode[STIX]{x1D702}}=0$
then
$SK_{1}(X_{\unicode[STIX]{x1D702}},0)=SK_{1}(X_{\unicode[STIX]{x1D702}})$
where
$SK_{1}(X_{\unicode[STIX]{x1D702}})$
is defined as $$\begin{eqnarray}\text{coker}\biggl(\bigoplus _{y\in (X_{\unicode[STIX]{x1D702}})_{1}}K_{2}^{M}(\unicode[STIX]{x1D705}(y))\xrightarrow[{}]{\unicode[STIX]{x2202}}\bigoplus _{x\in (X_{\unicode[STIX]{x1D702}})_{0}}\unicode[STIX]{x1D705}(x)^{\times }\biggr).\end{eqnarray}$$
(ii) If
$d=1$
and
$\text{Supp}(D)=X_{s}$
, then
$\widehat{SK}_{1}(U)=\widehat{SK}_{1}(X_{\unicode[STIX]{x1D702}})$
, which has been defined in [Reference Kato and SaitoKS83] via the idelic method.(iii) By Theorem 5.1.2 we get a canonical surjection
We do not know, whether this map is an isomorphism in general, but Theorem 5.3.7 suggests that it is so at least after tensoring with$$\begin{eqnarray}SK_{1}(X_{\unicode[STIX]{x1D702}},D_{\unicode[STIX]{x1D702}})\rightarrow C(X,D;X_{s}).\end{eqnarray}$$
$\mathbb{Z}/n\mathbb{Z}$
for any integer
$n>0$
.
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
$$\begin{eqnarray}Y_{i_{1},\ldots ,i_{s}}:=Y_{i_{1}}\times _{Y}\cdots \times _{Y}Y_{i_{s}}\end{eqnarray}$$
be the scheme-theoretic intersection of
$Y_{i_{1}},\ldots ,Y_{i_{s}}$
, and denote
$$\begin{eqnarray}Y^{[s]}:=\coprod _{1\leqslant i_{1}<\cdots <i_{s}\leqslant N}Y_{i_{1},\ldots ,i_{s}}\end{eqnarray}$$
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
$$\begin{eqnarray}i^{[s]}:Y^{[s]}\rightarrow Y,\qquad \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}:Y^{[s]}\rightarrow Y^{[s-1]}.\end{eqnarray}$$
For integer
$n>0,i\geqslant 0$
we define the following étale sheaves on
$Y$
:
(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.(ii) If
$n=mp^{r},r\geqslant 0$
with
$p\nmid m$
, then let where$$\begin{eqnarray}\mathbb{Z}/n\mathbb{Z}(i):=\unicode[STIX]{x1D708}_{r,Y}^{i}[-i]\oplus \unicode[STIX]{x1D707}_{m,Y}^{\otimes i}\end{eqnarray}$$
$\unicode[STIX]{x1D708}_{r,Y}^{i}(U):=\ker (\unicode[STIX]{x2202}:\bigoplus _{x\in U^{0}}W_{r}\unicode[STIX]{x1D6FA}_{x,\log }^{i}\rightarrow \bigoplus _{x\in U^{1}}W_{r}\unicode[STIX]{x1D6FA}_{x,\log }^{i-1})$
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:
$$\begin{eqnarray}\displaystyle & & \displaystyle \bigoplus _{y\in Y^{0}}H^{d+1}(y,\mathbb{Z}/n\mathbb{Z}(d))\rightarrow \bigoplus _{y\in Y^{1}}H^{d}(y,\mathbb{Z}/n\mathbb{Z}(d-1))\rightarrow \cdots \nonumber\\ \displaystyle & & \displaystyle \quad \cdots \rightarrow \bigoplus _{y\in Y^{a}}H^{d-a+1}(y,\mathbb{Z}/n\mathbb{Z}(d-a))\rightarrow \cdots \rightarrow \bigoplus _{y\in Y^{d}}H^{1}(y,\mathbb{Z}/n\mathbb{Z}),\nonumber\end{eqnarray}$$
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
$$\begin{eqnarray}\cdots \rightarrow (\mathbb{Z}/n\mathbb{Z})^{\unicode[STIX]{x1D70B}_{0}(Y^{[s+1]})}\xrightarrow[{}]{d_{s}}(\mathbb{Z}/n\mathbb{Z})^{\unicode[STIX]{x1D70B}_{0}(Y^{[s]})}\cdots \rightarrow (\mathbb{Z}/n\mathbb{Z})^{\unicode[STIX]{x1D70B}_{0}(Y^{[1]})},\end{eqnarray}$$
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
$\sum _{\unicode[STIX]{x1D708}=1}^{s+1}(-1)^{\unicode[STIX]{x1D708}+1}(\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})_{\ast }$
.
