# Reciprocity sheaves and logarithmic motives

Shuji Saito  Graduate School of Mathematical Sciences, University of Tokyo,
3-8-1 Komaba, Tokyo 153-8941, Japan<EMAIL_ADDRESS>
###### Abstract.

We connect two developments aiming at extending Voevodsky’s theory of motives
over a field in such a way to encompass non-$\mathbf{A}^{1}$-invariant
phenomina. One is theory of _reciprocity sheaves_ introduced by Kahn-Saito-
Yamazaki. Another is theory of the triangulated category
$\operatorname{\mathbf{logDM}}^{{\operatorname{eff}}}$ of _logarithmic
motives_ launched by Binda, Park and Østvær. We prove that the Nisnevich
cohomology of reciprocity sheaves is representable in
$\operatorname{\mathbf{logDM}}^{{\operatorname{eff}}}$.

The author is supported by the JSPS KAKENHI Grant (20H01791).

###### Contents

  1. 1 Preliminaries
  2. 2 Purity with reduced modulus
  3. 3 Review on higher local symbols
  4. 4 Logarithmic cohomology of reciprocity sheaves
  5. 5 Invariance of logarithmic cohomology under blowups
  6. 6 Relation with logarithmic sheaves with transfers

## Introduction

We fix once and for all a perfect base field $k$. The main purpose of this
paper is to connect two developments aiming at extending Voevodsky’s theory of
motives over $k$ in such a way to encompass non-$\mathbf{A}^{1}$-invariant
phenomina. One is theory of _reciprocity sheaves_ introduced by Kahn-Saito-
Yamazaki ([6] and [7]) and developed in [16] and [2]. Voevodsky’s theory is
based on the category ${\operatorname{\mathbf{PST}}}$ of _presheaves with
transers_ , defined as the category of additive presheaves of abelian groups
on the category $\operatorname{\mathbf{Cor}}$ of finite correspondences:
$\operatorname{\mathbf{Cor}}$ has the same objects as the category
$\operatorname{\mathbf{Sm}}$ of separated smooth schemes of finite type over
$k$ and morphisms in $\operatorname{\mathbf{Cor}}$ are finite correspondences.
Let $\operatorname{\mathbf{NST}}\subset{\operatorname{\mathbf{PST}}}$ be the
full subcategory of Nisnevich sheaves, i.e. those objects
$F\in{\operatorname{\mathbf{PST}}}$ whose restrictions $F_{X}$ to the small
étale site $X_{{\operatorname{\acute{e}t}}}$ over $X$ are Nisnevich sheaves
for all $X\in\operatorname{\mathbf{Sm}}$. Voevodsky proved that
$\operatorname{\mathbf{NST}}$ is a Grothendieck abelian category and defined
the triangulated category $\operatorname{\mathbf{DM}}^{{\operatorname{eff}}}$
of effective motives as the localization of the derived category
$D(\operatorname{\mathbf{NST}})$ of complexes in $\operatorname{\mathbf{NST}}$
with respect to an _$\mathbf{A}^{1}$ -weak equivalence_ (see [10, Def. 14.1]).
It is equipped with a functor
$M:\operatorname{\mathbf{Sm}}\to\operatorname{\mathbf{DM}}^{{\operatorname{eff}}}$
associating the motive $M(X)$ of $X\in\operatorname{\mathbf{Sm}}$.

Let
$\operatorname{\mathbf{HI}}_{\operatorname{Nis}}\subset\operatorname{\mathbf{NST}}$
be the full subcategory consisting of $\mathbf{A}^{1}$-invariant objects,
namely such $F\in\operatorname{\mathbf{NST}}$ that the projection
$\pi_{X}:X\times\mathbf{A}^{1}\to X$ induces an isomorphism
$\pi_{X}^{*}:F(X)\simeq F(X\times\mathbf{A}^{1})$ for any
$X\in\operatorname{\mathbf{Sm}}$. We say that
$F\in\operatorname{\mathbf{HI}}_{\operatorname{Nis}}$ is strictly
$\mathbf{A}^{1}$-invariant if $\pi_{X}$ induces isomorphisms

$\pi_{X}^{*}:H^{i}_{\operatorname{Nis}}(X,F_{X})\simeq
H^{i}_{\operatorname{Nis}}(X\times\mathbf{A}^{1},F_{X\times\mathbf{A}^{1}})\;\text{
for all }i\geq 0.$

The following theorem plays a fundamental role in Voevodsky’s theory.

######  Theorem 0.1.

(Voevodsky [17]) Any $F\in\operatorname{\mathbf{HI}}_{\operatorname{Nis}}$ is
strictly $\mathbf{A}^{1}$-invariant and we have a natural isomorphism

(0.1.1)
$H^{i}_{\operatorname{Nis}}(X,F_{X})\simeq\operatorname{Hom}_{\operatorname{\mathbf{DM}}^{{\operatorname{eff}}}}(M(X),L^{\mathbf{A}^{1}}F[i])\;\text{
for }X\in\operatorname{\mathbf{Sm}},$

where
$L^{\mathbf{A}^{1}}:D(\operatorname{\mathbf{NST}})\to\operatorname{\mathbf{DM}}^{{\operatorname{eff}}}$
is the localization functor.

Notice that there are interesting and important objects of
$\operatorname{\mathbf{NST}}$ which do not belong to
$\operatorname{\mathbf{HI}}_{\operatorname{Nis}}$. Such examples are given by
the sheaves $\Omega^{i}$ of (absolute or relative) differential forms, and the
$p$-typical de Rham-Witt sheaves $W_{m}\Omega^{i}$ of Bloch-Deligne-Illusie,
and smooth commutative $k$-groups schemes with a unipotent part (seen as
objects of $\operatorname{\mathbf{NST}}$), and the complexes
$R\varepsilon_{*}\mathbb{Z}/p^{r}(n)$ in case ${\operatorname{ch}}(k)=p>0$,
where $\mathbb{Z}/p^{r}(n)$ is the étale motivic complex of weight $n$ with
$\mathbb{Z}/p^{r}$ coefficients and $\varepsilon$ is the change of site
functor from the étale to the Nisnevich topology. For such examples, (0.1.1)
fails to hold since $\pi_{X}:X\times\mathbf{A}^{1}\to X$ induces an
isomorphism $M(X\times\mathbf{A}^{1})\simeq M(X)$ in
$\operatorname{\mathbf{DM}}^{{\operatorname{eff}}}$ but the maps induced on
cohomology of those sheaves are not isomorphism.

The category ${\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}$ of
reciprocity sheaves is a full abelian subcategory of
$\operatorname{\mathbf{NST}}$ that contains
$\operatorname{\mathbf{HI}}_{{\operatorname{Nis}}}$ as well as the
non-$\mathbf{A}^{1}$-invariant objects mentioned above. Heuristically, its
objects satisfy the property that for any $X\in\operatorname{\mathbf{Sm}}$,
each section $a\in F(X)$ “has bounded ramification at infinity” and the
objects of $\operatorname{\mathbf{HI}}_{\operatorname{Nis}}$ are special
reciprocity sheaves with the property that every section $a\in F(X)$ has
“tame” ramification at infinity111This heuristic viewpoint is manifested in
[13, Th. 2].. Slightly more exotic examples of reciprocity sheaves are given
by the sheaves ${\rm Conn}^{1}$ (in case ${\operatorname{ch}}(k)=0$), whose
sections over $X$ are rank $1$-connections, or ${\rm Lisse}^{1}_{\ell}$ (in
case ${\operatorname{ch}}(k)=p>0$), whose sections on $X$ are the lisse
$\overline{\mathbb{Q}}_{\ell}$-sheaves of rank $1$. Since
${\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}$ is an abelian category
equipped with a lax symmetric monoidal structure by [11], many more
interesting examples can be manufactured by taking kernels, quotients and
tensor products (see [2, §11.1] for more examples).

The main purpose of this article is to establish the formula (0.1.1) for all
$F\in{\operatorname{\mathbf{RSC}}}_{\operatorname{Nis}}$ in a new category
which enlarges $\operatorname{\mathbf{DM}}^{{\operatorname{eff}}}$ (see
(0.2.2)). It is the triangulated category
$\operatorname{\mathbf{logDM}}^{{\operatorname{eff}}}$ of _logarithmic
motives_ introduced by Binda, Park and Østvær in [1]. Let
${\operatorname{l\mathbf{Sm}}}$ be the category of log smooth and separated
$\mathrm{fs}$ log schemes of finite type over $k$ and
${\operatorname{l\mathbf{Cor}}}$ be the category with the same objects as
${\operatorname{l\mathbf{Sm}}}$ and whose morphisms are log finite
correspondences (see [1, Def. 2.1.1]). Let
${\operatorname{\mathbf{PSh}}^{ltr}}$ be the category of additive presheaves
of abelian groups on ${\operatorname{l\mathbf{Cor}}}$ and
${\operatorname{\mathbf{Shv}}^{ltr}_{dNis}}\subset{\operatorname{\mathbf{PSh}}^{ltr}}$
be the full subcategory consisting of those $\mathcal{F}$ whose restrictions
to ${\operatorname{l\mathbf{Sm}}}$ are dividing Nisnevich sheaves (see [1,
Def. 3.1.4]). It is shown in [1, §4 and Pr. 4.6.6] that
${\operatorname{\mathbf{Shv}}^{ltr}_{dNis}}$ is a Grothendieck abelian
category, and $\operatorname{\mathbf{logDM}}^{{\operatorname{eff}}}$ is
defined as the localization of the derived category
$D({\operatorname{\mathbf{Shv}}^{ltr}_{dNis}})$ of complexes in
${\operatorname{\mathbf{Shv}}^{ltr}_{dNis}}$ with respect to a
_${\overline{\square}}$ -weak equivalence_, where ${\overline{\square}}$ is
$\mathbf{P}^{1}$ with the log-structure associated to the effective divisor
$\infty\hookrightarrow\mathbf{P}^{1}$ (see [1, Def. 5.2.1]222 In fact it is
defined in loc.cite. as the localization of the homotopy category of complexes
in ${\operatorname{\mathbf{Shv}}^{ltr}_{dNis}}$ with respect to a
${\overline{\square}}$-local descent model structure.). It is equipped with a
functor
$M:{\operatorname{l\mathbf{Sm}}}\to\operatorname{\mathbf{logDM}}^{{\operatorname{eff}}}$
associating the logarithmic motive $M(\mathfrak{X})$ of
$\mathfrak{X}\in{\operatorname{l\mathbf{Sm}}}$.

Now we can state the main result of this paper.

######  Theorem 0.2.

(Theorems 6.1.1 and 6.3) There exists an exact and fully faithful functor

(0.2.1)
$\mathcal{L}og:{\operatorname{\mathbf{RSC}}}_{\operatorname{Nis}}\to{\operatorname{\mathbf{Shv}}^{ltr}_{dNis}}\;:\;F\to
F^{\log}=\mathcal{L}og(F)$

such that $F^{\log}$ for
$F\in{\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}$ is strictly
${\overline{\square}}$-invariant in the sense [1, Def. 5.2.2]333It is an
logarithmic analogue of Voevodsky’s strict $\mathbf{A}^{1}$-invariance.. For
$X\in\operatorname{\mathbf{Sm}}$ we have a natural isomorphism

(0.2.2)
$H^{i}_{\operatorname{Nis}}(X,F_{X})\simeq\operatorname{Hom}_{\operatorname{\mathbf{logDM}}^{{\operatorname{eff}}}}(M(X,\mathrm{triv}),L^{{\overline{\square}}}F^{\log}[i]),$

where
$L^{{\overline{\square}}}:D({\operatorname{\mathbf{Shv}}^{ltr}_{dNis}})\to\operatorname{\mathbf{logDM}}^{{\operatorname{eff}}}$
is the localization functor and $(X,\mathrm{triv})$ is the log-scheme with the
trivial log-structure.

We remark (see Remark 5.5) that for $F=\Omega^{i}$, $F^{\log}(\mathfrak{X})$
for $\mathfrak{X}\in{\operatorname{l\mathbf{Sm}}}$ whose underlying scheme is
smooth, agrees with the sheaf of logarithmic differential forms of
$\mathfrak{X}$ at least assuming ${\operatorname{ch}}(k)=0$ 444The assumption
is necessary to use [13, Cor. 6.8] proved in case ${\operatorname{ch}}(k)=0$.
We expect that it is removed by using a forthcoming work of K. Rülling
extending [13, Cor. 6.8] to the case ${\operatorname{ch}}(k)>0$..

We now explain the organization of the paper.

In §1 we discuss some preliminaries and fix the notation. We recall the
definitions and basic properties of _modulus (pre)sheaves with transfers_ from
[4], [5], [7] and [16]. It is a generalization of Voevodsky’s (pre)sheaves
with transfers to a version with modulus. The category
$\operatorname{\mathbf{\underline{M}Cor}}$ of _modulus correspondences_ is
introduced. Its objects are pairs $\mathcal{X}=(\overline{X},D)$, where
$\overline{X}$ is a separated scheme of finite type over $k$ equipped with an
effective Cartier divisor $D$ such that the _interior_ $\overline{X}-D=X$ is
smooth. The morphisms are finite correspondences on the interiors satisfying
some admissibility and a properness condition. Let
$\operatorname{\mathbf{\underline{M}PST}}$ be the category of additive
presheaves of abelian groups on $\operatorname{\mathbf{\underline{M}Cor}}$. A
full subcategory
$\operatorname{\mathbf{\underline{M}NST}}\subset\operatorname{\mathbf{\underline{M}PST}}$
of _Nisnevich sheaves_ is defined and there is a functor (see §1(20))

$\underline{\omega}^{\operatorname{\mathbf{CI}}}:{\operatorname{\mathbf{RSC}}}_{\operatorname{Nis}}\to\operatorname{\mathbf{\underline{M}NST}}.$

For every $F\in{\operatorname{\mathbf{RSC}}}_{\operatorname{Nis}}$ and
$X\in\operatorname{\mathbf{Sm}}$, it provides an exhaustive filtration on the
group $F(X)$ of sections over $X$ which measures depth of ramification along a
boundary of a partial compactification of $X$: For
$({\overline{X}},D)\in\operatorname{\mathbf{\underline{M}Cor}}$ with
${\overline{X}}-D=X$, we get the subgroups $\tilde{F}({\overline{X}},D)\subset
F(X)$ with $\tilde{F}=\underline{\omega}^{\operatorname{\mathbf{CI}}}F$ such
that $\tilde{F}({\overline{X}},D_{1})\subset\tilde{F}({\overline{X}},D_{2})$
if $D_{1}\leq D_{2}$.

In §2 we prove as a key technical input an analogue of Zariski-Nagata’s purity
theorem ([18, X 3.4]) for $\tilde{F}({\overline{X}},D)$ as above. It asserts
the exactness of the sequence

$0\to\tilde{F}({\overline{X}},D)\to
F(X)\to\underset{\xi\in{D}^{(0)}}{\bigoplus}\frac{F({\overline{X}}^{h}_{|\xi}-\xi)}{\tilde{F}({\overline{X}}^{h}_{|\xi},\xi)},$

in case ${\overline{X}}\in\operatorname{\mathbf{Sm}}$ and $D$ is reduced
simple normal crossing divisor, where ${D}^{(0)}$ is the set of the
irreducible components of $D$ and ${\overline{X}}^{h}_{|\xi}$ is the
henselization of ${\overline{X}}$ at $\xi$. In [14], this result is
generalized to the case where $D$ may not be reduced under the assumption that
${\overline{X}}$ admits a smooth compactification.

In §3 we review _higher local symbols_ for reciprocity sheaves constructed in
[14]. It is an effective tool with which one can decide when a given element
of $F(X)$ with $F\in{\operatorname{\mathbf{RSC}}}_{\operatorname{Nis}}$ and
$X\in\operatorname{\mathbf{Sm}}$ belongs to $\tilde{F}({\overline{X}},D)$ as
above. The construction of the pairing depends on pushforward maps for
cohomology of reciprocity sheaves constructed in [2] (which means that Theorem
0.2.2 depends on the result of [2]).

In §4, we prove the following result: Let
$\operatorname{\mathbf{\underline{M}Cor}}^{{\operatorname{fin}}}_{ls}$ be the
subcategory of $\operatorname{\mathbf{\underline{M}Cor}}$ whose objects are
pairs $(X,D)$ such that $X\in\operatorname{\mathbf{Sm}}$ and the reduced parts
$D_{{\operatorname{red}}}$ of $D$ is a SNCD on $X$ and whose morphisms are
modulus correspondences satisfying a finiteness conditions instead of the
properness condition (see §1(5)). Then, for
$F\in{\operatorname{\mathbf{RSC}}}_{\operatorname{Nis}}$, the association

$\tilde{F}^{\log}:(X,D)\to\underline{\omega}^{\operatorname{\mathbf{CI}}}F(X,D_{{\operatorname{red}}})$

gives a presheaf on
$\operatorname{\mathbf{\underline{M}Cor}}^{{\operatorname{fin}}}_{ls}$, which
gives rise to a cohomology theory $H^{i}_{\log}(-,\tilde{F}^{\log})$ on
$\operatorname{\mathbf{\underline{M}Cor}}^{{\operatorname{fin}}}_{ls}$, called
_the $i$-th logarithmic cohomology with coefficient $F$_ (see Definition 4.4).
The higher local symbols for $F$ plays a fundamental role in the proof of the
result .

