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 [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a, Reference SaitoSai20].

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 [Reference Voevodsky, Friedlander, Suslin and VoevodskyVoe00]). 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 $k$, that is, $X$ is a limit $\mathop {\varprojlim }_{i \in I} X_i$ over a filtered set $I$, where $X_i$ is smooth over $k$ and all transition maps are étale. Note that $\operatorname {Spec} K\in \operatorname {\mathbf {Sm}}^{\rm pro}$ 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 whose morphisms are defined as [Reference Rülling and SaitoRS21a, Definition 2,2]. We extend $F\in {\operatorname {\mathbf {PST}}}$ to a presheaf on $\operatorname {\mathbf {Cor}}^{\rm pro}$ by $F(X) := \mathop {\varinjlim }_{i \in I} F(X_i)$ for $X$ as above.

3. (3) We recall the definition of the category $\operatorname {\mathbf {\underline {M}Cor}}$ from [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a, Definition 1.3.1]. A pair $\mathcal {X} = (X,D)$ consisting of $X \in \operatorname {\mathbf {Sch}}$ and an effective Cartier divisor $D$ on $X$ is called a modulus pair if $X - D \in \operatorname {\mathbf {Sm}}$. 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 that $\Gamma$ is admissible (respectively, left proper) if $(D_X)_{\overline {\Gamma }^N}\geq (D_Y)_{\overline {\Gamma }^N}$ (respectively, 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 $\operatorname {\mathbf {\underline {M}Cor}}_{\rm ls}\subset \operatorname {\mathbf {\underline {M}Cor}}$ 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 $\operatorname {\mathbf {\underline {M}Cor}}^{{\operatorname {fin}}}_{\rm ls} = \operatorname {\mathbf {\underline {M}Cor}}^{{\operatorname {fin}}}\cap \operatorname {\mathbf {\underline {M}Cor}}_{\rm ls}$.

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

   \begin{gather*} \lambda: \operatorname{\mathbf{Cor}}\to \operatorname{\mathbf{\underline{M}Cor}}\quad X\mapsto (X,\emptyset),\end{gather*}
   \begin{gather*} \underline{\omega}: \operatorname{\mathbf{\underline{M}Cor}}\to\operatorname{\mathbf{Cor}}\quad (X,D)\mapsto X-D.\end{gather*}

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 $\operatorname {\mathbf {MPST}}$ (respectively, $\operatorname {\mathbf {\underline {M}PST}}$) be the category of additive presheaves of abelian groups on $\operatorname {\mathbf {MCor}}$ (respectively, $\operatorname {\mathbf {\underline {M}Cor}}$) 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 sometimes write $\mathcal {X}$ for ${\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}(\mathcal {X})$ for simplicity.

9. (9) In 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 [Reference Rülling and SaitoRS21a, § 3.7]).

10. (10) The adjunction $\lambda \dashv \underline {\omega }$ induces a string of four adjoint functors $(\lambda _!=\underline {\omega }^!,\lambda ^*=\underline {\omega }_!,\lambda _*=\underline {\omega }^*,\underline {\omega }_*)$ (see [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a, Proposition 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 localizations and $\underline {\omega }^!$ and $\underline {\omega }^*$ are fully faithful.

11. (11) The functor $\tau$ yields a string of three 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 localization; $\tau _!$ has a pro-left adjoint $\tau ^!$, hence is exact (see [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a, Proposition 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 that of $\mathbf {A}^1$ (see [Reference Kahn, Saito and YamazakiKSY22, Lemma 2.1.3]). We say that $F \in \operatorname {\mathbf {MPST}}$ 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 [Reference Kahn, Saito and YamazakiKSY22, 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