Theorem 5.2.1. [Reference Jannsen and SaitoJS03, Proposition 3.6 and Theorem 3.9]
(i) There is a spectral sequence
in which the differentials$$\begin{eqnarray}\displaystyle E_{s,t}^{1}(Y^{\bullet },\mathbb{Z}/n\mathbb{Z}) & = & \displaystyle H_{t}(C^{1,0}(Y^{[s+1]},\mathbb{Z}/n\mathbb{Z}(d-s)))\nonumber\\ \displaystyle & \Rightarrow & \displaystyle H_{s+t}(C^{1,0}(Y,\mathbb{Z}/n\mathbb{Z}(d)))\nonumber\end{eqnarray}$$
$d_{s,t}^{1}=\sum _{\unicode[STIX]{x1D708}=1}^{s+1}(-1)^{\unicode[STIX]{x1D708}+1}(\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})_{\ast }$
.(ii) We have
$E_{s,t}^{1}(Y,\mathbb{Z}/n\mathbb{Z})=0$
if
$t<0$
, and hence there are canonical edge morphisms $$\begin{eqnarray}e_{a}^{{\mathcal{Y}},p^{m}}:H_{a}(C^{1,0}(Y,\mathbb{Z}/n\mathbb{Z}(d)))\rightarrow E_{a,0}^{2}(Y^{\bullet },\mathbb{Z}/n\mathbb{Z}).\end{eqnarray}$$
(iii) The trace map induces a canonical isomorphism
$$\begin{eqnarray}\text{tr}:E_{a,0}^{2}(Y^{\bullet },\mathbb{Z}/n\mathbb{Z})\rightarrow H_{a}(C(Y^{\bullet },\mathbb{Z}/n\mathbb{Z})).\end{eqnarray}$$
(iv) The composite of edge and trace morphisms gives us a canonical map
which is an isomorphism if$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{a}^{Y,p^{m}}:H_{a}(C^{1,0}(Y,\mathbb{Z}/n\mathbb{Z}(d)))\rightarrow H_{a}(C(Y^{\bullet },\mathbb{Z}/n\mathbb{Z})),\end{eqnarray}$$
$0\leqslant a\leqslant 4$
.
Remark 5.2.2. In the following, we need the cases
$a=1$
and
$a=2$
, which will give us an explicit description of
$E_{2}$
-terms of certain coniveau spectral sequences.
5.3 The
$\ell$
-primary part
Let
$\ell$
be a prime number and
$\ell \neq p$
. The cup product induces the following morphism
$$\begin{eqnarray}R\jmath _{\ast }\unicode[STIX]{x1D707}_{\ell ^{m},U}^{\otimes r}\rightarrow R\jmath _{\ast }\mathscr{H}\text{om}_{U}(\unicode[STIX]{x1D707}_{\ell ^{m},U}^{\otimes d+1-r},\unicode[STIX]{x1D707}_{\ell ^{m},U}^{\otimes d+1}).\end{eqnarray}$$
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
$$\begin{eqnarray}R\jmath _{\ast }R\mathscr{H}\text{om}_{U}(\unicode[STIX]{x1D707}_{\ell ^{m},U}^{\otimes d+1-r},\unicode[STIX]{x1D707}_{\ell ^{m},U}^{\otimes d+1})=R\mathscr{H}\text{om}_{X}(\jmath _{!}\unicode[STIX]{x1D707}_{\ell ^{m},U}^{\otimes d+1-r},\unicode[STIX]{x1D707}_{\ell ^{m},X}^{\otimes d+1}).\end{eqnarray}$$
Using the adjoint pair
$(i_{\ast },Ri^{!})$
and these two maps above, we obtain a pairing on
$X_{\acute{\text{e}}\text{t}}$
:
$$\begin{eqnarray}i^{\ast }R\jmath _{\ast }\unicode[STIX]{x1D707}_{\ell ^{m},U}^{\otimes r}\otimes ^{L}Ri^{!}\jmath _{!}\unicode[STIX]{x1D707}_{\ell ^{m},U}^{\otimes d+1-r}\rightarrow Ri^{!}\unicode[STIX]{x1D707}_{\ell ^{m},X}^{\otimes d+1}.\end{eqnarray}$$
Therefore a pairing of cohomology groups:
$$\begin{eqnarray}H^{i}(U_{\acute{\text{e}}\text{t}},\unicode[STIX]{x1D707}_{\ell ^{m},U}^{\otimes r})\times H_{X_{s}}^{j}(X_{\acute{\text{e}}\text{t}},\jmath _{!}\unicode[STIX]{x1D707}_{\ell ^{m},U}^{\otimes d+1-r})\rightarrow H_{X_{s}}^{i+j}(X_{\acute{\text{e}}\text{t}},\unicode[STIX]{x1D707}_{\ell ^{m},X}^{\otimes d+1}).\end{eqnarray}$$
We have the following duality theorem; see [Reference GeisserGei10, Theorem 7.5].
Theorem 5.3.1.