In §5, we prove the invariance of logarithmic cohomology under blowups: Let
$\Lambda^{{\operatorname{fin}}}_{ls}$ be the subcategory of
$\operatorname{\mathbf{\underline{M}Cor}}^{{\operatorname{fin}}}_{ls}$ whose
objects are the same as
$\operatorname{\mathbf{\underline{M}Cor}}^{{\operatorname{fin}}}_{ls}$ and
whose morphisms are those $\rho:(Y,E)\to(X,D)$ where $E=\rho^{*}D$ and $\rho$
are induced by blowups of $X$ in smooth centers $Z\subset D$ which are normal
crossing to $D$ (see the beginning of the section). Then, for
$F\in{\operatorname{\mathbf{RSC}}}_{\operatorname{Nis}}$ and
$\rho:\mathcal{Y}\to\mathcal{X}$ in $\Lambda^{{\operatorname{fin}}}_{ls}$, we
have

$\rho^{*}:H_{\log}^{i}(\mathcal{X},F)\cong H_{\log}^{i}(\mathcal{Y},F)\;\text{
for }\forall i\geq 0.$

In §6, we prove Theorem 0.2.2, which is a formal consequence of the theorems
in §4 and §5.

Acknowledgements. The author would like to thank Kay Rülling and F. Binda for
many valuable discussions and comments. He is also grateful to A. Merici to
whom he owes crucial ideas for §2.

##  1\. Preliminaries

We fix once and for all a perfect base field $k$. In this section we recall
the definitions and basic properties of modulus sheaves with transfers from
[4] and [16] (see also [7] for a more detailed summary).

  1. (1)

Denote by $\operatorname{\mathbf{Sch}}$ the category of separated schemes of
finite type over $k$ and by $\operatorname{\mathbf{Sm}}$ the full subcategory
of smooth schemes. For $X,Y\in\operatorname{\mathbf{Sm}}$, an integral closed
subscheme of $X\times Y$ that is finite and surjective over a connected
component of $X$ is called a _prime correspondence from $X$ to $Y$_. The
category $\operatorname{\mathbf{Cor}}$ of finite correspondences has the same
objects as $\operatorname{\mathbf{Sm}}$, and for
$X,Y\in\operatorname{\mathbf{Sm}}$, $\operatorname{\mathbf{Cor}}(X,Y)$ is the
free abelian group on the set of all prime correspondences from $X$ to $Y$
(see [17]). We consider $\operatorname{\mathbf{Sm}}$ as a subcategory of
$\operatorname{\mathbf{Cor}}$ by regarding a morphism in
$\operatorname{\mathbf{Sm}}$ as its graph in $\operatorname{\mathbf{Cor}}$.

Let ${\operatorname{\mathbf{PST}}}$ be the category of additive presheaves of
abelian groups on $\operatorname{\mathbf{Cor}}$ whose objects are called
_presheaves with transfers_. Let
$\operatorname{\mathbf{NST}}\subseteq{\operatorname{\mathbf{PST}}}$ be the
category of Nisnevich sheaves with transfers and let

$a^{V}_{\operatorname{Nis}}:{\operatorname{\mathbf{PST}}}\to\operatorname{\mathbf{NST}}$

be Voevodsky’s Nisnevich sheafification functor, which is an exact left
adjoint to the inclusion
$\operatorname{\mathbf{NST}}\to{\operatorname{\mathbf{PST}}}$. Let
$\operatorname{\mathbf{HI}}\subseteq{\operatorname{\mathbf{PST}}}$ be the
category of $\mathbf{A}^{1}$-invariant presheaves and put
$\operatorname{\mathbf{HI}}_{\operatorname{Nis}}=\operatorname{\mathbf{HI}}\cap\operatorname{\mathbf{NST}}\subseteq\operatorname{\mathbf{NST}}$.

  2. (2)

Let $\operatorname{\mathbf{Sm}}^{\rm pro}$ be the category of $k$-schemes $X$
which are essentially smooth over $k$, i.e. $X$ is a limit
$\operatornamewithlimits{\varprojlim}_{i\in I}X_{i}$ over a filtered set $I$,
where $X_{i}$ is smooth over $k$ and all transition maps are étale. Note
$\operatorname{Spec}K\in\operatorname{\mathbf{Sm}}^{\rm pro}$ for a function
field $K$ over $k$ thanks to the assumption that $k$ is perfect. We define
$\operatorname{\mathbf{Cor}}^{\rm pro}$ whose objects are the same as
$\operatorname{\mathbf{Sm}}^{\rm pro}$ and morphisms are defined as [13, Def.
2,2]. We extend $F\in{\operatorname{\mathbf{PST}}}$ to a presheaf on
$\operatorname{\mathbf{Cor}}^{\rm pro}$ by
$F(X):=\operatornamewithlimits{\varinjlim}_{i\in I}F(X_{i})$ for $X$ as above.

  3. (3)

We recall the definition of the category
$\operatorname{\mathbf{\underline{M}Cor}}$ from [4, Definition 1.3.1]. A pair
$\mathcal{X}=(X,D)$ of $X\in\operatorname{\mathbf{Sch}}$ and an effective
Cartier divisor $D$ on $X$ is called a _modulus pair_ if
$M-|M_{\infty}|\in\operatorname{\mathbf{Sm}}$. Let $\mathcal{X}=(X,D_{X})$,
$\mathcal{Y}=(Y,D_{Y})$ be modulus pairs and
$\Gamma\in\operatorname{\mathbf{Cor}}(X-D_{X},Y-D_{Y})$ be a prime
correspondence. Let $\overline{\Gamma}\subseteq X\times Y$ be the closure of
$\Gamma$, and let $\overline{\Gamma}^{N}\to X\times Y$ be the normalization.
We say $\Gamma$ is _admissible_ (resp. _left proper_) if
$(D_{X})_{\overline{\Gamma}^{N}}\geq(D_{Y})_{\overline{\Gamma}^{N}}$ (resp. if
$\overline{\Gamma}$ is proper over $X$). Let
$\operatorname{\mathbf{\underline{M}Cor}}(\mathcal{X},\mathcal{Y})$ be the
subgroup of $\operatorname{\mathbf{Cor}}(X-D_{X},Y-D_{Y})$ generated by all
admissible left proper prime correspondences. The category
$\operatorname{\mathbf{\underline{M}Cor}}$ has modulus pairs as objects and
$\operatorname{\mathbf{\underline{M}Cor}}(\mathcal{X},\mathcal{Y})$ as the
group of morphisms from $\mathcal{X}$ to $\mathcal{Y}$.

  4. (4)

Let
$\operatorname{\mathbf{\underline{M}Cor}}_{ls}\subset\operatorname{\mathbf{\underline{M}Cor}}$
be the full subcategory of $(X,D)\in\operatorname{\mathbf{\underline{M}Cor}}$
with $X\in\operatorname{\mathbf{Sm}}$ and $|D|$ a normal crossing divisor on
$X$.

  5. (5)

Let
$\operatorname{\mathbf{\underline{M}Cor}}^{{\operatorname{fin}}}\subset\operatorname{\mathbf{\underline{M}Cor}}$
be the full subcategory of the same objects such that
$\operatorname{\mathbf{\underline{M}Cor}}^{{\operatorname{fin}}}(\mathcal{X},\mathcal{Y})$
are generated by all admissible _finite_ prime correspondences, where finite
prime correspondences are defined by replacing the left properness in (3) by
finiteness. We also define
$\operatorname{\mathbf{\underline{M}Cor}}^{{\operatorname{fin}}}_{ls}\subset\operatorname{\mathbf{\underline{M}Cor}}^{{\operatorname{fin}}}_{ls}\cap\operatorname{\mathbf{\underline{M}Cor}}_{ls}$.

  6. (6)

There is a canonical pair of adjoint functors
$\lambda\dashv\underline{\omega}$:

$\lambda:\operatorname{\mathbf{Cor}}\to\operatorname{\mathbf{\underline{M}Cor}}\quad
X\mapsto(X,\emptyset),$
$\underline{\omega}:\operatorname{\mathbf{\underline{M}Cor}}\to\operatorname{\mathbf{Cor}}\quad(X,D)\mapsto
X-|D|,$

  7. (7)

There is a full subcategory
$\mathbf{MCor}\subset\operatorname{\mathbf{\underline{M}Cor}}$ consisting of
_proper modulus pairs_ , where a modulus pair $(X,D)$ is _proper_ if $X$ is
proper. Let
$\tau:\operatorname{\mathbf{MCor}}\hookrightarrow\operatorname{\mathbf{\underline{M}Cor}}$
be the inclusion functor and $\omega=\underline{\omega}\tau$.

  8. (8)

Let $\operatorname{\mathbf{MPST}}$ (resp.
$\operatorname{\mathbf{\underline{M}PST}}$) be the category of additive
presheaves of abelian groups on $\operatorname{\mathbf{MCor}}$ (resp.
$\operatorname{\mathbf{\underline{M}Cor}}$) whose objects are called _modulus
presheaves with transfers_. For $\mathcal{X}\in\operatorname{\mathbf{MCor}}$,
let
${\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathcal{X})=\operatorname{\mathbf{\underline{M}Cor}}(-,\mathcal{X})$
be the representable object of $\operatorname{\mathbf{\underline{M}PST}}$. We
sometime write $\mathcal{X}$ for
${\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathcal{X})$ for
simplicity.

  9. (9)

By the same manner as (2), the category
$\operatorname{\mathbf{\underline{M}Cor}}^{\rm pro}$ is defined and
$F\in\operatorname{\mathbf{\underline{M}PST}}$ is extended to a presheaf on
$\operatorname{\mathbf{\underline{M}Cor}}^{\rm pro}$ (see [13, §3.7]).

  10. (10)

The adjunction $\lambda\dashv\underline{\omega}$ induce a string of $4$
adjoint functors
$(\lambda_{!}=\underline{\omega}^{!},\lambda^{*}=\underline{\omega}_{!},\lambda_{*}=\underline{\omega}^{*},\underline{\omega}_{*})$
(see [4, Pr. 2.3.1]):

$\operatorname{\mathbf{\underline{M}PST}}\begin{smallmatrix}\underline{\omega}^{!}\\\
\longleftarrow\\\ \underline{\omega}_{!}\\\ \longrightarrow\\\
\underline{\omega}^{*}\\\ \longleftarrow\\\ \underline{\omega}_{*}\\\
\longrightarrow\end{smallmatrix}{\operatorname{\mathbf{PST}}}$

where $\underline{\omega}_{!},\underline{\omega}_{*}$ are localisations and
$\underline{\omega}^{!}$ and $\underline{\omega}^{*}$ are fully faithful.

  11. (11)

The functor $\tau$ yields a string of $3$ adjoint functors
$(\tau_{!},\tau^{*},\tau_{*})$:

$\operatorname{\mathbf{MPST}}\begin{smallmatrix}\tau_{!}\\\ \longrightarrow\\\
\tau^{*}\\\ \longleftarrow\\\ \tau_{*}\\\
\longrightarrow\end{smallmatrix}\operatorname{\mathbf{\underline{M}PST}}$

where $\tau_{!},\tau_{*}$ are fully faithful and $\tau^{*}$ is a localisation;
$\tau_{!}$ has a pro-left adjoint $\tau^{!}$, hence is exact (see [4, Pr.
2.4.1]). We will denote by $\operatorname{\mathbf{\underline{M}PST}}^{\tau}$
the essential image of $\tau_{!}$ in
$\operatorname{\mathbf{\underline{M}PST}}$.

  12. (12)

The modulus pair ${\overline{\square}}:=(\mathbf{P}^{1},\infty)$ has an
interval structure induced by the one of $\mathbf{A}^{1}$ (see [7, Lem.
2.1.3]). We say $F\in\operatorname{\mathbf{MPST}}$ is
${\overline{\square}}$-invariant if $p^{*}:F(\mathcal{X})\to
F(\mathcal{X}\otimes{\overline{\square}})$ is an isomorphism for any
$\mathcal{X}\in\operatorname{\mathbf{MCor}}$, where
$p:\mathcal{X}\otimes{\overline{\square}}\to\mathcal{X}$ is the projection.
Let $\operatorname{\mathbf{CI}}$ be the full subcategory of
$\operatorname{\mathbf{MPST}}$ consisting of all
${\overline{\square}}$-invariant objects and
$\operatorname{\mathbf{CI}}^{\tau}\subset\operatorname{\mathbf{\underline{M}PST}}$
be the essential image of $\operatorname{\mathbf{CI}}$ under $\tau_{!}$.

  13. (13)

Recall from [7, Theorem 2.1.8] that $\operatorname{\mathbf{CI}}$ is a Serre
subcategory of $\operatorname{\mathbf{MPST}}$, and that the inclusion functor
$i^{\overline{\square}}:\operatorname{\mathbf{CI}}\to\operatorname{\mathbf{MPST}}$
has a left adjoint $h_{0}^{\overline{\square}}$ and a right adjoint
$h^{0}_{\overline{\square}}$ given for $F\in\operatorname{\mathbf{MPST}}$ and
$\mathcal{X}\in\operatorname{\mathbf{MCor}}$ by

$\displaystyle
h_{0}^{\overline{\square}}(F)(\mathcal{X})=\operatorname{Coker}(i_{0}^{*}-i_{1}^{*}:F(\mathcal{X}\otimes{\overline{\square}})\to
F(\mathcal{X})),$ $\displaystyle
h^{0}_{\overline{\square}}(F)(\mathcal{X})=\operatorname{Hom}(h_{0}^{\overline{\square}}(\mathcal{X}),F).$

For $\mathcal{X}\in\operatorname{\mathbf{MCor}}$, we write
$h_{0}^{\overline{\square}}(\mathcal{X})=h_{0}^{\overline{\square}}({\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathcal{X}))\in\operatorname{\mathbf{CI}}$,
and by abuse of notation, we let $h_{0}^{\overline{\square}}(\mathcal{X})$
denote also for
$\tau_{!}h_{0}^{\overline{\square}}(\mathcal{X})\in\operatorname{\mathbf{CI}}^{\tau}$.

  14. (14)

For $F\in\operatorname{\mathbf{\underline{M}PST}}$ and
$\mathcal{X}=(X,D)\in\operatorname{\mathbf{\underline{M}Cor}}$, write
$F_{\mathcal{X}}$ for the presheaf on the small étale site
$X_{{\operatorname{\acute{e}t}}}$ over $X$ given by $U\to F(\mathcal{X}_{U})$
for $U\to X$ étale, where
$\mathcal{X}_{U}=(U,D_{|U})\in\operatorname{\mathbf{\underline{M}Cor}}$. We
say $F$ is a Nisnevich sheaf if so is $F_{\mathcal{X}}$ for all
$\mathcal{X}\in\operatorname{\mathbf{\underline{M}Cor}}$ (see [4, Section 3]).
We write
$\operatorname{\mathbf{\underline{M}NST}}\subset\operatorname{\mathbf{\underline{M}PST}}$
for the full subcategory of Nisnevich sheaves and put

$\operatorname{\mathbf{MNST}}^{\tau}=\operatorname{\mathbf{\underline{M}NST}}\cap\operatorname{\mathbf{MPST}}^{\tau},\quad\operatorname{\mathbf{CI}}^{\tau}_{\operatorname{Nis}}=\operatorname{\mathbf{CI}}^{\tau}\cap\operatorname{\mathbf{MNST}}^{\tau}.$

By [4, Prop. 3.5.3] and [5, Theorem 2], the inclusion functor
$i_{\operatorname{Nis}}:\operatorname{\mathbf{\underline{M}NST}}\to\operatorname{\mathbf{\underline{M}PST}}$
has an exact left adjoint ${\underline{a}}_{\operatorname{Nis}}$ such that
${\underline{a}}_{\operatorname{Nis}}(\operatorname{\mathbf{MPST}}^{\tau})\subset\operatorname{\mathbf{MNST}}^{\tau}$.
The functor ${\underline{a}}_{\operatorname{Nis}}$ has the following
description: For $F\in\operatorname{\mathbf{\underline{M}PST}}$ and
$\mathcal{Y}\in\operatorname{\mathbf{\underline{M}Cor}}$, let
$F_{\mathcal{Y},{\operatorname{Nis}}}$ be the usual Nisnevich sheafification
of $F_{\mathcal{Y}}$. Then, for
$(X,D)\in\operatorname{\mathbf{\underline{M}Cor}}$ we have

${\underline{a}}_{\operatorname{Nis}}F(X,D)=\operatornamewithlimits{\varinjlim}_{f:Y\to
X}F_{(Y,f^{*}D),{\operatorname{Nis}}}(Y)$

where the colimit is taken over all proper maps $f:Y\to X$ that induce
isomorphisms $Y-|f^{*}D|\xrightarrow{\sim}X-|D|$.