    \begin{align*} &h_0^{\overline{\square}}(F)(\mathcal{X})=\operatorname{Coker}(i_0^* - i_1^* : F(\mathcal{X} \otimes {\overline{\square}}) \to F(\mathcal{X})),\\ &h^0_{\overline{\square}}(F)(\mathcal{X})=\operatorname{Hom}(h_0^{\overline{\square}}(\mathcal{X}), F). \end{align*}
    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 also write $h_0^{\overline {\square }}(\mathcal {X})$ 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 that $F$ is a Nisnevich sheaf if $F_{\mathcal {X}}$ is also one for all $\mathcal {X}\in \operatorname {\mathbf {\underline {M}Cor}}$ (see [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a, § 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 [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a, Proposition 3.5.3] and [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21b, 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
    \[ {\underline{a}}_{\operatorname{Nis}} F (X,D) = \mathop{\varinjlim}_{f:Y\to X} F_{(Y,f^*D),{\operatorname{Nis}}}(Y) \]
    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 [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21b, Proposition 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 [Reference Kahn, Saito and YamazakiKSY22, Lemma 2.3.1] and [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21b, Proposition 6.2.1a)], for $F\in {\operatorname {\mathbf {PST}}}$, we have $F\in \operatorname {\mathbf {HI}}$ (respectively, $F\in \operatorname {\mathbf {HI}}_{\operatorname {Nis}}$) if and only if $\underline {\omega }^*F\in \operatorname {\mathbf {CI}}^{\tau }$ (respectively, $\underline {\omega }^*F\in \operatorname {\mathbf {CI}}^{\tau }_{\operatorname {Nis}}$).

16. (16) We say that $F\in \operatorname {\mathbf {\underline {M}PST}}$ is semipure if the unit map

    \[ u: F\to \underline{\omega}^*\underline{\omega}_!F \]
    is injective. For $F\in \operatorname {\mathbf {\underline {M}PST}}$ (respectively, $F\in \operatorname {\mathbf {\underline {M}NST}}$), let $F^{{\rm sp}}\in \operatorname {\mathbf {\underline {M}PST}}$ (respectively, $F^{{\rm sp}}\in \operatorname {\mathbf {\underline {M}NST}}$) be the image of $F\to \underline {\omega }^*\underline {\omega }_! F$ (called the semipurification of $F$. See [Reference SaitoSai20, Lemma 1.30]). For $F\in \operatorname {\mathbf {\underline {M}PST}}$ we have
    \[ {\underline{a}}_{\operatorname{Nis}}(F^{{\rm sp}}) \simeq ({\underline{a}}_{\operatorname{Nis}} F)^{{\rm sp}}. \]
    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 $F^{{\rm sp}}\in \operatorname {\mathbf {MPST}}^{\tau }$ since $\tau$ is exact and $\underline {\omega }^*\underline {\omega }_!\tau _!=\tau _! \omega ^*\omega _!$.

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

    \[ \operatorname{\mathbf{CI}}^{\tau,{\rm sp}}_{\operatorname{Nis}} =\operatorname{\mathbf{CI}}^{\tau,{\rm sp}}\cap \operatorname{\mathbf{MNST}}^\tau \subset \operatorname{\mathbf{CI}}^{\tau}_{\operatorname{Nis}}. \]
    By [Reference SaitoSai20, Theorems 0.1 and 0.4], we have ${\underline {a}}_{\operatorname {Nis}}(\operatorname {\mathbf {CI}}^{\tau,{\rm sp}})\subset \operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{\operatorname {Nis}}$.

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 $\operatorname {\mathbf {CI}}^{\tau,{\rm sp}}$ under $\underline {\omega }_!$ since $\omega _!=\underline {\omega }_!\tau _!$ and $\underline {\omega }_!F=\underline {\omega }_!F^{{\rm sp}}$). Put ${\operatorname {\mathbf {RSC}}}_{{\operatorname {Nis}}}={\operatorname {\mathbf {RSC}}}\cap \operatorname {\mathbf {NST}}$. The objects of ${\operatorname {\mathbf {RSC}}}$ (respectively, ${\operatorname {\mathbf {RSC}}}_{\operatorname {Nis}}$) are called reciprocity presheaves (respectively, sheaves). By [Reference SaitoSai20, Theorem 0.1], we have

    (1.0.1)\begin{equation} a^V_{\operatorname{Nis}}({\operatorname{\mathbf{RSC}}})\subset {\operatorname{\mathbf{RSC}}}_{\operatorname{Nis}}. \end{equation}
    We have $\operatorname {\mathbf {HI}}\subseteq {\operatorname {\mathbf {RSC}}}$ which also contains smooth commutative group schemes (which may have non-trivial unipotent part), the sheaf $\Omega ^i$ of Kähler differentials, and the de Rham–Witt sheaves $W_n\Omega ^i$ (see [Reference Kahn, Saito and YamazakiKSY16, Reference Kahn, Saito and YamazakiKSY22]).