(i) There is a canonical isomorphism, so-called the trace map,
$$\begin{eqnarray}\text{Tr}:H_{X_{s}}^{2d+3}(X_{\acute{\text{e}}\text{t}},\unicode[STIX]{x1D707}_{\ell ^{m},X}^{\otimes d+1})\xrightarrow[{}]{\cong }\mathbb{Z}/\ell ^{m}\mathbb{Z}\end{eqnarray}$$
(ii) The trace map Tr and the pair (5.3.2) induce a perfect pairing of finite groups
$$\begin{eqnarray}\displaystyle & & \displaystyle H^{i}(U_{\acute{\text{e}}\text{t}},\unicode[STIX]{x1D707}_{\ell ^{m},U}^{\otimes r})\times H_{X_{s}}^{2d+3-i}(X_{\acute{\text{e}}\text{t}},\jmath _{!}\unicode[STIX]{x1D707}_{\ell ^{m},U}^{\otimes d+1-r})\nonumber\\ \displaystyle & & \displaystyle \quad \rightarrow \,H_{X_{s}}^{2d+3}(X_{\acute{\text{e}}\text{t}},\unicode[STIX]{x1D707}_{\ell ^{m},X}^{\otimes d+1})\xrightarrow[{}]{\text{Tr}}\mathbb{Z}/\ell ^{m}\mathbb{Z}.\nonumber\end{eqnarray}$$
For
$r=0,i=1$
, we obtain
$$\begin{eqnarray}H^{1}(U_{\acute{\text{e}}\text{t}},\mathbb{Z}/\ell ^{m}\mathbb{Z})\cong \text{Hom}(H_{X_{s}}^{2d+2}(X_{\acute{\text{e}}\text{t}},\jmath _{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d+1}),\mathbb{Z}/\ell ^{m}\mathbb{Z}),\end{eqnarray}$$
and by Pontryagin duality
$$\begin{eqnarray}H_{X_{s}}^{2d+2}(X_{\acute{\text{e}}\text{t}},\jmath _{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d+1})\cong \unicode[STIX]{x1D70B}_{1}^{\text{ab}}(U)/\ell ^{m}.\end{eqnarray}$$
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:
$$\begin{eqnarray}\displaystyle & \displaystyle E_{1,\acute{\text{e}}\text{t}}^{p,q}({\mathcal{F}}):=\bigoplus _{x\in X^{p}\cap X_{s}}H_{x}^{p+q}(X_{\acute{\text{e}}\text{t}},{\mathcal{F}})\Longrightarrow H_{X_{s}}^{p+q}(X_{\acute{\text{e}}\text{t}},{\mathcal{F}}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle E_{1,\text{Nis}}^{p,q}({\mathcal{F}}):=\bigoplus _{x\in X^{p}\cap X_{s}}H_{x}^{p+q}(X_{\text{Nis}},{\mathcal{F}})\Longrightarrow H_{X_{s}}^{p+q}(X_{\text{Nis}},{\mathcal{F}}). & \displaystyle \nonumber\end{eqnarray}$$
Proposition 5.3.2.
(i)
$E_{1,\acute{\text{e}}\text{t}}^{\bullet ,d+2}(\jmath _{!}\unicode[STIX]{x1D707}_{\ell ^{m},U}^{\otimes d+1})\cong E_{1,\acute{\text{e}}\text{t}}^{\bullet ,d+2}(\unicode[STIX]{x1D707}_{\ell ^{m},X}^{\otimes d+1})$
.(ii) The local Chern class map induces a surjection
$E_{1,\text{Nis}}^{\bullet ,0}({\mathcal{K}}_{d+1,X|D}^{M})/\ell ^{m}{\twoheadrightarrow}E_{1,\acute{\text{e}}\text{t}}^{\bullet ,d+1}(\jmath _{!}\unicode[STIX]{x1D707}_{\ell ^{m},U}^{\otimes d+1})$
and an isomorphism $$\begin{eqnarray}E_{1,\text{Nis}}^{d+1,0}({\mathcal{K}}_{d+1,X|D}^{M})/\ell ^{m}\cong E_{1,\acute{\text{e}}\text{t}}^{d+1,d+1}(\jmath _{!}\unicode[STIX]{x1D707}_{\ell ^{m},U}^{\otimes d+1}).\end{eqnarray}$$
Proof. The argument is analogous to that in Section 3.2. More precisely, part (i) corresponds to Proposition 3.2.1 and part (ii) corresponds to Theorem 3.2.3. ◻
Corollary 5.3.3. There are canonical isomorphisms
$$\begin{eqnarray}H_{X_{s}}^{d+1}(X_{\text{Nis}},{\mathcal{K}}_{d+1,X|D}^{M})/\ell ^{m}\cong E_{2,\text{Nis}}^{d+1,0}({\mathcal{K}}_{d+1,X|D}^{M})/\ell ^{m}\cong E_{2,\acute{\text{e}}\text{t}}^{d+1,d+1}(\jmath _{!}\unicode[STIX]{x1D707}_{\ell ^{m},U}^{\otimes d+1}).\end{eqnarray}$$
Proof. The degenerating coniveau spectral sequence on
$X_{\text{Nis}}$
gives the first isomorphism. The second isomorphism results from the same argument as in Corollary 3.2.4 using Proposition 5.3.2(ii).◻
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
$$\begin{eqnarray}\displaystyle & & \displaystyle \bigoplus _{y\in X_{s}^{0}}H^{d+1}(y,\mathbb{Z}/\ell ^{m}\mathbb{Z}(d))\rightarrow \bigoplus _{y\in X_{s}^{1}}H^{d}(y,\mathbb{Z}/\ell ^{m}\mathbb{Z}(d-1))\rightarrow \cdots \nonumber\\ \displaystyle & & \displaystyle \quad \cdots \rightarrow \bigoplus _{y\in X_{s}^{a}}H^{d-a+1}(y,\mathbb{Z}/\ell ^{m}\mathbb{Z}(d-a))\rightarrow \cdots \rightarrow \bigoplus _{y\in X_{s}^{d}}H^{1}(y,\mathbb{Z}/\ell ^{m}\mathbb{Z}),\nonumber\end{eqnarray}$$
where we set the last term in degree
$0$
as an object in the derived category.