  15. (15)

By [5, Pr. 6.2.1], $\underline{\omega}^{*}$ and $\underline{\omega}_{!}$ from
(10) respect $\operatorname{\mathbf{\underline{M}NST}}$ and
$\operatorname{\mathbf{NST}}$ and induce a pair of adjoint functors (which for
simplicity we write $\underline{\omega}_{!}$ and $\underline{\omega}^{*}$).
Moreover, we have

$\underline{\omega}_{!}{\underline{a}}_{\operatorname{Nis}}=a^{V}_{\operatorname{Nis}}\underline{\omega}_{!}.$

By [7, Lem. 2.3.1] and [5, Pr. 6.2.1a)], for
$F\in{\operatorname{\mathbf{PST}}}$, we have $F\in\operatorname{\mathbf{HI}}$
(resp $F\in\operatorname{\mathbf{HI}}_{\operatorname{Nis}}$) if and only if
$\underline{\omega}^{*}F\in\operatorname{\mathbf{CI}}^{\tau}$ (resp
$\underline{\omega}^{*}F\in\operatorname{\mathbf{CI}}^{\tau}_{\operatorname{Nis}}$).

  16. (16)

We say that $F\in\operatorname{\mathbf{\underline{M}PST}}$ is _semi-pure_ if
the unit map

$u:F\to\underline{\omega}^{*}\underline{\omega}_{!}F$

is injective. For $F\in\operatorname{\mathbf{\underline{M}PST}}$ (resp.
$F\in\operatorname{\mathbf{\underline{M}NST}}$), let
$F^{sp}\in\operatorname{\mathbf{\underline{M}PST}}$ (resp.
$F^{sp}\in\operatorname{\mathbf{\underline{M}NST}}$) be the image of
$F\to\underline{\omega}^{*}\underline{\omega}_{!}F$ (called the semi-
purification of $F$. See [16, Lem. 1.30]). For
$F\in\operatorname{\mathbf{\underline{M}PST}}$ we have

${\underline{a}}_{\operatorname{Nis}}(F^{sp})\simeq({\underline{a}}_{\operatorname{Nis}}F)^{sp}.$

This follows from the fact that ${\underline{a}}_{\operatorname{Nis}}$ is
exact and commutes with $\underline{\omega}^{*}\underline{\omega}_{!}$. For
$F\in\operatorname{\mathbf{MPST}}^{\tau}$ we have
$F^{sp}\in\operatorname{\mathbf{MPST}}^{\tau}$ since $\tau$ is exact and
$\underline{\omega}^{*}\underline{\omega}_{!}\tau_{!}=\tau_{!}\omega^{*}\omega_{!}$.

  17. (17)

Let
$\operatorname{\mathbf{CI}}^{\tau,sp}\subset\operatorname{\mathbf{CI}}^{\tau}$
be the full subcategory of semipure objects and consider the full subcategory

$\operatorname{\mathbf{CI}}^{\tau,sp}_{\operatorname{Nis}}=\operatorname{\mathbf{CI}}^{\tau,sp}\cap\operatorname{\mathbf{MNST}}^{\tau}\subset\operatorname{\mathbf{CI}}^{\tau}_{\operatorname{Nis}}.$

By [16, Th. 0.1 and 0.4], we have
${\underline{a}}_{\operatorname{Nis}}(\operatorname{\mathbf{CI}}^{\tau,sp})\subset\operatorname{\mathbf{CI}}^{\tau,sp}_{\operatorname{Nis}}$.

  18. (18)

We write ${\operatorname{\mathbf{RSC}}}\subseteq{\operatorname{\mathbf{PST}}}$
for the essential image of $\operatorname{\mathbf{CI}}$ under $\omega_{!}$
(which is the same as the essential image of
$\operatorname{\mathbf{CI}}^{\tau,sp}$ under $\underline{\omega}_{!}$ since
$\omega_{!}=\underline{\omega}_{!}\tau_{!}$ and
$\underline{\omega}_{!}F=\underline{\omega}_{!}F^{sp}$). Put
${\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}={\operatorname{\mathbf{RSC}}}\cap\operatorname{\mathbf{NST}}$.
The objects of ${\operatorname{\mathbf{RSC}}}$ (resp.
${\operatorname{\mathbf{RSC}}}_{\operatorname{Nis}}$) are called reciprocity
presheaves (resp. sheaves). By [16, Th. 0.1], we have

(1.0.1)
$a^{V}_{\operatorname{Nis}}({\operatorname{\mathbf{RSC}}})\subset{\operatorname{\mathbf{RSC}}}_{\operatorname{Nis}}.$

We have $\operatorname{\mathbf{HI}}\subseteq{\operatorname{\mathbf{RSC}}}$ and
it contains also smooth commutative group schemes (which may have non-trivial
unipotent part), and the sheaf $\Omega^{i}$ of Kähler differentials, and the
de Rham-Witt sheaves $W\Omega^{i}$ (see [6] and [7]).

  19. (19)

$\operatorname{\mathbf{NST}}$ is a Grothendieck abelian category by [17, Lem.
3.1.6] and we can make ${\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}$
its full sub-abelian category as follows: We define the kernel (resp.
cokernel) of a map $\phi:F\to G$ in
${\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}$ to be that of $\phi$ as
a map in $\operatorname{\mathbf{NST}}$. Here we need (1.0.1) to ensure that
the cokernel of $\phi$ in $\operatorname{\mathbf{NST}}$ stays in
${\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}$. By definition, a
sequence $0\to F\to G\to H\to 0$ is exact in
${\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}$ if and only if it is
exact in $\operatorname{\mathbf{NST}}$.

  20. (20)

By [7, Prop. 2.3.7] we have a pair of adjoint functors:

(1.0.2)
$\operatorname{\mathbf{CI}}\begin{smallmatrix}\omega^{\operatorname{\mathbf{CI}}}\\\
\longleftarrow\\\ \omega_{!}\\\
\longrightarrow\end{smallmatrix}{\operatorname{\mathbf{RSC}}},$

where
$\omega^{\operatorname{\mathbf{CI}}}=h^{0}_{\overline{\square}}\omega^{*}$ and
it is fully faithful. It induces a pair of adjoint functors:

(1.0.3)
$\operatorname{\mathbf{CI}}^{\tau}\begin{smallmatrix}\underline{\omega}^{\operatorname{\mathbf{CI}}}\\\
\longleftarrow\\\ \underline{\omega}_{!}\\\
\longrightarrow\end{smallmatrix}{\operatorname{\mathbf{RSC}}},$

where
$\underline{\omega}^{\operatorname{\mathbf{CI}}}=\tau_{!}h^{0}_{\overline{\square}}\omega^{*}$
and it is fully faithful. Indeed, let $F=\tau_{!}\hat{F}$ for
$\hat{F}\in\operatorname{\mathbf{CI}}$ and
$G\in{\operatorname{\mathbf{RSC}}}$. In view of (13) and the exactness and
full faithfulness of $\tau_{!}$, we have

$\operatorname{Hom}_{\operatorname{\mathbf{CI}}^{\tau}}(F,\tau_{!}h^{0}_{\overline{\square}}\omega^{*}G)\simeq\operatorname{Hom}_{\operatorname{\mathbf{CI}}}(\hat{F},h^{0}_{\overline{\square}}\omega^{*}G)\simeq\\\
\operatorname{Hom}_{\operatorname{\mathbf{MPST}}}(\hat{F},\omega^{*}G)\simeq\operatorname{Hom}_{\operatorname{\mathbf{\underline{M}PST}}}(\tau_{!}\hat{F},\underline{\omega}^{*}G)\simeq\operatorname{Hom}_{{\operatorname{\mathbf{RSC}}}}(\underline{\omega}_{!}F,G).$

In view of (15), (1.0.3) induce pair of adjoint functors:

(1.0.4)
$\operatorname{\mathbf{CI}}^{\tau,sp}_{\operatorname{Nis}}\begin{smallmatrix}\underline{\omega}^{\operatorname{\mathbf{CI}}}\\\
\longleftarrow\\\ \underline{\omega}_{!}\\\
\longrightarrow\end{smallmatrix}{\operatorname{\mathbf{RSC}}}_{\operatorname{Nis}},$

##  2\. Purity with reduced modulus

For $F\in\operatorname{\mathbf{\underline{M}PST}}$, we put

$F_{-1}=\operatorname{Ker}\big{(}\operatorname{\underline{Hom}}_{\operatorname{\mathbf{\underline{M}PST}}}((\mathbf{P}^{1}-0,\infty),F)\overset{i_{1}^{*}}{\longrightarrow}F\big{)},$
$F_{-1}^{(1)}=\operatorname{Ker}\big{(}\operatorname{\underline{Hom}}_{\operatorname{\mathbf{\underline{M}PST}}}((\mathbf{P}^{1},0+\infty),F)\overset{i_{1}^{*}}{\longrightarrow}F\big{)},$

Note that if
$F\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$, we have for
$\mathcal{X}\in\operatorname{\mathbf{\underline{M}Cor}}$

(2.0.1) $\displaystyle
F_{-1}^{(1)}(\mathcal{X})=\operatorname{Hom}_{\operatorname{\mathbf{\underline{M}PST}}}(h_{0,{\operatorname{Nis}}}^{{\overline{\square}},{\rm
sp}}(\mathbf{P}^{1},0+\infty)^{0},\operatorname{\underline{Hom}}_{\operatorname{\mathbf{\underline{M}PST}}}({\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathcal{X}),F)),$
$\displaystyle
F_{-1}(\mathcal{X})=\operatornamewithlimits{\varinjlim}_{n}\operatorname{Hom}_{\operatorname{\mathbf{\underline{M}PST}}}(h_{0,{\operatorname{Nis}}}^{{\overline{\square}},{\rm
sp}}(\mathbf{P}^{1},n\cdot
0+\infty)^{0},\operatorname{\underline{Hom}}_{\operatorname{\mathbf{\underline{M}PST}}}({\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathcal{X}),F)),$

where

$h_{0,{\operatorname{Nis}}}^{{\overline{\square}},{\rm
sp}}(\mathbf{P}^{1},n\cdot
0+\infty)^{0}=\operatorname{Coker}\big{(}\mathbb{Z}={\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\operatorname{Spec}k,\emptyset)\overset{i_{1}}{\longrightarrow}h_{0,{\operatorname{Nis}}}^{{\overline{\square}},{\rm
sp}}(\mathbf{P}^{1},n\cdot 0+\infty)\big{)}.$

######  Definition 2.1.

For $e_{1},\dots,e_{r}\in\\{0,1\\}$, put

$\tau^{(e_{1},\dots,e_{r})}F=\tau^{(e_{r})}\cdots\tau^{(e_{1})}F,$

where

$\tau^{(0)}F=F_{-1}\;\text{ and }\;\tau^{(1)}F=F_{-1}/F_{-1}^{(1)}.$

The existence of retractions in the following lemma was suggested by A.
Merici. It implies
$\tau^{(e_{1},\dots,e_{r})}F\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$
if $F\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$.

######  Lemma 2.2.

For $F\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$, the
inclusion $F_{-1}^{(1)}\to F_{-1}$ admits a retraction $s_{F}:F_{-1}\to
F_{-1}^{(1)}$ such that for any map $\phi:F\to G$ in
$\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$, the following
diagram is commutative:

$\textstyle{F_{-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{F}}$$\scriptstyle{\phi}$$\textstyle{F_{-1}^{(1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{G_{-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{F}}$$\textstyle{G_{-1}^{(1)}}$

In particular
$\tau^{(1)}F\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$ if
$F\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$.

###### Proof.

In view of (2.0.1), this follows from [2, Lem. 2.4]. ∎

######  Theorem 2.3.

Let $F\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$. Let
$\mathcal{X}=\operatorname{Spec}K\\{t_{1},\dots,t_{n}\\}$ and
$D=\\{t_{1}^{e_{1}}\cdots t_{n}^{e_{n}}=0\\}\subset\mathcal{X}$ with
$e_{1},\dots,e_{n}\in\\{0,1\\}$. For a subset $I\subset[1,n]$ let
$i_{\mathcal{H}}:\mathcal{H}\hookrightarrow\mathcal{X}$ be the closed
immersion defined by $\\{t_{i}=0\\}_{i\in I}$ and
$D_{\mathcal{H}}=\\{\underset{j\in[1,n]-I}{\prod}t_{j}^{e_{j}}=0\\}\subset\mathcal{H}$.
Then

(2.3.1) $R^{\nu}i_{\mathcal{H}}^{!}F_{(\mathcal{X},D)}=0\;\text{ for
}\nu\not=q:=|I|,$

and there is an isomorphism

(2.3.2) $(\tau^{(e_{I})}F)_{(\mathcal{H},D_{\mathcal{H}})}\simeq
R^{q}i_{\mathcal{H}}^{!}F_{(\mathcal{X},D)}\;\text{ with }e_{I}=(e_{i})_{i\in
I}\in\mathbb{Z}_{\geq 0}^{q}.$

###### Proof.

The proof is divided into two steps.

Step 1: We prove (2.3.1) and (2.3.2) in case $q=|I|=1$.

For $\nu=0$ (2.3.1) follows from the semipurity of $F$. Thus it suffices to
show (2.3.1) only for $\nu>1$. Let $J=\\{j\in[1,n]\;|\;e_{j}\not=0\\}$ and
$r=|J|$. If $\dim(\mathcal{X})=0$, the assertion is trivial. If $r=0$, the
assertion follows from [16, Cor. 8.6(3)]. Assume $r>0$ and
$\dim(\mathcal{X})\geq 1$, and proceed by the double induction on $r$ and
$\dim(\mathcal{X})$. Without loss of generality, we may assume

  1. $(\spadesuit)$

$e_{1}\not=0$, and $\mathcal{H}=\\{t_{1}=0\\}$ if $\mathcal{H}\subset|D|$.

Let $\iota:\mathcal{Z}\hookrightarrow\mathcal{X}$ be the closed immersion
defined by $\\{t_{1}=0\\}$ and $D_{\mathcal{Z}}=\\{t_{2}^{e_{2}}\cdots
t_{r}^{e_{r}}=0\\}\subset\mathcal{Z}$ and $D^{\prime}=\\{t_{2}^{e_{2}}\cdots
t_{r}^{e_{r}}=0\\}\subset\mathcal{X}$. By [16, Lem. 7.1], we have an exact
sequence sheaves on $\mathcal{X}_{{\operatorname{Nis}}}$:

$0\to F_{(\mathcal{X},D^{\prime})}\to
F_{(\mathcal{X},D)}\to\iota_{*}(F^{(e_{1})}_{-1})_{(\mathcal{Z},D_{\mathcal{Z}})}\to
0,$

which gives rise to a long exact sequence of sheaves on
$\mathcal{H}_{{\operatorname{Nis}}}$:

(2.3.3) $\cdots\to R^{\nu}i_{\mathcal{H}}^{!}F_{(\mathcal{X},D^{\prime})}\to
R^{\nu}i_{\mathcal{H}}^{!}F_{(\mathcal{X},D)}\to
R^{\nu}i_{\mathcal{H}}^{!}\iota_{*}(F^{(e_{1})}_{-1})_{(\mathcal{Z},D_{\mathcal{Z}})}\to\cdots.$

By the induction hypothesis,
$R^{\nu}i_{\mathcal{H}}^{!}F_{(\mathcal{X},D^{\prime})}=0$ for $\nu>1$. In
case $\mathcal{H}\not=\mathcal{Z}$, we have a Cartesian diagram of closed
immersions

$\textstyle{\mathcal{H}\cap\mathcal{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{\prime}}$$\scriptstyle{i_{\mathcal{H}\cap\mathcal{Z}}}$$\textstyle{\mathcal{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\mathcal{H}}}$$\textstyle{\mathcal{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{\mathcal{X}}$

and we have an isomorphism

$R^{\nu}i_{\mathcal{H}}^{!}\iota_{*}(F^{(e_{1})}_{-1})_{(\mathcal{Z},D_{\mathcal{Z}})}\simeq\iota^{\prime}_{*}R^{\nu}i_{\mathcal{H}\cap\mathcal{Z}}^{!}(F^{(e_{1})}_{-1})_{(\mathcal{Z},D_{\mathcal{Z}})}.$

By the induction hypothesis,
$R^{\nu}i_{\mathcal{H}\cap\mathcal{Z}}^{!}(F^{(e_{1})}_{-1})_{(\mathcal{Z},D_{\mathcal{Z}})}=0$
for $\nu>1$ noting
$F^{(e_{1})}_{-1}\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$
by Lemma 2.2. So the desired vanishing follows from (2.3.3). Moreover, the
assumptions $(\spadesuit)$ and $\mathcal{H}\not=\mathcal{Z}$ imply that
$\mathcal{H}\not\subset|D|$. Then (2.3.2) (with $q=1$) follows from [16, Lem.
7.1(2)].