19. (19) $\operatorname {\mathbf {NST}}$ is a Grothendieck abelian category by [Reference Voevodsky, Friedlander, Suslin and VoevodskyVoe00, Lemma 3.1.6] and we can make ${\operatorname {\mathbf {RSC}}}_{{\operatorname {Nis}}}$ its full subabelian category as follows. We define the kernel (respectively, 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 [Reference Kahn, Saito and YamazakiKSY22, Proposition 2.3.7] we have a pair of adjoint functors

    (1.0.2)\begin{equation} \operatorname{\mathbf{CI}}\begin{smallmatrix}\omega^{\operatorname{\mathbf{CI}}}\\\longleftarrow\\\omega_!\\\longrightarrow \end{smallmatrix}{\operatorname{\mathbf{RSC}}}, \end{equation}
    where $\omega ^{\operatorname {\mathbf {CI}}}=h^0_{\overline {\square }}\omega ^*$ and is fully faithful. It induces a pair of adjoint functors
    (1.0.3)\begin{equation} \operatorname{\mathbf{CI}}^{\tau}\begin{smallmatrix}\underline{\omega}^{\operatorname{\mathbf{CI}}}\\\longleftarrow\\\underline{\omega}_!\\\longrightarrow \end{smallmatrix}{\operatorname{\mathbf{RSC}}}, \end{equation}
    where $\underline {\omega }^{\operatorname {\mathbf {CI}}}=\tau _!h^0_{\overline {\square }}\omega ^*$ and 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
    \begin{align*} \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). \end{align*}
    In view of (15), (1.0.3) induces a pair of adjoint functors
    (1.0.4)\begin{equation} \operatorname{\mathbf{CI}}^{\tau,{\rm sp}}_{\operatorname{Nis}}\begin{smallmatrix}\underline{\omega}^{\operatorname{\mathbf{CI}}}\\\longleftarrow\\\underline{\omega}_!\\\longrightarrow \end{smallmatrix}{\operatorname{\mathbf{RSC}}}_{\operatorname{Nis}}. \end{equation}

2. Purity with reduced modulus

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

\begin{align*} 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),\end{align*}

\begin{eqnarray*} \\ 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)\end{eqnarray*}

Note that if $F\in \operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{{\operatorname {Nis}}}$, then $F_{-1},F_{-1}^{(1)}\in \operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{{\operatorname {Nis}}}$ and

(2.0.1)\begin{equation} \begin{aligned}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)),\\ F_{-1}(\mathcal{X}) & =\mathop{\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)) \end{aligned} \end{equation}

for $\mathcal {X}\in \operatorname {\mathbf {\underline {M}Cor}}$, 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

\[ {\tau^{(0)}} F=F_{-1}\quad\text{and}\quad {\tau^{(1)}} F = F_{-1}/F_{-1}^{(1)}, \]

where the quotient is taken in $\operatorname {\mathbf {\underline {M}PST}}$.

The existence of retractions in the following lemma was suggested by A. Merici. It implies ${\tau^{(e_1,\dots,e_r)}}F\in \operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{{\operatorname {Nis}}}$ if $F\in \operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{{\operatorname {Nis}}}$.

Lemma 2.2 For $F\in \operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{{\operatorname {Nis}}}$, 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 $\operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{{\operatorname {Nis}}}$, the following diagram is commutative:

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

Theorem 2.3 Let $F\in \operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{{\operatorname {Nis}}}$. Let $K\{t_1,\dots,t_n\}$ be the henselization of $K[t_1,\dots,t_n]$ at $(t_1,\dots,t_n)$ and $\mathcal {X}=\operatorname {Spec} K\{t_1,\dots,t_n\}$ 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 $D_\mathcal {H}=\bigl \{ \prod _{j\in [1,n]-I}t_j^{e_j}=0\bigr \}\subset \mathcal {H}$. Then

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

and there is an isomorphism

(2.3.2)\begin{equation} ({\tau^{(e_I)}} F)_{(\mathcal{H},D_\mathcal{H})} \simeq R^q i_\mathcal{H}^! F_{(\mathcal{X},D)}\quad \text{with } e_I=(e_i)_{i\in I}\in \mathbb{Z}_{\geq 0}^q. \end{equation}

Proof. The proof is divided into two steps.