Theorem 5.3.4. The canonical morphism
$$\begin{eqnarray}H_{X_{s}}^{d+1}(X_{\text{Nis}},{\mathcal{K}}_{d+1,X|D}^{M})/\ell ^{m}\rightarrow H_{X_{s}}^{2d+2}(X_{\acute{\text{e}}\text{t}},\jmath _{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d+1})\end{eqnarray}$$
fits into an exact sequence
$$\begin{eqnarray}\displaystyle H_{2}(C(X_{s}^{\bullet },\mathbb{Z}/\ell ^{m}\mathbb{Z})) & \rightarrow & \displaystyle H_{X_{s}}^{d+1}(X_{\text{Nis}},{\mathcal{K}}_{d+1,X|D}^{M})/\ell ^{m}\rightarrow H_{X_{s}}^{2d+2}(X_{\acute{\text{e}}\text{t}},\jmath _{!}\unicode[STIX]{x1D707}_{\ell ^{m}}^{\otimes d+1})\nonumber\\ \displaystyle & \rightarrow & \displaystyle H_{1}(C(X_{s}^{\bullet },\mathbb{Z}/\ell ^{m}\mathbb{Z}))\rightarrow 0.\nonumber\end{eqnarray}$$
Proof. By the coniveau spectral sequence for
${\mathcal{F}}=\jmath _{!}\unicode[STIX]{x1D707}_{\ell ^{m},U}^{\otimes d+1}$
on
$X_{\acute{\text{e}}\text{t}}$
, we have an exact sequence:
$$\begin{eqnarray}E_{2,\acute{\text{e}}\text{t}}^{d-1,d+2}({\mathcal{F}})\rightarrow E_{2,\acute{\text{e}}\text{t}}^{d+1,d+1}({\mathcal{F}})\rightarrow H_{X_{s}}^{2d+2}(X_{\acute{\text{e}}\text{t}},{\mathcal{F}})\rightarrow E_{2,\acute{\text{e}}\text{t}}^{d,d+2}({\mathcal{F}})\rightarrow 0.\end{eqnarray}$$
Using Proposition 5.3.2, we have
$$\begin{eqnarray}\displaystyle E_{2,\acute{\text{e}}\text{t}}^{d+1,d+1}({\mathcal{F}}) & = & \displaystyle E_{2,\text{Nis}}^{d+1,0}({\mathcal{K}}_{d+1,X|D}^{M}/\ell ^{m})=H_{X_{s}}^{d+1}(X_{\text{Nis}},{\mathcal{K}}_{d+1,X|D}^{M}/\ell ^{m})\nonumber\\ \displaystyle & = & \displaystyle H_{X_{s}}^{d+1}(X_{\text{Nis}},{\mathcal{K}}_{d+1,X|D}^{M})/\ell ^{m}.\nonumber\end{eqnarray}$$
Moreover combining with Theorem 5.2.1, we obtain
$$\begin{eqnarray}\displaystyle & \displaystyle E_{2,\acute{\text{e}}\text{t}}^{d-1,d+2}({\mathcal{F}})=E_{2,\acute{\text{e}}\text{t}}^{d-1,d+2}(\unicode[STIX]{x1D707}_{\ell ^{m},X}^{\otimes d+1})=H_{2}(C(X_{s}^{\bullet },\mathbb{Z}/\ell ^{m}\mathbb{Z})); & \displaystyle \nonumber\\ \displaystyle & \displaystyle \hspace{51.50002pt}E_{2,\acute{\text{e}}\text{t}}^{d,d+2}({\mathcal{F}})=E_{2,\acute{\text{e}}\text{t}}^{d,d+2}(\unicode[STIX]{x1D707}_{\ell ^{m},X}^{\otimes d+1})=H_{1}(C(X_{s}^{\bullet },\mathbb{Z}/\ell ^{m}\mathbb{Z})).\hspace{28.50009pt}\square & \displaystyle \nonumber\end{eqnarray}$$
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.