In case $\mathcal{Z}=\mathcal{H}$, we have

$R^{\nu}i_{\mathcal{H}}^{!}\iota_{*}(F^{(e_{1})}_{-1})_{(\mathcal{Z},D_{\mathcal{Z}})}=R^{\nu}\iota^{!}\iota_{*}(F^{(e_{1})}_{-1})_{(\mathcal{Z},D_{\mathcal{Z}})},$

which vanishes for $\nu>0$. Hence (2.3.3) gives the desired vanishing together
with an exact sequence:

$0\to(F^{(e_{1})}_{-1})_{(\mathcal{H},D_{\mathcal{H}})}\overset{\delta}{\longrightarrow}R^{1}i_{\mathcal{H}}^{!}F_{(\mathcal{X},D^{\prime})}\to
R^{1}i_{\mathcal{H}}^{!}F_{(\mathcal{X},D)}\to 0.$

By [16, Lem. 7.1(2)] we have an isomorphism

$(F_{-1})_{(\mathcal{H},D_{\mathcal{H}})}\simeq
R^{1}i_{\mathcal{H}}^{!}F_{(\mathcal{X},D^{\prime})}$

through which $\delta$ is identified with the map induced by the canonical map
$F^{(e_{1})}_{-1}\to F_{-1}$. This proves the desired isomorphism (2.3.2) in
case $\mathcal{Z}=\mathcal{H}$ and completes Step 1.

Step 2: We prove the theoerm by the induction on $q$ assuming $q>0$. Let
$I=\\{i_{1},\dots,i_{q}\\}\subset[1,n]$ and $\mathcal{Y}\subset\mathcal{X}$ be
the closed subscheme defined by $\\{t_{i_{1}}=0\\}$. Let
$i_{\mathcal{Y}}:\mathcal{Y}\hookrightarrow\mathcal{X}$ and
$i_{\mathcal{H},\mathcal{Y}}:\mathcal{H}\to\mathcal{Y}$ be the induced closed
immersions. By Step 1 we have
$R^{\nu}i_{\mathcal{Y}}^{!}F_{(\mathcal{X},D)}=0$ for $\nu\not=1$ and we have
an isomorphism

$(\tau^{(e_{i_{1}})}F)_{(\mathcal{Y},D_{\mathcal{Y}})}\simeq
R^{1}i_{\mathcal{Y}}^{!}F_{(\mathcal{X},D)}\;\text{ with
}D_{\mathcal{Y}}=\\{t_{1}^{e_{1}}\cdots\overset{\vee}{t_{i_{1}}^{e_{i_{1}}}}\cdots
t_{n}^{e_{n}}=0\\}\subset\mathcal{Y}.$

Note
$\tau^{(e_{i_{1}})}F\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$
by Lemma 2.2. Thus, by the induction hypothesis, we have
$R^{\nu}i_{\mathcal{H},\mathcal{Y}}^{!}\tau^{(e_{i_{1}})}F_{(\mathcal{Y},D_{\mathcal{Y}})}=0$
for $\nu\not=q-1$. By the spectral sequence

$E_{2}^{a,b}=R^{b}i_{\mathcal{H},\mathcal{Y}}^{!}R^{a}i_{\mathcal{Y}}^{!}F_{(\mathcal{X},D)}\Rightarrow
R^{a+b}i_{\mathcal{H}}^{!}F_{(\mathcal{X},D)},$

we get the desired vanishing (2.3.1) and an isomorphism

$R^{q}i_{\mathcal{H}}^{!}F_{(\mathcal{X},D)}\simeq
R^{q-1}i_{\mathcal{H},\mathcal{Y}}^{!}R^{1}i_{\mathcal{Y}}^{!}F_{(\mathcal{X},D)}\simeq
R^{q-1}i_{\mathcal{H},\mathcal{Y}}^{!}(\tau^{(e_{i_{1}})}F)_{(\mathcal{Y},D_{\mathcal{Y}})}\\\
\simeq(\tau^{(e_{i_{2}},\dots,e_{i_{q}})}(\tau^{(e_{i_{1}})}F))_{(\mathcal{H},D_{\mathcal{H}})}\simeq(\tau^{(e_{i_{1}},e_{i_{2}},\dots,e_{i_{q}})}F)_{(\mathcal{H},D_{\mathcal{H}})},$

where the third isomorphism holds by the induction hypothesis. This completes
the proof of the theorem. ∎

We say $\mathcal{X}=(X,D)\in\operatorname{\mathbf{\underline{M}Cor}}$ reduced
if so is $D$. The following corollaries 2.4 and 2.5 are immediate consequences
of Theorem 2.3.

######  Corollary 2.4.

Take $F\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$ and
$(X,D)\in\operatorname{\mathbf{\underline{M}Cor}}_{ls}$ reduced. Let
$x\in{X}^{(n)}$ with $K=k(x)$ and let $\mathcal{X}=X^{h}_{|x}$ be the
henselization of $X$ at $x$. Then

$H^{i}_{x}(X_{{\operatorname{Nis}}},F_{(X,D)})=0\;\text{ for }i\not=n.$

Choosing an isomorphism

$\varepsilon:\mathcal{X}\simeq\operatorname{Spec}K\\{t_{1},\dots,t_{n}\\}$

such that $D_{|\mathcal{X}}=\\{t_{1}^{e_{1}}\cdots
t_{n}^{e_{n}}=0\\}\subset\mathcal{X}$ with $e_{1},\dots,e_{n}\in\\{0,1\\}$,
there exists an isomorphism depending on $\varepsilon$:

$\theta_{\varepsilon}:\tau^{(e_{1},e_{2},\dots,e_{n})}F(x)\simeq
H^{n}_{x}(X_{{\operatorname{Nis}}},F_{(X,D)}).$

######  Corollary 2.5.

For $F\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$ and
$\mathcal{X}=(X,D)\in\operatorname{\mathbf{\underline{M}Cor}}_{ls}$ reduced,
the following sequence is exact:

$0\to F(X,D)\to
F(X-D,\emptyset)\to\underset{\xi\in{D}^{(0)}}{\bigoplus}\frac{F(X^{h}_{|\xi}-\xi,\emptyset)}{F(X^{h}_{|\xi},\xi)}.$

The idea of deducing the following corollary from the above is due to A.
Merici.

######  Corollary 2.6.

Let $\mathcal{X}=(X,D)\in\operatorname{\mathbf{\underline{M}Cor}}_{ls}$ be
reduced.

  * (1)

Assume given an exact sequence in $\operatorname{\mathbf{\underline{M}NST}}$:

(2.6.1) $0\to
H\overset{\phi}{\longrightarrow}G\overset{\psi}{\longrightarrow}F$

such that
$F,G,H\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$ and that
$\underline{\omega}_{!}\psi$ is surjective in $\operatorname{\mathbf{NST}}$.
If $X$ is henselian local,

$0\to H(\mathcal{X})\to G(\mathcal{X})\to F(\mathcal{X})\to 0$

is exact.

  * (2)

Let $\gamma:F\to G$ be a map in
$\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$ such that
$\underline{\omega}_{!}\gamma$ is an isomorphism. Then $F(\mathcal{X})\to
G(\mathcal{X})$ is an isomorphism.

  * (3)

For $F\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$, the
unit map
$u:F\to\underline{\omega}^{\operatorname{\mathbf{CI}}}\underline{\omega}_{!}F$
induces an isomorphism
$F(\mathcal{X})\cong\underline{\omega}^{\operatorname{\mathbf{CI}}}\underline{\omega}_{!}F(\mathcal{X})$.

###### Proof.

To show (1), it suffices to show the surjectivity of $G(\mathcal{X})\to
F(\mathcal{X})$. Let $\eta\in X$ be the generic point and consider the
following commutative diagram of the Cousin complexes:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H(\mathcal{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H(\eta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi(\eta)}$$\textstyle{\underset{x\in{X}^{(1)}}{\bigoplus}H^{1}_{x}(X,H_{\mathcal{X}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H^{1}_{x}(\phi)}$$\textstyle{\underset{y\in{X}^{(2)}}{\bigoplus}H^{2}_{y}(X,H_{\mathcal{X}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H^{2}_{y}(\phi)}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G(\mathcal{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G(\eta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi(\eta)}$$\textstyle{\underset{x\in{X}^{(1)}}{\bigoplus}H^{1}_{x}(X,G_{\mathcal{X}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H^{1}_{x}(\psi)}$$\textstyle{\underset{y\in{X}^{(2)}}{\bigoplus}H^{2}_{y}(X,G_{\mathcal{X}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H^{2}_{y}(\psi)}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F(\mathcal{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F(\eta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{x\in{X}^{(1)}}{\bigoplus}H^{1}_{x}(X,F_{\mathcal{X}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{y\in{X}^{(2)}}{\bigoplus}H^{2}_{y}(X,F_{\mathcal{X}})}$

By Corollary 2.4, the horizontal sequences are exact. By the assumption,
$\psi(\eta)$ is surjective. By a diagram chase we are reduced to showing the
following.

######  Claim 2.6.1.

  * (i)

For $x\in{X}^{(1)}$, the sequence

$H^{1}_{x}(X,H_{\mathcal{X}})\to H^{1}_{x}(X,G_{\mathcal{X}})\to
H^{1}_{x}(X,F_{\mathcal{X}})$

is exact.

  * (ii)

For $y\in{X}^{(2)}$, $H^{2}_{y}(\phi)$ is injective.

To show (i), by Corollary 2.4, it suffices to show the exactness of
$\tau^{(e)}H\to\tau^{(e)}G\to\tau^{(e)}F$ for $e\in\\{0,1\\}$. The case $e=0$
follows from the left exactness of the endofunctor
$\operatorname{\underline{Hom}}_{\operatorname{\mathbf{\underline{M}PST}}}(\mathcal{X},-)$
on $\operatorname{\mathbf{\underline{M}NST}}$ for any
$\mathcal{X}\in\operatorname{\mathbf{\underline{M}Cor}}$. We have a
commutative diagram

$\textstyle{\tau^{(1)}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{s_{H}}$$\textstyle{\tau^{(1)}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{s_{G}}$$\textstyle{\tau^{(1)}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{F}}$$\textstyle{\tau^{(0)}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{p_{H}}$$\textstyle{\tau^{(0)}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{p_{G}}$$\textstyle{\tau^{(0)}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{F}}$

where $p_{*}$ are the projections and $s_{*}$ is a right inverse of $p_{*}$
coming from the retractions from Lemma 2.2. We have

$\phi\circ p_{H}=p_{G}\circ\phi,\;\psi\circ p_{G}=p_{F}\circ\psi,\;\phi\circ
s_{H}=s_{G}\circ\phi,\;\psi\circ s_{G}=s_{F}\circ\psi.$

By a diagram chase, the case $e=1$ follows from the case $e=0$.

To show (ii), by Corollary 2.4, it suffices to show the injectivity of
$\tau^{({\underline{e}})}H\to\tau^{({\underline{e}})}G$ for
${\underline{e}}\in\\{(0,0),(0,1),(1,0),(1.1)\\}$. The case
${\underline{e}}=(0,0)$ follows from the same left exactness as above, and the
other cases from this case thanks to Lemma 2.2.

To show (2), we may assume $\mathcal{X}$ is henselian local. Then it follows
from (1). (3) follows from (2) since $\underline{\omega}_{!}u$ is an
isomorphism. This completes the proof of the corollary. ∎

##  3\. Review on higher local symbols

In this section we recall from [14] the higher local symbols for reciprocity
sheaves, which is a fundamental tool to prove Theorem 4.2, one of the main
theorems of this paper.

######  3.1.

Let $X$ be an excellent separate scheme of pure dimension $d$. We denote by
$X^{(c)}$ the set of points of codimension $c$. Let $x,y\in X$ be two points.
We write

$x>y:\Longleftrightarrow\overline{\\{x\\}}\supset\overline{\\{y\\}},\text{
i.e. },y\in\overline{\\{x\\}}.$

A specialization chain (or just chain) in $X$ is a sequence

$x=(x_{1},\ldots,x_{n})\quad\text{with }x_{1}>x_{2}>\ldots>x_{n}.$

We say a specialization chain $x$ is maximal or a Parsin chain if $n=d$ and
$x_{i}\in X^{(i)}$555 The notation is different from [14] in which a
specialization chain $x=(x_{1},\ldots,x_{n})$ satisfies $x_{n}>\ldots>x_{1}$
and $x$ is maximal if the closures of $x_{i}$ are of dimension $i$.. We denote
by $c(X)$ the set of specialization chains in $X$ and

$\mathrm{mc}(X):=\\{x\in c(X)|\;\text{$x$ is maximal}\\}.$

If $d=0$, the only element in $\mathrm{mc}(X)$ is the empty chain.

We say a specialization chain $x=(x_{1},\dots,x_{n})$ is a $Q$-chain with
break at $r$ if $0\leq r\leq n=d-1$ and $x_{i}\in{X}^{(i)}$ for $i\leq r$ and
$x_{i}\in{X}^{(i+1)}$ for $i>r$. We denote

$Q_{r}(X):=\\{\text{$Q$-chains with break at $r$ in }X\\}.$

For $x\in Q_{r}(X)$, we denote by $B(x)$ the set of all $y\in X$ such that

$x(y):=(x_{1},\dots,x_{r},y,x_{r+1},\dots,x_{d-1})\in\mathrm{mc}(X).$

For a specialization chain $x=(x_{1},\ldots,x_{n})$ in $X$, we let
$\mathcal{O}_{X,x}^{h}$ be the henselization of $X$ along $x$ as defined in
[3, Def. 1.6.2]: If $n=1$, $\mathcal{O}_{X,x}^{h}$ is the henselization of
$\mathcal{O}_{X,x}$. If $n>1$, let $y=(x_{2},\dots,x_{n})$ and assume that we
have defined $R=\mathcal{O}_{X,y}^{h}$. Let $T$ be the set of all prime ideals
of $R$ lying over $x_{1}$. Then we define $\mathcal{O}_{X,x}^{h}$ as the
product of the henselization of $R$ at $\mathfrak{p}$ for all $\mathfrak{p}\in
T$. By definition, $\mathcal{O}_{X,x}^{h}$ is a finite product of henselian
local rings and we let $\mathfrak{m}_{x}\subset\mathcal{O}^{h}_{X,x}$ denote
the product of its maximal ideals and $K^{h}_{X,x}$ denote its total fraction
ring.

For $x\in Q_{r}(X)$ and $x(y)$ with $y\in B(x)$ as above, we have a natural
inclusion of rings

(3.1.1) $\iota_{y}:K^{h}_{X,x}\to K^{h}_{X,x(y)}.$

In what follows in this section, we fix
$F\in{\operatorname{\mathbf{RSC}}}_{\operatorname{Nis}}$ and write
$\tilde{F}=\underline{\omega}^{\operatorname{\mathbf{CI}}}F\in\operatorname{\mathbf{CI}}^{\tau,sp}_{\operatorname{Nis}}$
(cf. (1.0.4)). We also fix a function field $K$ over the base field $k$. Let
$X$ be an integral $K$-scheme of dimension $d$. Recall from [14] that we have
a collection of bilinear pairings

(3.1.2) $\\{(-,-)_{X/K,x}:F(K(X))\otimes K^{M}_{d}(K^{h}_{X,x})\to
F(K)\\}_{x\in\mathrm{mc}(X)},$

where we recall $K^{h}_{X,x}$ is a finite product of fields and
$K^{M}_{n}(K^{h}_{X,x})$ for an integer $n>0$ denotes the product of the
Milnor $K$-groups of the fields. We also note that $\mathcal{O}^{h}_{X,x}$ is
a finite product of henselian dvr’s and let $K^{M}_{n}(\mathcal{O}^{h}_{X,x})$
denote the product of the Milnor $K$-groups of the dvr’s.

For a local ring $R$ and an ideal $I\subset R$, let $K^{M}_{n}(R,I)\subset
K^{M}_{n}(R)$ denote the subgroup generated by symbols

$\\{1+a,b_{1},\dots,b_{n-1}\\}\;\text{ with }a\in I,\,b_{i}\in R^{\times}.$

The following properties hold for all $a\in F(K(X))$:

  1. (HS1)

Let $X\hookrightarrow X^{\prime}$ be an open immersion where $X^{\prime}$ is
an integral $K$-scheme of dimension $d$. Then we have
$(a,\beta)_{X/K,x}=(a,\beta)_{X^{\prime}/K,x}$ for all $\beta\in
K^{M}_{d}(K^{h}_{X,x})$.