Theorem 5.3.5. There is a canonical map
$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{X,D}:C(X,D;X_{s})/\ell ^{m}\rightarrow \unicode[STIX]{x1D70B}_{1}^{\text{ab}}(U)/\ell ^{m},\end{eqnarray}$$
which fits into an exact sequence of finite groups
$$\begin{eqnarray}\displaystyle H_{2}(C(X_{s}^{\bullet },\mathbb{Z}/\ell ^{m}\mathbb{Z})) & \rightarrow & \displaystyle C(X,D;X_{s})/\ell ^{m}\rightarrow \unicode[STIX]{x1D70B}_{1}^{\text{ab}}(U)/\ell ^{m}\nonumber\\ \displaystyle & \rightarrow & \displaystyle H_{1}(C(X_{s}^{\bullet },\mathbb{Z}/\ell ^{m}\mathbb{Z}))\rightarrow 0.\nonumber\end{eqnarray}$$
Equivalently, there is an exact sequence:
$$\begin{eqnarray}\displaystyle H_{2}(C(X_{s}^{\bullet },\mathbb{Z}/\ell ^{m}\mathbb{Z})) & \rightarrow & \displaystyle \widehat{SK}_{1}(U)/\ell ^{m}\rightarrow \unicode[STIX]{x1D70B}_{1}^{\text{ab}}(U)/\ell ^{m}\nonumber\\ \displaystyle & \rightarrow & \displaystyle H_{1}(C(X_{s}^{\bullet },\mathbb{Z}/\ell ^{m}\mathbb{Z}))\rightarrow 0.\end{eqnarray}$$
Proof. The map is defined by the following diagram
So the first exact sequence is a direct consequence of Theorem 5.3.4. The second exact sequence results from the fact that
$$\begin{eqnarray}\widehat{SK}_{1}(U)/\ell ^{m}=H_{X_{s}}^{d+1}(X_{\text{Nis}},{\mathcal{K}}_{d+1,X|D}^{M})/\ell ^{m}\end{eqnarray}$$
for any
$D$
with
$\text{Supp}(D)=X\setminus U$
. Indeed, we denote by
$D_{0}=X\setminus U$
the reduced divisor, it suffices to show the following claim.◻
Claim 5.3.6. We have
$$\begin{eqnarray}\Bigl(\underset{D}{\varprojlim }H_{X_{s}}^{d+1}(X_{\text{Nis}},{\mathcal{K}}_{d+1,X|D}^{M})\Bigr)\otimes _{\mathbb{ Z}}\mathbb{Z}/\ell ^{m}\mathbb{Z}\xrightarrow[{}]{\cong }H_{X_{s}}^{d+1}(X_{\text{Nis}},{\mathcal{K}}_{d+1,X|D_{0}}^{M})/\ell ^{m}.\end{eqnarray}$$
Proof of Claim.
The canonical surjective map
$$\begin{eqnarray}\unicode[STIX]{x1D711}_{D}:H_{X_{s}}^{d+1}(X_{\text{Nis}},{\mathcal{K}}_{d+1,X|D}^{M})\rightarrow H_{X_{s}}^{d+1}(X_{\text{Nis}},{\mathcal{K}}_{d+1,X|D_{0}}^{M})\end{eqnarray}$$
fits into the exact sequence
Applying
$\mathop{\varprojlim }\nolimits_{D}$
to the above exact sequence, we obtain an exact sequence
By the long exact sequence associated to the short exact sequence
$$\begin{eqnarray}0\rightarrow {\mathcal{K}}_{d+1,X|D}^{M}\rightarrow {\mathcal{K}}_{d+1,X|D_{0}}^{M}\rightarrow {\mathcal{K}}_{d+1,X|D_{0}}^{M}/{\mathcal{K}}_{d+1,X|D}^{M}\rightarrow 0,\end{eqnarray}$$
we see that
$H_{X_{s}}^{d}(X_{\text{Nis}},{\mathcal{K}}_{d+1,X|D_{0}}^{M}/{\mathcal{K}}_{d+1,X|D}^{M}){\twoheadrightarrow}\ker (\unicode[STIX]{x1D711}_{D})$
is surjective. Proposition 2.2.5(ii) tells us that
$H_{X_{s}}^{d}(X_{\text{Nis}},{\mathcal{K}}_{d+1,X|D}^{M}/{\mathcal{K}}_{d+1,X|D_{0}}^{M})$
is
$p$
-primary torsion group, therefore in particular
$\ker (\unicode[STIX]{x1D711}_{D})$
is a
$\mathbb{Z}_{(p)}$
-module, so is the inverse limit
$\mathop{\varprojlim }\nolimits_{D}\ker (\unicode[STIX]{x1D711}_{D})$
. It follows that
$$\begin{eqnarray}\mathbb{Z}/\ell ^{m}\mathbb{Z}\otimes _{\mathbb{ Z}}\underset{D}{\varprojlim }\ker (\unicode[STIX]{x1D711}_{D})=0.\end{eqnarray}$$
Tensoring the exact sequence (5.3.7) with
$\mathbb{Z}/\ell ^{m}\mathbb{Z}$
, we obtain the claim.◻
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.
Theorem 5.3.7. If
$\text{Supp}(D)=X_{s}$
is smooth, then the map
$\unicode[STIX]{x1D719}:SK_{1}(X_{\unicode[STIX]{x1D702}})/\ell ^{m}\rightarrow \widehat{SK}_{1}(X_{\unicode[STIX]{x1D702}})/\ell ^{m}$
is an isomorphism.
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
$$\begin{eqnarray}\unicode[STIX]{x1D6EC}(i)_{X}\rightarrow j_{\ast }\unicode[STIX]{x1D6EC}(i)_{X_{\unicode[STIX]{x1D702}}}\xrightarrow[{}]{\cdot \unicode[STIX]{x1D70B}}j_{\ast }\unicode[STIX]{x1D6EC}(i+1)_{X_{\unicode[STIX]{x1D702}}}[1]\rightarrow i_{\ast }\unicode[STIX]{x1D6EC}(i)_{X_{s}},\end{eqnarray}$$
where the middle arrow is given by multiplication by
$\unicode[STIX]{x1D70B}$
, and the last arrow is the localization map.
Let
$$\begin{eqnarray}\unicode[STIX]{x1D6EC}(i)_{X|X_{s}}:=\text{hofib}(r_{i}:\unicode[STIX]{x1D6EC}(i)_{X}\rightarrow i_{\ast }\unicode[STIX]{x1D6EC}(i)_{X_{s}})\end{eqnarray}$$
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:
Proposition 5.3.8. There is a canonical isomorphism
$$\begin{eqnarray}{\mathcal{K}}_{i,X|X_{s}}^{M}/\ell ^{m}\cong {\mathcal{H}}^{i}(\unicode[STIX]{x1D6EC}(i)_{X|X_{s}})\end{eqnarray}$$
and
${\mathcal{H}}^{j}(\unicode[STIX]{x1D6EC}(i)_{X|X_{s}})=0$
for
$j>i$
.
Note that Proposition 5.3.8 implies that the canonical map
$$\begin{eqnarray}H_{X_{s}}^{2d+2}(X_{\text{Nis}},\unicode[STIX]{x1D6EC}(d+1)_{X|X_{s}})\xrightarrow[{}]{\cong }H_{X_{s}}^{d+1}(X_{\text{Nis}},{\mathcal{K}}_{d+1,X|X_{s}}^{M})/\ell ^{m}\end{eqnarray}$$
is an isomorphism.
To finish the proof of Theorem 5.3.7, we also need the following result:
Proposition 5.3.9. The group
$H^{2d+1}(X_{\text{Nis}},\unicode[STIX]{x1D6EC}(d+1)_{X|X_{s}})=0$
.
Proof. By the definition of
$\unicode[STIX]{x1D6EC}(d+1)_{X|X_{s}}$
, there is a long exact sequence
$$\begin{eqnarray}\displaystyle & & \displaystyle H^{2d}(X_{\text{Nis}},\unicode[STIX]{x1D6EC}(d+1)_{X})\xrightarrow[{}]{\unicode[STIX]{x1D6FC}}H^{2d}(X_{s,\text{Nis}},\unicode[STIX]{x1D6EC}(d+1)_{X_{s}})\nonumber\\ \displaystyle & & \displaystyle \quad \rightarrow \,H^{2d+1}(X_{\text{Nis}},\unicode[STIX]{x1D6EC}(d+1)_{X|X_{s}})\nonumber\\ \displaystyle & & \displaystyle \quad \rightarrow \,H^{2d+1}(X_{\text{Nis}},\unicode[STIX]{x1D6EC}(d+1)_{X})\xrightarrow[{}]{\unicode[STIX]{x1D6FD}}H^{2d+1}(X_{s,\text{Nis}},\unicode[STIX]{x1D6EC}(d+1)_{X_{s}}).\nonumber\end{eqnarray}$$
It suffices to show that
$\unicode[STIX]{x1D6FC}$
is surjective and
$\unicode[STIX]{x1D6FD}$
is injective. In fact, using the relation between motivic cohomology and higher Chow groups, we will show that both
$\unicode[STIX]{x1D6FC}$
and
$\unicode[STIX]{x1D6FD}$
are isomorphisms. More precisely, the fact that
$\unicode[STIX]{x1D6FC}$
is an isomorphism follows from the diagram:
where the equalities in the rows are the definitions of higher Chow groups with coefficients in
$\mathbb{Z}/\ell ^{m}\mathbb{Z}$
(cf. [Reference Geisser and LevineGL01]), the two horizontal arrows are isomorphisms by the known Kato conjecture [Reference Kerz and SaitoKS12, Theorem 9.3], and the right vertical is the proper base change theorem (SGA4
$\frac{1}{2}$
, [Reference DeligneDel77, Arcata IV]). The assertion for
$\unicode[STIX]{x1D6FD}$
is similar:
Proof of Theorem 5.3.7.
The assertion follows directly from the diagram:
where the first row is the exact localization sequence, note that
$j^{\ast }\unicode[STIX]{x1D6EC}(d+1)_{X|X_{s}}=\unicode[STIX]{x1D6EC}(d+1)_{X_{\unicode[STIX]{x1D702}}}$
. The first vertical isomorphism is given by (5.3.8) and the second vertical isomorphism is given by Proposition 5.3.8 and (5.3.5).◻
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.