  2. (HS2)

Let $x=(x_{1},\ldots,x_{d})\in\mathrm{mc}(X)$ and $X_{1}\subset X$ be the
closure of $x_{1}$ and $\delta=(x_{2},\ldots,x_{d})\in\mathrm{mc}(X_{1})$.
Then for all $\beta\in K^{M}_{d}(K^{h}_{X,x})$

$(a,\beta)_{X/K,x}=\begin{cases}\beta\cdot\operatorname{Tr}_{E/K}(a),&\text{if
$d=0$ and $E=K(X)$};\\\
(a_{|X_{1}},\partial_{x_{1}}\beta)_{X_{1}/K,\delta},&\text{if }d\geq 1\text{
and }a\in F(\mathcal{O}_{X,x_{1}});\end{cases}$

where $a_{|X_{1}}\in F(K(X_{1}))$ is the restriction of $a$, and

$\partial_{x_{1}}:K^{M}_{d}(K^{h}_{X,x})\to K^{M}_{d-1}(K^{h}_{X_{1},\delta})$

is the tame symbol coming from the fact that $\mathcal{O}_{X,x}^{h}$ is a
product of dvr’s and $K^{h}_{X_{1},\delta}$ is identified with the product of
its residue fields.

  3. (HS3)

Let $D\subset X$ be an effective Cartier divisor such that
$(X,D)\in\operatorname{\mathbf{\underline{M}Cor}}^{\rm pro}$ (cf. §1(9) ). Let
$I_{D}\subset\mathcal{O}_{X}$ be the ideal sheaf of $D$. Assume
$a\in\tilde{F}(X,D)$. Then, for all $x\in\mathrm{mc}(X)$, we have

$(a,\beta)_{X/K,x}=0\;\text{ for }\forall\beta\in
K^{M}_{d}(\mathcal{O}_{X,x}^{h},I_{D}\mathcal{O}_{X,x}^{h}).$

  4. (HS4)

Let $x\in Q_{r}(X)$ with $0\leq r\leq d-1$. Then
$(a,\iota_{y}(\beta))_{X/K,x(y)}=0$ for almost all $y\in B(x)$. If $r<d-1$ or
$X$ is projective over $K$,

$\sum_{y\in B(x)}(a,\iota_{y}(\beta))_{X/K,x(y)}=0,\quad\text{for all
}\beta\in K^{M}_{d}(K^{h}_{X,x}),$

where $\iota_{y}:K^{h}_{X,x}\to K^{h}_{X,x(y)}$ is the natural map.

  5. (HS5)

Let $f:Y\to X$ be a finite surjective $K$-morphism between two integral
$K$-schemes. Let $x\in\mathrm{mc}(X)$ and $y\in\mathrm{mc}(Y)$ with $f(y)=x$.
Then we have

$(f^{*}a,\beta)_{Y/K,y}=(a,\operatorname{Nm}_{y/x}(\beta))_{X/K,x}\;(a\in
F(K(X)),\;\beta\in K^{M}_{d}(K^{h}_{Y,y})),$
$(f_{*}b,\alpha)_{X/K,x}=\underset{\begin{subarray}{c}z\in\mathrm{mc}(X^{\prime})\\\
f(z)=x\end{subarray}}{\sum}(b,i_{x/z}\alpha)_{Y/K,z}\;(b\in
F(K(X^{\prime})),\;\alpha\in K^{M}_{d}(K^{h}_{X,x})),$

where $\operatorname{Nm}_{y/x}:K^{M}_{d}(K^{h}_{Y,y})\to
K^{M}_{d}(K^{h}_{X,x})$ is the norm map on Milnor $K$-theory and
$i_{x/z}:K^{M}_{d}(K^{h}_{X,x})\to K^{M}_{d}(K^{h}_{Y,z})$ is the inclusion
map.

Now assume that $X\in\operatorname{\mathbf{Sm}}$ of dimension $d$ and
$D\subset X$ be a reduced SNCD on $X$. For a function field $K$ over $k$ and a
scheme $Z$ over $k$, write $Z_{K}=Z\otimes_{k}K$ with $\phi_{Z}:Z_{K}\to Z$
the projection. If $Z$ is integral, we denote by $K(Z)$ the function field of
$Z_{K}$. We quote the following result from [14, Pr. 8.4].

######  Proposition 3.2.

Let $F\in{\operatorname{\mathbf{RSC}}}_{\operatorname{Nis}}$ and $a\in
F(X-|D|)$. Assume that there exists an open subset $U\subset X$ which contains
all generic points of $D$ such that the following condition holds: For any
function field $K$ over $k$ and any
$x=(x_{1},\dots,x_{d})\in\mathrm{mc}(U_{K})$ with $x_{1}\in D_{K}^{(0)}$, we
have

$(\phi_{X}^{*}(a),\beta)_{U_{K}/K,x}=0\;\text{ for }\forall\beta\in
K_{d}^{M}(\mathcal{O}_{X_{K},x}^{h},\mathfrak{m}_{x}).$

Then we have $a\in F(X,D)$.

##  4\. Logarithmic cohomology of reciprocity sheaves

For $\mathcal{X}=(X,D)\in\operatorname{\mathbf{\underline{M}Cor}}_{ls}$, we
write
$\mathcal{X}_{{\operatorname{red}}}=(X,|D|)\in\operatorname{\mathbf{\underline{M}Cor}}_{ls}$.
We say $\mathcal{X}=(X,D)\in\operatorname{\mathbf{\underline{M}Cor}}_{ls}$ is
reduced if $\mathcal{X}=\mathcal{X}_{{\operatorname{red}}}$.

######  Definition 4.1.

Let $F\in\operatorname{\mathbf{\underline{M}PST}}$.

  * (1)

We say that $F$ is _log-semipure_ if for any
$\mathcal{X}\in\operatorname{\mathbf{\underline{M}Cor}}_{ls}$, the map
$F(\mathcal{X}_{{\operatorname{red}}})\to F(\mathcal{X})$ is injective. Note
that if $F$ is semipure, $F$ is log-semipure.

  * (2)

We say that $F$ is _logarithmic_ if it is log-semipure and satisfies the
condition that for
$\mathcal{X},\mathcal{Y}\in\operatorname{\mathbf{\underline{M}Cor}}_{ls}$ with
$\mathcal{X}$ reduced and
$\alpha\in\operatorname{\mathbf{\underline{M}Cor}}^{{\operatorname{fin}}}(\mathcal{Y},\mathcal{X})$,
the image of $\alpha^{*}:F(\mathcal{X})\to F(\mathcal{Y})$ is contained in
$F(\mathcal{Y}_{{\operatorname{red}}})\subset F(\mathcal{Y})$, where
$\mathcal{Y}_{{\operatorname{red}}}=(Y,E_{{\operatorname{red}}})$ for
$\mathcal{Y}=(Y,E)$.

Let $\operatorname{\mathbf{\underline{M}PST}}_{\log}$ be the full subcategory
of $\operatorname{\mathbf{\underline{M}PST}}$ consisting of logarithmic
objects and put
$\operatorname{\mathbf{\underline{M}NST}}_{\log}=\operatorname{\mathbf{\underline{M}NST}}\cap\operatorname{\mathbf{\underline{M}PST}}_{\log}$.

######  Theorem 4.2.

Any $F\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$ is
logarithmic, i.e.
$\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}\subset\operatorname{\mathbf{\underline{M}NST}}_{\log}$.

We need a preliminary for the proof of the theorem.

######  Lemma 4.3.

Let $F\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$. Let
$\mathbf{A}^{n}_{K}=\operatorname{Spec}K[x_{1},\dots,x_{n}]$ be the affine
space over a function field $K$ over $k$ and
$\mathcal{X}=\operatorname{Spec}K\\{x_{1},\dots,x_{n}\\}$ be the henselization
of $\mathbf{A}^{n}_{K}$ at the origin. Let
$L_{i}=\\{x_{i}=0\\}\subset\mathbf{A}^{n}$ and
$\mathcal{L}_{i}=L_{i}\times_{\mathbf{A}^{n}_{K}}X$ for $i\in[1,n]$. For an
integer $0<r\leq n$, the natural map

$K\\{x_{r+1},\dots,x_{n}\\}[x_{1},\dots,x_{r}]\to K\\{x_{1},\dots,x_{n}\\}$

induces a map in $\operatorname{\mathbf{\underline{M}Cor}}_{ls}$:

$\rho_{r}:(\mathcal{X},\mathcal{L}_{1}+\cdots+\mathcal{L}_{r})\to(\mathbf{A}^{r}_{S},\\{x_{1}\cdots
x_{r}=0\\})\simeq(\mathbf{A}^{1},0)^{\otimes r}\otimes(S,\emptyset),$

where $S=\operatorname{Spec}K\\{x_{r+1},\dots,x_{n}\\}$. It induces

(4.3.1) $\rho_{r}^{*}:F(\mathbf{A}^{r}_{S},\\{x_{1}\cdots x_{r}=0\\})\to
F(\mathcal{X},\mathcal{L}_{1}+\cdots+\mathcal{L}_{r})$

Then $F(\mathcal{X},\mathcal{L}_{1}+\cdots+\mathcal{L}_{r})$ is generated by
the image of $\rho_{r}^{*}$ and

$F(\mathcal{X},\mathcal{L}_{1}+\cdots+\overset{\vee}{\mathcal{L}_{i}}+\cdots\mathcal{L}_{r})\;\text{
for }i=1,\dots,r.$

###### Proof.

For $\mathcal{Y}\in\operatorname{\mathbf{\underline{M}Cor}}$, let
$F^{\mathcal{Y}}\in\operatorname{\mathbf{\underline{M}PST}}$ be defined by
$F^{\mathcal{Y}}(\mathcal{Z})=F(\mathcal{Y}\otimes\mathcal{Z})$. By [2, Lem.
1.5(3)],
$F^{\mathcal{Y}}\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$
for $F\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$. We
prove the lemma by the induction on $r$. The case $r=1$ holds since by [16,
Lem. 7.1 and Lem 5.9], $\rho_{1}$ induces an isomorphism

$F^{(\mathbf{A}^{1},0)}(S)/F^{(\mathbf{A}^{1},\emptyset)}(S)\overset{\simeq}{\longrightarrow}F(\mathcal{X},\mathcal{L}_{1})/F(\mathcal{X}).$

By definition $\mathcal{L}_{1}=\operatorname{Spec}K\\{x_{2},\dots,x_{n}\\}$
and we have a map in $\operatorname{\mathbf{\underline{M}Cor}}$:

$(\mathcal{X},\mathcal{L}_{1}+\cdots+\mathcal{L}_{r})\to(\mathbf{A}^{1},0)\otimes(\mathcal{L}_{1},\mathcal{L}_{1}\cap(\mathcal{L}_{2}+\cdots+\mathcal{L}_{r}))$

induced by the natural map $K\\{x_{2},\dots,x_{n}\\}[x_{1}]\to
K\\{x_{1},\dots,x_{n}\\}$. By [16, Lem. 7.1 and Lem 5.9], it induces an
isomorphism

$F^{(\mathbf{A}^{1},0)}(\mathcal{L}_{1},E)/F^{(\mathbf{A}^{1},\emptyset)}(\mathcal{L}_{1},E)\overset{\simeq}{\longrightarrow}F(\mathcal{X},\mathcal{L}_{1}+\cdots+\mathcal{L}_{r})/F(\mathcal{X},\mathcal{L}_{2}+\cdots+\mathcal{L}_{r})$

with $E=\mathcal{L}_{1}\cap(\mathcal{L}_{2}+\cdots+\mathcal{L}_{r})$. By the
induction hypothesis, $F^{(\mathbf{A}^{1},0)}(\mathcal{L}_{1},E)$ is generated
by $F^{(\mathbf{A}^{1},0)}(\mathcal{L}_{1},E_{j})$ with
$E_{j}=\mathcal{L}_{1}\cap(\mathcal{L}_{2}\cdots+\overset{\vee}{\mathcal{L}_{j}}+\cdots\mathcal{L}_{r})$
for $j=2,\dots,r$ together with the image of the map

$(F^{(\mathbf{A}^{1},0)})^{(\mathbf{A}^{1},0)^{\otimes
r-1}}(S)=F^{(\mathbf{A}^{1},0)^{\otimes r}}(S)\to
F^{(\mathbf{A}^{1},0)}(\mathcal{L}_{1},E)$

induced by

$(\mathcal{L}_{1},E)\to(\mathbf{A}^{r-1}_{S},\\{x_{2}\cdots
x_{r}=0\\})\simeq(\mathbf{A}^{1},0)^{\otimes r-1}\otimes(S,\emptyset)$

coming from the map $K\\{x_{r+1},\dots,x_{n}\\}[x_{2},\dots,x_{r}]\to
K\\{x_{2},\dots,x_{d}\\}$. This proves the lemma. ∎

Proof of Theorem 4.2 : By Corollary 2.6(3), we may assume
$F=\underline{\omega}^{\operatorname{\mathbf{CI}}}G$ for
$G\in{\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}$. Take
$\mathcal{X}=(X,D),\mathcal{Y}=(Y,E)\in\operatorname{\mathbf{\underline{M}Cor}}_{ls}$
with $\mathcal{X}$ reduced and let
$\alpha\in\operatorname{\mathbf{\underline{M}Cor}}^{{\operatorname{fin}}}(\mathcal{Y},\mathcal{X})$
be an elementary correspondence. We need to show that
$\alpha^{*}(F(\mathcal{X}))\subset F(\mathcal{Y}_{{\operatorname{red}}})$. The
question is Nisnevich local over $X$ and $Y$. Hence we may assume
$(X,D)=(\mathcal{X},\mathcal{L}_{1}+\cdots+\mathcal{L}_{r})\in\operatorname{\mathbf{\underline{M}Cor}}^{\rm
pro}$ under the notation from Lemma 4.3. If $r=0$, we have
$\alpha\in\operatorname{\mathbf{\underline{M}Cor}}((Y,\emptyset),(X,\emptyset))$
by the assumption
$\alpha\in\operatorname{\mathbf{\underline{M}Cor}}^{{\operatorname{fin}}}(\mathcal{Y},\mathcal{X})$
so that

$\alpha^{*}(F(\mathcal{X}))=\alpha^{*}(F(X,\emptyset))\subset
F(Y,\emptyset)\subset F(\mathcal{Y}_{{\operatorname{red}}}).$

Assume $r>0$ and proceed by the induction on $r$. By Lemma 4.3, we may assume
then

$(X,D)=\mathcal{M}:=(\mathbf{A}^{1},0)^{\otimes r}\otimes(S,\emptyset)\;\text{
for }S\in\operatorname{\mathbf{Sm}}.$

On the other hand, by Corollary 2.5, we have an exact sequence

$0\to F(Y,E_{{\operatorname{red}}})\to
F(Y-E_{{\operatorname{red}}},\emptyset)\to\underset{\xi\in{E}^{(0)}}{\bigoplus}\frac{F(Y^{h}_{|\xi}-\xi,\emptyset)}{F(Y^{h}_{|\xi},\xi)}.$

Hence we may replace $Y$ with its Nisnevich neighborhood of a generic point
$\xi$ of $E$. Using the assumption that $k$ is perfect, we may then assume the
following condition $(\spadesuit)$. Recall that $\alpha$ is by definition an
integral closed subschem of $(Y-E)\times(X-D)$ finite surjective over $Y-E$
and its closure $\overline{\alpha}$ in $Y\times X$ is finite surjective over
$Y$.

  1. $(\spadesuit)$

$X$ and $Y$ are irreducible and $\alpha$ is smooth. The normalization
$Y^{\prime}$ of $\overline{\alpha}$ is smooth and
$E^{\prime}:=E\times_{Y}Y^{\prime}\subset Y^{\prime}$ is irreducible and
$E^{\prime}_{{\operatorname{red}}}$ is smooth.