Theorem 5.4.1. [Reference ZhaoZha16, Theorem 3.4.2]
Let
$R=\mathbb{F}_{q}[[t]]$
. There is a perfect pairing of topological abelian groups
$$\begin{eqnarray}\displaystyle & & \displaystyle H^{i}(U_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{U,\log }^{r})\times \underset{E}{\varprojlim }H_{X_{s}}^{d+2-i}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|E,\log }^{d+1-r})\nonumber\\ \displaystyle & & \displaystyle \quad \rightarrow \,H_{X_{s}}^{d+2}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d+1})\xrightarrow[{}]{\text{Tr}}\mathbb{Z}/p^{m}\mathbb{Z},\nonumber\end{eqnarray}$$
where the inverse limit runs over the set of effective divisors
$D$
such that
$\text{Supp}(D)\subset X-U$
. The first group is endowed with the discrete topology, and the second is with profinite topology.
For
$r=0,i=1$
, we get
$$\begin{eqnarray}H^{1}(U_{\acute{\text{e}}\text{t}},\mathbb{Z}/p^{m}\mathbb{Z})\cong \underset{E}{\varinjlim }\text{Hom}(H_{X_{s}}^{d+1}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|E,\log }^{d+1}),\mathbb{Z}/p^{m}\mathbb{Z}).\end{eqnarray}$$
Similar to Corollary 3.3.5, the transition maps are surjective in the projective limit, for our divisor
$D$
we define
$$\begin{eqnarray}\text{Fil}_{D}H^{1}(U_{\acute{\text{e}}\text{t}},\mathbb{Z}/p^{m}\mathbb{Z}):=\text{Hom}(H_{X_{s}}^{d+1}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d+1}),\mathbb{Z}/p^{m}\mathbb{Z});\end{eqnarray}$$
by Pontryagin duality, we also define
$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{1}^{\text{ab}}(X,D)/p^{m}:=\text{Hom}(\text{Fil}_{D}H^{1}(U_{\acute{\text{e}}\text{t}},\mathbb{Z}/p^{m}\mathbb{Z}),\mathbb{Z}/p^{m}\mathbb{Z}).\end{eqnarray}$$
Therefore Theorem 5.4.1 gives us an isomorphism
$$\begin{eqnarray}H_{X_{s}}^{d+1}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d+1})\xrightarrow[{}]{\cong }\unicode[STIX]{x1D70B}_{1}^{\text{ab}}(X,D)/p^{m}.\end{eqnarray}$$
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:
$$\begin{eqnarray}\displaystyle & \displaystyle E_{1,\acute{\text{e}}\text{t}}^{p,q}({\mathcal{F}}):=\bigoplus _{x\in X^{p}\cap X_{s}}H_{x}^{p+q}(X_{\acute{\text{e}}\text{t}},{\mathcal{F}})\Longrightarrow H_{X_{s}}^{p+q}(X_{\acute{\text{e}}\text{t}},{\mathcal{F}}) & \displaystyle \nonumber\\ \displaystyle & \displaystyle E_{1,\text{Nis}}^{p,q}({\mathcal{F}}):=\bigoplus _{x\in X^{p}\cap X_{s}}H_{x}^{p+q}(X_{\text{Nis}},{\mathcal{F}})\Longrightarrow H_{X_{s}}^{p+q}(X_{\text{Nis}},{\mathcal{F}}). & \displaystyle \nonumber\end{eqnarray}$$
Proposition 5.4.2. We have the following isomorphisms:
(i)
$E_{1,\acute{\text{e}}\text{t}}^{\bullet ,1}(W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d+1})\cong E_{1,\acute{\text{e}}\text{t}}^{\bullet ,1}(W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d+1})$
;(ii)
$E_{1,\text{Nis}}^{\bullet ,0}(W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d+1})\cong E_{1,\acute{\text{e}}\text{t}}^{\bullet ,0}(W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d+1})$
.
Proof. This is a local question. The first claim follows by the same argument as in Proposition 3.3.2, and the second as in Proposition 3.3.3. ◻
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
$$\begin{eqnarray}\displaystyle & & \displaystyle \bigoplus _{y\in X_{s}^{0}}H_{y}^{d+1}(X_{s,\acute{\text{e}}\text{t}},\mathbb{Z}/p^{m}\mathbb{Z}(d))\rightarrow \bigoplus _{y\in X_{s}^{1}}H_{y}^{d+2}(X_{s,\acute{\text{e}}\text{t}},\mathbb{Z}/p^{m}\mathbb{Z}(d))\rightarrow \cdots \nonumber\\ \displaystyle & & \displaystyle \quad \cdots \rightarrow \bigoplus _{y\in X_{s}^{a}}H_{y}^{d+a+1}(X_{s,\acute{\text{e}}\text{t}},\mathbb{Z}/p^{m}\mathbb{Z}(d))\rightarrow \cdots \nonumber\\ \displaystyle & & \displaystyle \quad \rightarrow \,\bigoplus _{y\in X_{s}^{d}}H_{y}^{2d+1}(X_{s,\acute{\text{e}}\text{t}},\mathbb{Z}/p^{m}\mathbb{Z}(d)),\nonumber\end{eqnarray}$$
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
$0$
.
Theorem 5.4.3. The canonical map
$$\begin{eqnarray}H_{X_{s}}^{d+1}(X_{\text{Nis}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d+1})\rightarrow H_{X_{s}}^{d+1}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d+1})\end{eqnarray}$$
fits into an exact sequence of finite groups
$$\begin{eqnarray}\displaystyle H_{2}(C(X_{s}^{\bullet },\mathbb{Z}/p^{m}\mathbb{Z})) & \rightarrow & \displaystyle H_{X_{s}}^{d+1}(X_{\text{Nis}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d+1})\nonumber\\ \displaystyle & \rightarrow & \displaystyle H_{X_{s}}^{d+1}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d+1})\rightarrow H_{1}(C(X_{s}^{\bullet },\mathbb{Z}/p^{m}\mathbb{Z}))\rightarrow 0.\nonumber\end{eqnarray}$$
Proof. By the coniveau spectral sequence for
${\mathcal{F}}=W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d+1}$
on
$X_{\acute{\text{e}}\text{t}}$
, we have the following exact sequence
$$\begin{eqnarray}E_{2,\acute{\text{e}}\text{t}}^{d-1,1}({\mathcal{F}})\rightarrow E_{2,\acute{\text{e}}\text{t}}^{d+1,0}({\mathcal{F}})\rightarrow H_{X_{s}}^{d+1}(X_{\acute{\text{e}}\text{t}},{\mathcal{F}})\rightarrow E_{2,\acute{\text{e}}\text{t}}^{d,1}({\mathcal{F}})\rightarrow 0.\end{eqnarray}$$
By Proposition 5.4.2, we have
$$\begin{eqnarray}E_{2,\acute{\text{e}}\text{t}}^{d+1,0}({\mathcal{F}})=E_{2,\text{Nis}}^{d+1,0}({\mathcal{F}})=H_{X_{s}}^{d+1}(X_{\text{Nis}},{\mathcal{F}}).\end{eqnarray}$$
Moreover combining with Theorem 5.2.1, we obtain
$$\begin{eqnarray}\displaystyle & \displaystyle E_{2,\acute{\text{e}}\text{t}}^{d-1,1}(W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d+1})=E_{2,\acute{\text{e}}\text{t}}^{d-1,1}(W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d+1})=H_{2}(C(X_{s}^{\bullet },\mathbb{Z}/p^{m}\mathbb{Z})); & \displaystyle \nonumber\\ \displaystyle & \displaystyle \hspace{28.00006pt}E_{2,\acute{\text{e}}\text{t}}^{d,1}(W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d+1})=E_{2,\acute{\text{e}}\text{t}}^{d,1}(W_{m}\unicode[STIX]{x1D6FA}_{X,\log }^{d+1})=H_{1}(C(X_{s}^{\bullet },\mathbb{Z}/p^{m}\mathbb{Z})).\hspace{5.59998pt}\square & \displaystyle \nonumber\end{eqnarray}$$
Remark 5.4.4. In particular, if
$X$
has good reduction, then
$$\begin{eqnarray}H_{X_{s}}^{d+1}(X_{\text{Nis}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d+1})\cong H_{X_{s}}^{d+1}(X_{\acute{\text{e}}\text{t}},W_{m}\unicode[STIX]{x1D6FA}_{X|D,\log }^{d+1}).\end{eqnarray}$$
The
$p$
-primary part of class field theory in this setting can be reformulated as follows:
Theorem 5.4.5. There is a canonical map
$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{X,D}:C(X,D;X_{s})/p^{m}\rightarrow \unicode[STIX]{x1D70B}_{1}^{\text{ab}}(X,D)/p^{m},\end{eqnarray}$$
which fits into an exact sequence of finite groups
$$\begin{eqnarray}\displaystyle H_{2}(C(X_{s}^{\bullet },\mathbb{Z}/p^{m}\mathbb{Z})) & \rightarrow & \displaystyle C(X,D;X_{s})/p^{m}\rightarrow \unicode[STIX]{x1D70B}_{1}^{\text{ab}}(X,D)/p^{m}\nonumber\\ \displaystyle & \rightarrow & \displaystyle H_{1}(C(X_{s}^{\bullet },\mathbb{Z}/p^{m}\mathbb{Z}))\rightarrow 0.\nonumber\end{eqnarray}$$
In particular, we have
$$\begin{eqnarray}\displaystyle H_{2}(C(X_{s}^{\bullet },\mathbb{Z}/p^{m}\mathbb{Z})) & \rightarrow & \displaystyle \underset{D}{\varprojlim }(C(X,D;X_{s})/p^{m})\rightarrow \unicode[STIX]{x1D70B}_{1}^{\text{ab}}(U)/p^{m}\nonumber\\ \displaystyle & \rightarrow & \displaystyle H_{1}(C(X_{s}^{\bullet },\mathbb{Z}/p^{m}\mathbb{Z}))\rightarrow 0.\nonumber\end{eqnarray}$$