Let $g:Y^{\prime}\to Y\;\text{ and }\;f:Y^{\prime}\to X$ be the induced maps.
We have $E^{\prime}=g^{*}E\geq f^{*}D$ as Cartier divisors on $Y^{\prime}$ by
the modulus condition for $\alpha$. Hence these maps induce

$F(X,D)\overset{f^{*}}{\longrightarrow}F(Y^{\prime},E^{\prime})\overset{g_{*}}{\longrightarrow}F(Y,E).$

We claim that $\alpha^{*}:F(X,D)\to F(Y,E)$ agrees with this map. Indeed, this
follows from the equality

$\Gamma_{f}\circ^{t}\Gamma_{g}=\alpha\in\operatorname{\mathbf{Cor}}(Y-E,X-D),$

where
${}^{t}\Gamma_{g}\in\operatorname{\mathbf{Cor}}(Y-E,Y^{\prime}-E^{\prime})$ is
the transpose of the graph of $g$ and
$\Gamma_{f}\in\operatorname{\mathbf{Cor}}(Y^{\prime}-E^{\prime},X-D)$ is the
graph of $f$. By definition this follows from the equality

${}^{t}\Gamma_{g}\times_{Y^{\prime}-E^{\prime}}\Gamma_{f}=\alpha\subset(Y-E)\times(X-D)$

which one can check easily noting $Y^{\prime}\to\overline{\alpha}$ is an
isomorphism over $\alpha$ assumed to be smooth. Then we get a commutative
diagram

$\textstyle{F(Y^{\prime},E^{\prime}_{{\operatorname{red}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hookrightarrow}$$\textstyle{F(Y^{\prime},E_{{\operatorname{red}}}\times_{Y}Y^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{*}}$$\scriptstyle{\hookrightarrow}$$\textstyle{F(Y,E_{{\operatorname{red}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hookrightarrow}$$\textstyle{F(X,D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\textstyle{F(Y^{\prime},E^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{*}}$$\textstyle{F(Y,E)}$

where the top inclusion comes from
$E_{{\operatorname{red}}}\times_{Y}Y^{\prime}\geq
E^{\prime}_{{\operatorname{red}}}$ as Cartier divisors on $Y^{\prime}$ thanks
to the sempurity of $F$ (cf. §1(16)). Hence it suffices to show
$f^{*}(F(X,D))\subset F(Y^{\prime},E^{\prime}_{{\operatorname{red}}})$. By
replacing $(Y,E)$ with $(Y^{\prime},E^{\prime})$, we may now assume that
$\alpha$ is induced by a morphism $f:Y\to X=\mathbf{A}^{r}\times S$. Then
$\alpha$ factors in $\operatorname{\mathbf{\underline{M}Cor}}$ as

$(Y,E)\to(\mathbf{A}^{1},0)^{\otimes
r}\otimes(Y,\emptyset)\to(\mathbf{A}^{1},0)^{\otimes r}\otimes(S,\emptyset).$

where the first map is induced by the map

$i=(pr_{\mathbf{A}^{r}}\circ f,id_{Y}):Y\to\mathbf{A}^{r}\times Y,$

and the second induced by

$id_{\mathbf{A}^{r}}\times(pr_{S}\circ f):\mathbf{A}^{r}\times
Y\to\mathbf{A}^{r}\times S.$

Note that $i$ is a section of the projection $\mathbf{A}^{r}\times Y\to Y$.
Thus we are reduced to showing $i^{*}(F((\mathbf{A}^{1},0)^{\otimes
r}\otimes(Y,\emptyset))\subset F(Y,E_{{\operatorname{red}}})$. By Proposition
3.2 this follows from the following.

######  Claim 4.3.1.

Assume $F=\underline{\omega}^{\operatorname{\mathbf{CI}}}G$ for some
$G\in{\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}$. Take $a\in
F((\mathbf{A}^{1},0)^{\otimes r}\otimes(Y,\emptyset))$. Let $K$ be a function
field over $k$. After replacing $Y$ by an open subset containing $\xi$, we
have

$(i^{*}(a)_{K},\gamma)_{Y_{K}/K,\delta}=0\;\text{ for }\forall\gamma\in
K^{M}_{e}(\mathcal{O}^{h}_{Y_{K},\delta},\mathfrak{m}_{\delta})$

and for any $\delta=(\xi,\delta_{1},\dots,\delta_{e-1})\in\mathrm{mc}(Y_{K})$,
where $e=\dim(Y)$ and $\xi\in E$ is the generic point, and

$(-,-)_{Y_{K}/K,\delta}:F(K(Y))\otimes K^{M}_{d}(K^{h}_{Y_{K},\delta})\to
F(K)$

is from (3.1.2).

###### Proof.

Write

$\mathbf{A}^{r}\times
Y=\operatorname{Spec}A[x_{1},\dots,x_{r}],\;(\mathbf{A}^{1},0)^{\otimes
r}\otimes(Y,\emptyset)=(\mathbf{A}^{r}_{Y},\\{x_{1}\cdots x_{r}=0\\}).$

After replacing $Y$ by an open subset containing $\xi$, we can write

$i(Y)=\underset{1\leq i\leq r}{\bigcap}\\{x_{i}-u_{i}\pi^{m_{i}}=0\\}\;\text{
with }m_{i}\in\mathbb{Z}_{\geq 0},\;u_{i}\in A^{\times}.$

Let $\delta=(\xi,\delta_{1},\dots,\delta_{e-1})$ be as in the claim and put
$\delta^{\prime}=(\delta_{1},\dots,\delta_{e-1})\in\mathrm{mc}((E_{{\operatorname{red}}})_{K})$.
Let $X_{K}=\mathbf{A}^{r}\times Y_{K}$ and $z_{j}$ for $1\leq j\leq r$ be the
generic point of

$Z_{j}=\underset{1\leq i\leq j}{\bigcap}\\{x_{i}-u_{i}\pi^{m_{i}}=0\\}\subset
X_{K},$

and $w_{j}$ for $1\leq j\leq r$ be the generic point of

$W_{j}=\\{\pi=x_{1}=\cdots=x_{j}=0\\}=F\cap Z_{j}\;\text{ with
}F=\\{\pi=0\\}\subset X_{K}.$

The section $i$ induces isomorphisms

(4.3.2) $Y_{K}\simeq Z_{r}\;\text{ and }\;(E_{{\operatorname{red}}})_{K}\simeq
W_{r}.$

Let
$\sigma=(\eta_{1},w_{1},\dots,w_{r},i(\delta^{\prime}))\in\mathrm{mc}(X_{K})$,
where $\eta_{1}$ is the generic point of $D_{1}=\\{x_{1}=0\\}\subset X_{K}$
and $i(\delta^{\prime})\in\mathrm{mc}(W_{r})$ is the image of
$\delta^{\prime}$ under (4.3.2). Let $\tilde{\sigma}$ be the chain
$(w_{r},i(\delta^{\prime}))$ in $X_{K}$. By the definition of henselization
along chains, the projection $X=\mathbf{A}^{r}\times Y\to Y$ induces
$\iota:\mathcal{O}^{h}_{Y_{K},\delta}\to\mathcal{O}^{h}_{X_{K},\tilde{\sigma}}$.
Take any $\gamma\in
K^{M}_{e}(\mathcal{O}^{h}_{Y_{K},\delta},\mathfrak{m}_{\delta})$ and put

(4.3.3)
$\beta=\\{\iota(\gamma),\frac{u_{1}\pi^{m_{1}}-x_{1}}{u_{1}\pi^{m_{1}}},\dots,\frac{u_{r}\pi^{m_{r}}-x_{r}}{u_{r}\pi^{m_{r}}}\\}\in
K^{M}_{d}(K^{h}_{X_{K},\tilde{\sigma}})\;(d=e+r),$

For $a\in F((\mathbf{A}^{1},0)^{\otimes r}\otimes(Y,\emptyset))$ and its
restriction $a_{K}\in F((\mathbf{A}^{1},0)^{\otimes
r}\otimes(Y_{K},\emptyset))$, we have666By abuse of notation, for
$\rho\in\mathrm{mc}(X_{K})$ containing $\tilde{\sigma}$ as a sub-chain,
$\beta$ in $(a_{K},\beta)_{X_{K}/K,\rho}$ denotes the image of $\beta$ from
(4.3.3) under the natural map $K^{M}_{d}(K^{h}_{X_{K},\tilde{\sigma}})\to
K^{M}_{d}(K^{h}_{X_{K},\rho})$. The same convention for $\beta_{1}$ and
$\beta_{2}$ below.

$\displaystyle 0=(a_{K},\beta)_{X_{K}/K,\sigma}$
$\displaystyle=-\underset{\begin{subarray}{c}\tau\in X_{K}^{(1)}\\\
\tau>w_{1},\tau\not=\eta_{1}\end{subarray}}{\sum}(a_{K},\beta)_{X_{K}/K,(\tau,w_{1},\dots,w_{r},i(\delta^{\prime}))}$
$\displaystyle=-(a_{K},\beta)_{X_{K}/K,(z_{1},w_{1},\dots,w_{r},i(\delta^{\prime}))}$
$\displaystyle=\pm((a_{K})_{|Z_{1}},\beta_{1})_{Z_{1}/K,(w_{1},\dots,w_{r},i(\delta^{\prime}))},$
$\beta_{1}=\\{\iota_{1}(\gamma),\frac{u_{2}\pi^{m_{2}}-x_{2}}{u_{2}\pi^{m_{2}}},\dots,\frac{u_{r}\pi^{m_{r}}-x_{r}}{u_{r}\pi^{m_{r}}}\\}\in
K^{M}_{d-1}(K^{h}_{Z_{1},\tilde{\sigma}_{1}}))$

where $\tilde{\sigma}_{1}$ is the chain $(w_{r},i(\delta^{\prime}))$ in
$Z_{1}$ and
$\iota_{1}:\mathcal{O}^{h}_{Y_{K},\delta}\to\mathcal{O}^{h}_{Z_{1},\tilde{\sigma}_{1}}$
is induced by $Z_{1}\hookrightarrow X_{K}\to Y_{K}$. The first equality
follows from §3 (HS3) applied to $D_{1}\subset X_{K}$ noting that $\beta$ lies
in $K^{M}_{d}(\mathcal{O}^{h}_{X_{K},\sigma},\mathfrak{m}_{\sigma})$ since
$(u_{1}\pi^{m_{1}}-x_{1})/u_{1}\pi^{m_{1}}\in
1+x_{1}\mathcal{O}_{X_{K},\eta_{1}}$. The second follows from (HS4) applied to
$x=(w_{1},\dots.w_{r},i(\delta^{\prime}))\in Q_{0}(X_{K})$. The third equality
holds since $z_{1}$ is the unique $\tau\in X_{K}^{(1)}-\\{\eta_{1}\\}$ such
that $\tau>w_{1}$ and
$(a_{K},\beta)_{X_{K}/K,(\tau,w_{1},\dots,w_{r},i(\delta^{\prime}))}$ may not
vanish, which follows from (HS2) noting $\iota(\gamma)_{|F}=0$. Finally the
last equality follows from (HS2). We further get

$\displaystyle
0=((a_{K})_{|Z_{1}},\beta_{1})_{Z_{1}/K,(w_{1},w_{2},\dots,w_{r},i(\delta^{\prime}))}$
$\displaystyle=-\underset{\begin{subarray}{c}\tau\in Z_{1}^{(1)}\\\
\tau>w_{2},\tau\not=w_{1}\end{subarray}}{\sum}((a_{K})_{|Z_{1}},\beta_{1})_{Z_{1}/K,(\tau,w_{2},\dots,w_{r},i(\delta^{\prime}))}$
$\displaystyle=-((a_{K})_{|Z_{1}},\beta_{1})_{Z_{1}/K,(z_{2},w_{2},\dots,w_{r},i(\delta^{\prime}))}$
$\displaystyle=\pm((a_{K})_{|Z_{2}},\beta_{2})_{Z_{2}/K,(w_{2},\dots,w_{r},i(\delta^{\prime}))},$
$\beta_{2}=\\{\iota_{2}(\gamma),\frac{u_{3}\pi^{m_{3}}-x_{3}}{u_{3}\pi^{m_{3}}},\dots,\frac{u_{r}\pi^{m_{r}}-x_{r}}{u_{r}\pi^{m_{r}}}\\}\in
K^{M}_{d-1}(K^{h}_{Z_{2},\tilde{\sigma}_{2}})),$

where $\tilde{\sigma}_{2}$ is the chain $(w_{r},i(\delta^{\prime}))$ in
$Z_{2}$ and
$\iota_{2}:\mathcal{O}^{h}_{Y_{K},\delta}\to\mathcal{O}^{h}_{Z_{2},\tilde{\sigma}_{2}}$
is induced by $Z_{2}\hookrightarrow X_{K}\to Y_{K}$. The above equalities hold
by the same arguments as above except that for the third equality, there are a
priori two $\tau\in Z_{1}^{(1)}-\\{w_{1}\\}$ with $\tau>w_{2}$ for which
$((a_{K})_{|Z_{1}},\beta_{1})_{Z_{1}/K,(\tau,w_{2},\dots,w_{r},i(\delta^{\prime}))}$
may not vanish. One is $z_{2}$ and another is the generic point $\eta_{2}$ of
$Z_{1}\cap D_{2}$ with $D_{2}=\\{x_{2}=0\\}\subset X_{K}$, but
$((a_{K})_{|Z_{1}},\beta_{1})_{Z_{1}/K,(\eta_{2},w_{2},\dots,w_{r},i(\delta^{\prime}))}=0$.
Indeed, $(a_{K})_{|Z_{1}}\in
F(\operatorname{Spec}(\mathcal{O}_{Z_{1},\eta_{2}}),\eta_{2})$ since $Z_{1}$
and $D_{2}$ intersect transversally in $X_{K}$. Hence the vanishing follows
from (HS3) applied to $Z_{1}\cap D_{2}\subset Z_{1}$ noting $\beta_{1}\in
K^{M}_{d}(\mathcal{O}_{Z_{1},(\eta_{2},\tilde{\sigma}_{1})},\mathfrak{m}_{\eta_{2}})$.
Repeating the same arguments, we finally get

$0=((a_{K})_{|Z_{r}},\iota_{r}(\gamma))_{Z_{r}/K,(w_{r},i(\delta^{\prime}))}=((a_{K})_{|Y_{K}},\gamma)_{Y_{K}/K,\delta},$

where
$\iota_{r}:\mathcal{O}^{h}_{Y_{K},\delta}\to\mathcal{O}^{h}_{Z_{r},(w_{r},i(\delta^{\prime}))}$
is induced by $Z_{r}\hookrightarrow X_{K}\to Y_{K}$ and the second equality
follows from (4.3.2). This completes the proof of the claim and Theorem 4.2. ∎

######  Definition 4.4.

For $F\in\operatorname{\mathbf{\underline{M}NST}}_{\log}$ and an integer
$i\geq 0$, consider the association

$H_{\log}^{i}(-,F):\operatorname{\mathbf{\underline{M}Cor}}^{{\operatorname{fin}}}_{ls}\to\operatorname{\mathbf{Ab}}\;;\;(X,D)\to
H^{i}(X_{\operatorname{Nis}},F_{(X,D_{{\operatorname{red}}})}).$

By the definition this gives a presheaf on
$\operatorname{\mathbf{\underline{M}Cor}}^{{\operatorname{fin}}}_{ls}$, which
we call _the $i$-th logarithmic cohomology with coefficient $F$_.

##  5\. Invariance of logarithmic cohomology under blowups

Let the notation be as in §4. Let $\Lambda^{{\operatorname{fin}}}_{ls}$ be the
subcategory of
$\operatorname{\mathbf{\underline{M}Cor}}^{{\operatorname{fin}}}_{ls}$ whose
objects are the same as
$\operatorname{\mathbf{\underline{M}Cor}}^{{\operatorname{fin}}}_{ls}$ and
whose morphisms are those $\rho:(Y,E)\to(X,D)$ where $E=\rho^{*}D$ and $\rho$
are induced by blowups of $X$ in smooth centers $Z\subset D$ which are normal
crossing to $D$ in the following sense: For any point $x$ of $D$, there exists
a system $z_{1},\cdots,z_{d}$ of regular parameters of $X$ at $x$ satisfying
the conditions:

  * •

Locally at $x$, $Z=\\{z_{1}=\dots=z_{r}=0\\}$ with
$r={\operatorname{codim}}_{X}Z$.

  * •

Locally at $x$, $|D|=\\{\prod_{j\in J}z_{j}=0\\}$ for some
$J\subset\\{1,\dots,r\\}$.

######  Theorem 5.1.

For $F\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$ and
$\rho:\mathcal{Y}\to\mathcal{X}$ in $\Lambda^{{\operatorname{fin}}}_{ls}$, we
have

(5.1.1) $\rho^{*}:H_{\log}^{i}(\mathcal{X},F)\cong
H_{\log}^{i}(\mathcal{Y},F)\;\text{ for }\forall i\geq 0.$

###### Proof.

Writing $\mathcal{Y}=(Y,E)$ and $\mathcal{X}=(X,D)$, $\rho$ is induced by a
blowup $\rho:Y\to X$ in a smooth center $Z\subset D$ normal crossing to $D$.
First we prove the theorem in case $i=0$. We may assume that $D$ is reduced
and $E=\rho^{*}D$. Then $\rho$ is invertible in
$\operatorname{\mathbf{\underline{M}Cor}}$ so that
$\rho^{*}:F(\mathcal{X})\cong F(\mathcal{Y})$. Since this factors through
$F(Y,E_{{\operatorname{red}}})$, we get (5.1.1) for $i=0$.

To show (5.1.1) for $i>0$, it now suffices to prove
$R^{i}\rho_{*}F_{(Y,E_{{\operatorname{red}}})}=0$. By [8, Lem. 9], Nisnevich
locally around a point of $Z$, $(X,D)$ is isomorphic to

$(\mathbf{A}^{c},L_{1}+\cdots+L_{r})\otimes\mathcal{W}\;\text{ with
}\mathcal{W}=(W,W^{\infty})\in\operatorname{\mathbf{\underline{M}Cor}}_{ls},$

where $\mathbf{A}^{c}=\operatorname{Spec}k[t_{1},\dots,t_{c}]$ with
$c={\operatorname{codim}}_{z}(Z,X)$ and $L_{i}=V(t_{i})$ for $i=1,\dots,r$
with $1\leq r\leq c$, and $Z$ corresponds to $0\times W$. Hence the theorem
follows from the following proposition. ∎

######  Proposition 5.2.

Let $F\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$ and
$\mathcal{W}=(W,W^{\infty})\in\operatorname{\mathbf{\underline{M}Cor}}_{ls}$.
Let $\mathbf{A}^{n}=\operatorname{Spec}k[t_{1},\dots,t_{n}]$ and put
$L_{i}=V(t_{i})$ for $1\leq i\leq n$. Let $\rho:Y\to\mathbf{A}^{n}$ be the
blow-up at the origin $0\in\mathbf{A}^{n}$ and $\tilde{L}_{i}\subset Y$ be the
strict transforms of $L_{i}$ for $i=1,\dots,r$ and $E=\rho^{-1}(0)\subset Y$.
For any $1\leq r\leq n$, we have

(5.2.1)
$R^{i}\rho_{W*}F_{(Y,\tilde{L}_{1}+\cdots+\tilde{L}_{r}+E)\otimes\mathcal{W}}=0\;\text{
for }i\geq 1,$

where $\rho_{W}:=\rho\times{\operatorname{id}}_{W}:Y\times
W\to\mathbf{A}^{2}\times W$.

######  Lemma 5.3.

Proposition 5.2 holds for $n=2$.

###### Proof.

We can assume $W$ is henselian local. The case $r=1$ is proved in [2, Lem.
2.13] and we show the case $r=2$.777The following argument is adopted from [2,
Lem. 2.13], but the present case is easier. Put $D=L_{1}+L_{2}$. By the case
$i=0$ of Theorem 5.1, we get

(5.3.1)
$F_{(\mathbf{A}^{2},D)\otimes\mathcal{W}}\cong\rho_{W*}F_{(Y,\tilde{L}_{1}+\tilde{L}_{2}+E)\otimes\mathcal{W}}.$

Set

$\mathcal{F}:=F_{(Y,\tilde{L}_{1}+\tilde{L}_{2}+E)\otimes\mathcal{W}}.$

Since $R^{i}\rho_{W*}\mathcal{F}$ for $i\geq 1$ is supported in $0\times W$ we
have

$R^{i}\rho_{W*}\mathcal{F}=0\Longleftrightarrow
H^{0}(\mathbf{A}^{2}_{W},R^{i}\rho_{W*}\mathcal{F})=0,$

where $\mathbf{A}^{2}_{W}=\mathbf{A}^{2}\times W$, and

$H^{j}(\mathbf{A}^{2}_{W},R^{i}\rho_{W*}\mathcal{F})=0,\quad\text{for all
}i,j\geq 1.$

By (5.3.1) and [2, Lem. 2.11]

$H^{i}(\mathbf{A}^{2}_{W},\rho_{W*}\mathcal{F})=H^{i}(\mathbf{A}^{2}_{W},F_{(\mathbf{A}^{2},D)\otimes\mathcal{W}})=0.$

Thus the Leray spectral sequence yields

$H^{0}(\mathbf{A}_{W}^{2},R^{i}\rho_{W*}\mathcal{F})=H^{i}(Y\times
W,\mathcal{F}),\quad i\geq 0,$

and we have to show, that this group vanishes for $i\geq 1$. We can write

$\mathbf{A}^{2}=\operatorname{Spec}k[x,y]\;\text{ and
}\;L_{1}=V(x),\;L_{2}=V(y)\subset\mathbf{A}^{2}.$

Then we have

$Y=\operatorname{Proj}k[x,y][S,T]/(xT-
yS)\subset\mathbf{A}^{2}\times\mathbf{P}^{1}.$

Denote by

$\pi_{0}:Y\hookrightarrow\mathbf{A}^{2}\times\mathbf{P}^{1}\to\mathbf{P}^{1}=\operatorname{Proj}k[S,T]$

the morphism induced by projection and let $\pi:Y\times
W\to\mathbf{P}^{1}_{W}$ be its base change. Then $\pi_{0}$ induces an
isomorphism $E\simeq\mathbf{P}^{1}$, and we have

(5.3.2)
$\tilde{L}_{1}=\pi_{0}^{-1}(0),\;\;\tilde{L}_{2}=\pi_{0}^{-1}(\infty).$

Set $s=S/T=x/y$ and write

$\mathbf{P}^{1}\setminus\\{\infty\\}=\mathbf{A}^{1}_{s}:=\operatorname{Spec}k[s],\quad\mathbf{P}^{1}\setminus\\{0\\}=\operatorname{Spec}k[\tfrac{1}{s}].$

Set $U:=\mathbf{A}^{1}_{s}\times W$ and
$V:=(\mathbf{P}^{1}\setminus\\{0\\})\times W$ and

$\mathcal{U}:=(\mathbf{A}^{1}_{s},0)\otimes\mathcal{W},\quad\mathcal{V}:=(\mathbf{P}^{1}\setminus\\{0\\},\infty)\otimes\mathcal{W}.$

We have

$\pi^{-1}(U)=\mathbf{A}^{1}_{y}\times
U,\quad\pi^{-1}(V)=\mathbf{A}^{1}_{x}\times V,$

and the restriction of $\pi$ to these open subsets is given by projection.
Furthermore, $E\times W\subset Y$ is defined by $y=0$ on $\pi^{-1}(U)$ and by
$x=0$ on $\pi^{-1}(V)$. In view of (5.3.2), we have

(5.3.3)
$\mathcal{F}_{|\pi^{-1}(U)}=F_{(\mathbf{A}^{1}_{y},0)\otimes\mathcal{U}},\quad\mathcal{F}_{|\pi^{-1}(V)}=F_{(\mathbf{A}^{1}_{x},0)\otimes\mathcal{V}}.$

Thus [2, Lem. 2.11] yields

$R^{j}\pi_{*}\mathcal{F}=0\;\text{ for }j\geq 1,$

and it remains to show

(5.3.4) $H^{i}(\mathbf{P}^{1}_{W},\pi_{*}\mathcal{F})=0\;\text{ for }i\geq 1.$

where $\mathbf{P}^{1}_{W}=\mathbf{P}^{1}\times W$. For this consider the map

$a_{0}:Y\to\mathbf{A}^{1}_{x}\times\mathbf{P}^{1}$

which is the closed immersion
$Y\hookrightarrow\mathbf{A}^{2}\times\mathbf{P}^{1}$ followed by the
projection $\mathbf{A}^{2}\to\mathbf{A}^{1}_{x}$. Let $a:Y\times
W\to\mathbf{A}^{1}_{x}\times\mathbf{P}^{1}\times W$ be its base change. In
view of (5.3.3), the map $a$ induces a morphism in
$\operatorname{\mathbf{\underline{M}Cor}}$:

$\alpha:(Y,\tilde{L}_{1}+\tilde{L}_{2}+E)\otimes\mathcal{W}\to(\mathbf{A}^{1}_{x},0)\otimes(\mathbf{P}^{1},\infty)\otimes\mathcal{W},$

which is an isomorphism over
$(\mathbf{A}^{1}_{x},0)\otimes(\mathbf{P}^{1}\backslash\\{0\\},\infty)\otimes\mathcal{W}$.
Setting

$F_{1}:=\operatorname{\underline{Hom}}({\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathbf{A}^{1}_{x},0),F)\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}},$

it induces a map of Nisnevich sheaves on $\mathbf{P}^{1}_{W}$:

$\pi_{*}(\alpha^{*}):F_{1,(\mathbf{P}^{1},\infty)\otimes\mathcal{W}}\to\pi_{*}\mathcal{F},$

which becomes an isomorphism over $(\mathbf{P}^{1}-\\{0\\})\times W$. Hence
(5.3.4) follows from

$H^{i}(\mathbf{P}^{1}_{W},F_{1,(\mathbf{P}^{1},\infty)\otimes\mathcal{W}})=0\;\text{
for }i\geq 1,$

which follows from [16, Th. 0.6]. ∎

######  Lemma 5.4.

Let $N>2$ be an integer and assume that Proposition 5.2 holds for $n<N$. Let
$(X,D)\in\operatorname{\mathbf{\underline{M}Cor}}_{ls}$ and $Z\subset X$ be a
smooth integral closed subscheme with $2\leq{\operatorname{codim}}(Z,Y)=:c<N$.
Assume

$D=D_{1}+\cdots+D_{r}+D^{\prime}\;\text{ with }r\leq c,$

where $D_{1},\dots,D_{r}$ are distinct and reduced irreducible components of
$D$ containing $Z$ and $D^{\prime}$ is an effective divisor on $X$ such that
none of the component of $D^{\prime}$ contains $Z$ and $Z$ is transversal to
$|D^{\prime}|$. Let $\rho:Y\to X$ be the blow-up of $X$ in $Z$ and
$\tilde{D}_{i},\tilde{D}^{\prime}\subset Y$ be the strict transforms of
$D_{i}$ and $D^{\prime}$ respectively and $E_{Z}=\rho^{-1}(Z)$. Then, for all
$\mathcal{W}=(W,W^{\infty})\in\operatorname{\mathbf{\underline{M}Cor}}_{ls}$,

$R^{i}\rho_{W*}F_{(Y,\tilde{D}_{1}+\cdots+\tilde{D}_{r}+E_{Z}+\tilde{D}^{\prime})\otimes\mathcal{W}}=0\;\text{
for }i\geq 1,$

where $\rho_{W}:Y\times W\to X\times W$ denotes the base change of $\rho$.

###### Proof.

888The proof is adopted from [2, Lem. 2.14].

The question is Nisnevich local around the points in $Z\times W$. Let $z\in
Z\times W$ be a point and set $A:=\mathcal{O}_{X\times W,z}^{h}$. For
$V\subset Y\times W$ we denote by $V_{(z)}:=V\times_{X\times
W}\operatorname{Spec}A$. By assumption we find a system of local parameters
$t_{1},\ldots,t_{m}$ of $A$, such that

$(D_{i}\times W)_{(z)}=V(t_{i})\;\text{ for }1\leq i\leq r,\;(Z\times
W)_{(z)}=V(t_{1},\ldots,t_{c}),$ $(D^{\prime}\times
W)_{(z)}=V(t_{c+1}^{e_{c+1}}\cdots t_{m_{0}}^{e_{m_{0}}})\;\text{ with
}c+1\leq m_{0}\leq m,$ $(X\times
W^{\infty})_{(z)}=V(t_{m_{0}+1}^{e_{m_{0}+1}}\cdots
t_{m_{1}}^{e_{m_{1}}})\;\text{ with }m_{0}\leq m_{1}\leq m.$

Letting $K$ be the residue field of $A$, we can choose a ring homomorphism
$K\hookrightarrow A$ which is a section of $A\to K$. Then we obtain an
isomorphism

$K\\{t_{1},\ldots,t_{m}\\}\xrightarrow{\simeq}A.$

Let $\rho_{1}:\widetilde{\mathbf{A}^{c}}\to\mathbf{A}^{c}$ be the blow-up in
$0$. By the above

$\rho_{W}:(Y,\tilde{D}_{1}+\cdots+\tilde{D}_{r}+E_{Z}+\tilde{D}^{\prime})\otimes\mathcal{W}\to(X,D)\otimes\mathcal{W}$

is Nisnevich locally around $z$ isomorphic over $k$ to the morphism

$(\widetilde{\mathbf{A}^{c}},\tilde{L}_{1}+\cdots+\tilde{L}_{r}+E)\otimes\mathcal{W}^{\prime}\to(\mathbf{A}^{c},L_{1}+\cdots+L_{r})\otimes\mathcal{W}^{\prime},$
$(\mathcal{W}^{\prime}=(\mathbf{A}_{K}^{m-c},(\prod_{i=c+1}^{m_{1}}t_{i}^{e_{i}})))$

induced by a map
$(\widetilde{\mathbf{A}^{c}},\tilde{L}_{1}+\cdots+\tilde{L}_{r}+E)\to(\mathbf{A}^{c},L_{1}+\cdots+L_{r})$
as in Proposition 5.2. Hence the statement follows from the proposition for
$n=c<N$. ∎

###### Proof of Proposition 5.2..

The proof is by induction on $n\geq 2$. The case $n=2$ follows from Lemma 5.3.
Assume $n>2$ and the theorem is proven for $\mathbf{A}^{m}$ with $m<n$. In
case $r=1$, Proposition 5.2 is proved in [2, Th. 2.12]. Assume $r\geq 2$. Let
$Z:=L_{1}\cap L_{2}\subset\mathbf{A}^{n}$ and $\tilde{Z}\subset Y$ be the
strict transform of $Z$. Denote by $\rho^{\prime}:Y^{\prime}\to Y$ the blow-up
of $Y$ in $\tilde{Z}$ and $\tilde{L}_{i}^{\prime},E^{\prime}\subset
Y^{\prime}$ be the strict transforms of $\tilde{L},E$ respectively and
$E^{\prime\prime}=(\rho^{\prime})^{-1}(\tilde{Z})$. Note that
$\tilde{Z}=\tilde{L}_{1}\cap\tilde{L}_{2}$ intersecting transversally with
$\tilde{L}_{3}+\cdots+\tilde{L}_{r}+E$ and
${\operatorname{codim}}(\tilde{Z},Y)=2$. Hence, by Lemma 5.4

$R^{i}\rho^{\prime}_{W*}F_{(Y^{\prime},\tilde{L}_{1}^{\prime}+\cdots+\tilde{L}_{r}^{\prime}+E^{\prime}+E^{\prime\prime})\otimes\mathcal{W}}=0\;\text{
for }i\geq 1.$

Since Theorem 5.1 has been proved for $i=0$, we have

$\rho^{\prime}_{*}F_{(Y^{\prime},\tilde{L}_{1}^{\prime}+\cdots+\tilde{L}_{r}^{\prime}+E^{\prime}+E^{\prime\prime})\otimes\mathcal{W}}=F_{(Y,\tilde{L}_{1}+\cdots+\tilde{L}_{r}+E)\otimes\mathcal{W}}.$

Hence we obtain

(5.4.1)
$R^{i}\rho_{W*}F_{(Y,\tilde{L}_{1}+\cdots+\tilde{L}_{r}+E)\otimes\mathcal{W}}=R^{i}(\rho\rho^{\prime})_{W*}F_{(Y^{\prime},\tilde{L}_{1}^{\prime}+\cdots+\tilde{L}_{r}^{\prime}+\tilde{E}+E^{\prime})\otimes\mathcal{W}}.$

Denote by $\sigma:\hat{Y}\to\mathbf{A}^{n}$ the blow-up in $Z$ and
$\hat{L}_{i}\subset\hat{Y}$ be the strict transform of $L_{i}$ and
$\Xi=\sigma^{-1}(Z)$. By Lemma 5.4 we get

(5.4.2)
$R^{i}\sigma_{W*}F_{(\hat{Y},\hat{L}_{1}+\cdots+\hat{L}_{r}+\Xi)\otimes\mathcal{W}}=0\;\text{
for }i\geq 1.$

Denote by $\sigma^{\prime}:\hat{Y}^{\prime}\to\hat{Y}$ the blow-up in
$\hat{Z}=\sigma^{-1}(0)\subset\Xi$ and
$\hat{L}^{\prime}_{i},\;\Xi^{\prime}\subset\hat{Y}^{\prime}$ be the strict
transforms of $\hat{L}_{i},\;\Xi$ respectively and
$\Xi^{\prime\prime}=\sigma^{\prime-1}(\hat{Z})$. Note that
$\tilde{Z}\subset\hat{L}_{3}\cap\cdots\cap\hat{L}_{n}\cap\Xi$ and
${\operatorname{codim}}(\tilde{Z},\hat{Y})=n-1$ and $\tilde{Z}$ intersects
transversally with $\hat{L}_{1}+\hat{L}_{2}$. Thus by Lemma 5.4 and the case
$i=0$ of Theorem 5.1, we obtain

(5.4.3)
$R\sigma^{\prime}_{W*}F_{(\hat{Y}^{\prime},\hat{L}_{1}^{\prime}+\cdots+\hat{L}_{r}^{\prime}+\Xi^{\prime}+\Xi^{\prime\prime})\otimes\mathcal{W}}=F_{(\hat{Y},\hat{L}_{1}+\cdots+\hat{L}_{r}+\Xi)\otimes\mathcal{W}}.$

Finally, by [2, Lem. 2.15], there is an isomorphism of $\mathbf{A}^{n}\times
W$-schemes

(5.4.4)
$(\hat{Y}^{\prime},\hat{L}_{1}^{\prime},\dots,\hat{L}_{r},\Xi^{\prime},\Xi^{\prime\prime})\cong(Y^{\prime},\tilde{L}_{1}^{\prime},\dots,\tilde{L}_{r}^{\prime},E^{\prime},E^{\prime\prime}).$

Altogether we obtain for $i\geq 1$

$\displaystyle
R^{i}\rho_{W*}F_{(Y,\tilde{L}_{1}+\cdots+\tilde{L}_{r}+E)\otimes\mathcal{W}}$
$\displaystyle=R^{i}(\rho\rho^{\prime})_{W*}F_{(Y^{\prime},\tilde{L}_{1}^{\prime}+\cdots+\tilde{L}_{r}^{\prime}+E^{\prime}+E^{\prime\prime})\otimes\mathcal{W}},$
$\displaystyle\text{by }\eqref{thm:blow-upAn1},$
$\displaystyle=R^{i}(\sigma\sigma^{\prime})_{W*}F_{(\hat{Y}^{\prime},\hat{L}_{1}^{\prime}+\cdots+\hat{L}_{r}^{\prime}+\Xi^{\prime}+\Xi^{\prime\prime})\otimes\mathcal{W}},$
$\displaystyle\text{by }\eqref{thm:blow-upAn4},$
$\displaystyle=R^{i}\sigma_{W*}F_{(\hat{Y},\hat{L}_{1}+\cdots+\hat{L}_{r}+\Xi)\otimes\mathcal{W}},$
$\displaystyle\text{by }\eqref{thm:blow-upAn3},$ $\displaystyle=0,$
$\displaystyle\text{by }\eqref{thm:blow-upAn2}.$

This completes the proof of the proposition. ∎

######  Remark 5.5.

For simplicity, we write

$H_{\log}^{i}(-,F)=H_{\log}^{i}(-,\underline{\omega}^{\operatorname{\mathbf{CI}}}F)\;\text{
for }F\in{\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}.$

By [13, Cor. 6.8], if ${\operatorname{ch}}(k)=0$ and $F=\Omega^{i}$, we have

$H_{\log}^{i}(-,\Omega^{i})=H^{i}(X,\Omega^{i}(\log|D|)\;\text{ for
}(X,D)\in\operatorname{\mathbf{\underline{M}Cor}}_{ls}.$

Hence $H_{\log}^{i}(-,F)$ for
$F\in{\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}$ is a generalization
of cohomology of sheaves of logarithmic differentials.

##  6\. Relation with logarithmic sheaves with transfers

In this section we use the same notations as [1].

Let ${\operatorname{l\mathbf{Sm}}}$ be the category of log smooth and
separated $\mathrm{fs}$ log schemes of finite type over the base field $k$ and
${\operatorname{\mathbf{Sm}l\mathbf{Sm}}}\subset{\operatorname{l\mathbf{Sm}}}$
be the full subcategory consisting of objects whose underlying schemes are
smooth over $k$. Let ${\operatorname{l\mathbf{Cor}}}$ be the category with the
same objects as ${\operatorname{l\mathbf{Sm}}}$ and whose morphisms are log
correspondences defined in [1, Def. 2.1.1]. Let
${\operatorname{l\mathbf{Cor}}}_{{\operatorname{\mathbf{Sm}l\mathbf{Sm}}}}\subset{\operatorname{l\mathbf{Cor}}}$
be the full subcategory consisting of all objects in
${\operatorname{\mathbf{Sm}l\mathbf{Sm}}}$.

Let ${\operatorname{\mathbf{PSh}}^{ltr}}$ be the category of additive
presheaves of abelian groups on ${\operatorname{l\mathbf{Cor}}}$ and
${\operatorname{\mathbf{Shv}}^{ltr}_{dNis}}\subset{\operatorname{\mathbf{PSh}}^{ltr}}$
be the full subcategory consisting of those $\mathcal{F}$ whose restrictions
to ${\operatorname{l\mathbf{Sm}}}$ are dividing Nisnevich sheaves (see [1,
Def. 3.1.4]). It is shown in [1, §4 and Pr. 4.6.6] that
${\operatorname{\mathbf{Shv}}^{ltr}_{dNis}}$ is a Grothendieck abelian
category and there is an equivalence of categories

(6.0.1)
${\operatorname{\mathbf{Shv}}^{ltr}_{dNis}}\simeq{\operatorname{\mathbf{Shv}}^{ltr}_{dNis}}({\operatorname{\mathbf{Sm}l\mathbf{Sm}}}),$

where the right hand side denotes the full subcategory of the category
${\operatorname{\mathbf{PSh}}^{ltr}}({\operatorname{\mathbf{Sm}l\mathbf{Sm}}})$
of additive presheaves of abelian groups on
${\operatorname{l\mathbf{Cor}}}_{{\operatorname{\mathbf{Sm}l\mathbf{Sm}}}}$
consisting of those $\mathcal{F}$ whose restrictions to
${\operatorname{\mathbf{Sm}l\mathbf{Sm}}}$ are dividing Nisnevich sheaves.

Now we construct a functor

(6.0.2)
$\mathcal{L}og:\operatorname{\mathbf{\underline{M}NST}}_{\log}\to{\operatorname{\mathbf{Shv}}^{ltr}_{dNis}}.$

For $\mathfrak{X}=(X,\mathcal{M})\in{\operatorname{\mathbf{Sm}l\mathbf{Sm}}}$,
we put $\mathfrak{X}^{MP}=(X,\partial\mathfrak{X})$, where
$\partial\mathfrak{X}\subset X$ is the closed subscheme consisting of the
points where the log-structure $\mathcal{M}$ is not trivial. By [12, Theorem
III.1.11.12], $\partial\mathfrak{X}$ with reduced structure is a normal
crossing divisor on $X$ so that we can view $\mathfrak{X}^{MP}$ as an objects
of $\operatorname{\mathbf{\underline{M}Cor}}_{ls}$. For
$F\in\operatorname{\mathbf{\underline{M}PST}}_{\log}$ and
$\mathfrak{X}\in{\operatorname{\mathbf{Sm}l\mathbf{Sm}}}$, we put

(6.0.3) $F^{\log}(\mathfrak{X})=F(\mathfrak{X}^{MP}).$

Take $\mathfrak{Y}\in{\operatorname{\mathbf{Sm}l\mathbf{Sm}}}$ and
$\alpha\in{\operatorname{l\mathbf{Cor}}}(\mathfrak{Y},\mathfrak{X})$. By [1,
Def. 2.1.1 and Rem. 2.1.1(iv)], we have

$\alpha\in\operatorname{\mathbf{\underline{M}Cor}}^{{\operatorname{fin}}}((Y,n\cdot\partial\mathfrak{Y}),(X,\partial\mathfrak{X}))\;\text{
for some }n>0,$

where $n\cdot\partial\mathfrak{Y}\hookrightarrow Y$ is the $n$-th thickening
of $\partial\mathfrak{Y}\hookrightarrow Y$. By the assumption
$F\in\operatorname{\mathbf{\underline{M}PST}}_{\log}$, the induced map

$F^{\log}(\mathfrak{X})=F(\mathfrak{X}^{MP})\overset{\alpha^{*}}{\longrightarrow}F(Y,n\cdot\partial\mathfrak{Y})$

factors through $F^{\log}(\mathfrak{Y})=F(Y,\partial\mathfrak{Y})\subset
F(Y,n\cdot\partial\mathfrak{Y})$ and we get a map

$\alpha^{*\log}:F^{\log}(\mathfrak{X})\to F^{\log}(\mathfrak{Y}).$

Moreover, for a map $\gamma:F\to G$ in
$\operatorname{\mathbf{\underline{M}PST}}_{\log}$, the diagram

$\textstyle{F^{\log}(\mathfrak{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\alpha^{*\log}}$$\textstyle{G^{\log}(\mathfrak{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{*\log}}$$\textstyle{F^{\log}(\mathfrak{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{G^{\log}(\mathfrak{Y})}$

is obviously commutative. Hence the assignment $\mathcal{X}\to
F^{\log}(\mathcal{X})$ gives an object $F^{\log}$ of
${\operatorname{\mathbf{PSh}}^{ltr}}({\operatorname{\mathbf{Sm}l\mathbf{Sm}}})$
and we get a functor

(6.0.4)
$\mathcal{L}og:\operatorname{\mathbf{\underline{M}PST}}_{\log}\to{\operatorname{\mathbf{PSh}}^{ltr}}({\operatorname{\mathbf{Sm}l\mathbf{Sm}}})\;;\;F\to
F^{\log}.$

By the definitions of sheaves ([4, Def. 1] and [1, Def. 3.1.4]) and [4, Pr.
1.9.2], this induces a functor

$\operatorname{\mathbf{\underline{M}NST}}_{\log}\to{\operatorname{\mathbf{Shv}}^{ltr}_{dNis}}({\operatorname{\mathbf{Sm}l\mathbf{Sm}}})$

which induces the desired functor (6.0.2) using (6.0.1). By the construction,
for $F\in\operatorname{\mathbf{\underline{M}NST}}_{\log}$ and
$\mathfrak{X}\in{\operatorname{\mathbf{Sm}l\mathbf{Sm}}}$ with
$\mathcal{X}=\mathfrak{X}^{MP}\in\operatorname{\mathbf{\underline{M}Cor}}_{ls}$,
we have

(6.0.5)
$H^{i}_{\operatorname{Nis}}(X,F_{\mathcal{X}})=H^{i}_{s\mathrm{Nis}}(\mathfrak{X},F^{\log})\;(F^{\log}=\mathcal{L}og(F)),$

where the right hand side is the cohomology for the strict Nisnevich topology
(see [1, Def. 4.3.1]).

######  Theorem 6.1.

For $F\in\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}$,
$F^{\log}=\mathcal{L}og(F)\in{\operatorname{\mathbf{Shv}}^{ltr}_{dNis}}$ is
strictly ${\overline{\square}}$-invariant in the sense [1, Def. 5.2.2]. For
$\mathfrak{X}\in{\operatorname{\mathbf{Sm}l\mathbf{Sm}}}$ with
$\mathcal{X}=\mathfrak{X}^{MP}\in\operatorname{\mathbf{\underline{M}Cor}}_{ls}$,
we have a natural isomorphism

(6.1.1)
$H^{i}_{\operatorname{Nis}}(X,F_{\mathcal{X}})\simeq\operatorname{Hom}_{\operatorname{\mathbf{logDM}}^{{\operatorname{eff}}}}(M(\mathfrak{X}),F^{\log}[i]),$

where $\operatorname{\mathbf{logDM}}^{{\operatorname{eff}}}$ is the
triangulated category of logarithmic motives defined in [1, Def. 5.2.3].

###### Proof.

We have isomorphisms

$H^{i}_{\operatorname{Nis}}(X,F_{\mathcal{X}})\overset{\eqref{comparisonlogcoh}}{\simeq}H^{i}_{s\mathrm{Nis}}(\mathfrak{X},F^{\log})\overset{(*1)}{\simeq}\operatornamewithlimits{\varinjlim}_{\mathfrak{Y}\in\mathfrak{X}_{div}^{Sm}}H^{i}_{s\mathrm{Nis}}(\mathfrak{X},F^{\log})\overset{(*2)}{\simeq}H^{i}_{d\mathrm{Nis}}(\mathfrak{X},F^{\log}),$

where $(*2)$ comes from [1, Th. 5.1.8] and $(*1)$ is a consequence of Theorem
5.1 in view of (6.0.5). Hence the strict ${\overline{\square}}$-invariance of
$F^{\log}$ follows from [16, Th. 0.6]. Finally (6.1.1) follows from [1, Pr.
5.2.8]. ∎

Now we consider the composite functor

$\mathcal{L}og^{\prime}:{\operatorname{\mathbf{RSC}}}_{\operatorname{Nis}}\overset{\underline{\omega}^{\operatorname{\mathbf{CI}}}}{\longrightarrow}\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}\overset{\mathcal{L}og}{\longrightarrow}{\operatorname{\mathbf{Shv}}^{ltr}_{dNis}}.$

######  Lemma 6.2.

$\mathcal{L}og$ and $\mathcal{L}og^{\prime}$ have the same essential image.

###### Proof.

This follows directly from the construction and Corollary 2.6(3). ∎

In what follows, we let

(6.2.1)
$\mathcal{L}og:{\operatorname{\mathbf{RSC}}}_{\operatorname{Nis}}\to{\operatorname{\mathbf{Shv}}^{ltr}_{dNis}}\;:\;F\to
F^{\log}$

denote $\mathcal{L}og^{\prime}$ defined as above. By (6.0.3), we have

(6.2.2) $F^{\log}(X,\mathrm{triv})=F(X)\;\text{ for
}F\in{\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}},\;X\in\operatorname{\mathbf{Sm}},$

where $(X,\mathrm{triv})$ denotes the log-scheme with the trivial log
structure.

######  Theorem 6.3.

$\mathcal{L}og$ is exact and fully faithful.

###### Proof.

First we prove the full faithfulness. The faithfulness follows from (6.2.2).
Let $F,G\in{\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}$ and
$\gamma:F^{\log}\to G^{\log}$ be a map in
${\operatorname{\mathbf{Shv}}^{ltr}_{dNis}}$. By (6.2.2) it induces maps
$\gamma_{X}:F(X)\to G(X)$ for all $X\in\operatorname{\mathbf{Sm}}$. They are
compatible with the action of $\operatorname{\mathbf{Cor}}$ since by [1, Rem
2.1.3(3)],

$\operatorname{\mathbf{Cor}}(Y,X)={\operatorname{l\mathbf{Cor}}}(Y,\mathrm{triv}),(X,\mathrm{triv}))\;\text{
for }X,Y\in\operatorname{\mathbf{Sm}}.$

Thus $\gamma_{X}$ for $X\in\operatorname{\mathbf{Sm}}$ give a map
$\gamma_{{\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}}:F\to G$ in
${\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}$. To see
$\mathcal{L}og(\gamma_{{\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}})=\gamma$,
it suffices by (6.0.1) to show that
$\mathcal{L}og(\gamma_{{\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}})$
and $\gamma$ induce the same map $F^{\log}(\mathfrak{X})\to
G^{\log}(\mathfrak{X})$ for
$\mathfrak{X}\in{\operatorname{\mathbf{Sm}l\mathbf{Sm}}}$. If $\mathfrak{X}$
has the trivial log-structure, this follows immediately from the construction
of $\gamma_{{\operatorname{\mathbf{RSC}}}}$. The general case follows from
this in view of the commutative diagram

$\textstyle{F^{\log}(\mathfrak{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{j^{*}}$$\textstyle{G^{\log}(\mathfrak{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{F^{\log}(X\backslash\partial\mathfrak{X},\mathrm{triv})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{G^{\log}(X\backslash\partial\mathfrak{X},\mathrm{triv})}$

where $j^{*}$ are induced by the natural map
$(X\backslash\partial\mathfrak{X},\mathrm{triv})\to\mathfrak{X}$ of log-
schemes and injective by the construction and the semipurity of
$\underline{\omega}^{\operatorname{\mathbf{CI}}}F$. This completes the proof
of the full faithfulness.

Next we show the exactness of $\mathcal{L}og$. It suffices to show the
following.

######  Claim 6.3.1.

Given an exact sequence $0\to F\to G\to H\to 0$ in
${\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}$, the induced sequence

$0\to F^{\log}(\mathfrak{X})\to G^{\log}(\mathfrak{X})\to
H^{\log}(\mathfrak{X})\to 0$

is exact for every $\mathfrak{X}\in{\operatorname{\mathbf{Sm}l\mathbf{Sm}}}$
with $X$ henselian local.

Indeed, by the definition of $\mathcal{L}og$, this is reduced to the exactness
of

$0\to\underline{\omega}^{\operatorname{\mathbf{CI}}}F(\mathfrak{X}^{MP})\to\underline{\omega}^{\operatorname{\mathbf{CI}}}G(\mathfrak{X}^{MP})\to\underline{\omega}^{\operatorname{\mathbf{CI}}}H(\mathfrak{X}^{MP})\to
0,$

which follows from Corollary 2.6(2). This completes the proof of Theorem 6.3.
∎

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