Step 1: we prove (2.3.1) and (2.3.2) for $q=|I|=1$. For $\nu =0$, (2.3.1) follows from the semipurity of $F$ and [Reference SaitoSai20, Theorem 3.1]. 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 [Reference SaitoSai20, Corollary 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

• $(\spadesuit )$ $e_1\not =0$, 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 $D'=\{t_2^{e_2}\cdots t_r^{e_r}=0\}\subset \mathcal {X}$. By [Reference SaitoSai20, Lemma 7.1], we have an exact sequence sheaves on $\mathcal {X}_{{\operatorname {Nis}}}$,

\[ 0\to F_{(\mathcal{X},D')} \to F_{(\mathcal{X},D)} \to \iota_*({F^{(e)}_{-1}})_{(\mathcal{Z},D_\mathcal{Z})} \to 0, \]

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

(2.3.3)\begin{equation} \cdots \to R^\nu i_\mathcal{H}^!F_{(\mathcal{X},D')}\to R^\nu i_\mathcal{H}^! F_{(\mathcal{X},D)}\to R^\nu i_\mathcal{H}^!\iota_*({F^{(e)}_{-1}})_{(\mathcal{Z},D_\mathcal{Z})}\to\cdots. \end{equation}

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

and we have an isomorphism

\[ R^\nu i_\mathcal{H}^!\iota_*({F^{(e)}_{-1}})_{(\mathcal{Z},D_\mathcal{Z})} \simeq \iota'_* R^\nu i_{\mathcal{H}\cap \mathcal{Z}}^!({F^{(e)}_{-1}})_{(\mathcal{Z},D_\mathcal{Z})}. \]

By the induction hypothesis, $R^\nu i_{\mathcal {H}\cap \mathcal {Z}}^!({F^{(e)}_{-1}})_{(\mathcal {Z},D_\mathcal {Z})}=0$ for $\nu >1$, noting that ${{F^{(e)}_{-1}}}\in \operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{{\operatorname {Nis}}}$ 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 [Reference SaitoSai20, Lemma 7.1(2)].

If $\mathcal {Z}=\mathcal {H}$, we have

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

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

\[ 0\to ({F^{(e)}_{-1}})_{(\mathcal{H},D_\mathcal{H})} \overset{\delta}{\longrightarrow} R^1 i_\mathcal{H}^!F_{(\mathcal{X},D')}\to R^1 i_\mathcal{H}^! F_{(\mathcal{X},D)}\to 0. \]

By [Reference SaitoSai20, Lemma 7.1(2)] we have an isomorphism

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

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

Step 2: we prove the theorem by 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

\[ ({\tau^{(e_{i_1})}} F)_{(\mathcal{Y},D_\mathcal{Y})} \simeq R^1 i_\mathcal{Y}^! F_{(\mathcal{X},D)}\quad {\rm with}\ D_\mathcal{Y}=\{t_1^{e_1}\cdots \overset{\vee}{t_{i_1}^{e_{i_1}}}\cdots t_n^{e_n}=0\}\subset \mathcal{Y}. \]

Note ${\tau^{(e_{i_1})}} F\in \operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{{\operatorname {Nis}}}$ 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

\begin{align*} 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})}, \end{align*}

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

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

Corollary 2.4 Take $F\in \operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{{\operatorname {Nis}}}$ and $(X,D)\in \operatorname {\mathbf {\underline {M}Cor}}_{\rm ls}$ 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

\[ H^i_x(X_{{\operatorname{Nis}}},F_{(X,D)})=0 \quad \text{for } i\not=n. \]

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 $F\in \operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{{\operatorname {Nis}}}$ and $\mathcal {X}=(X,D) \in \operatorname {\mathbf {\underline {M}Cor}}_{\rm ls}$ 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.

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:

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.

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

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

\[ \phi\circ p_H= p_G\circ \phi,\quad \psi\circ p_G= p_F\circ \psi, \quad \phi\circ s_H= s_G\circ \phi,\quad \psi\circ s_G= s_F\circ \psi. \]

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).

Finally, (3) follows from (2) since $\underline {\omega }_! u$ is an isomorphism. This completes the proof of the corollary.

3. Review of higher local symbols

In this section we recall from [Reference Rülling and SaitoRS21c] the higher local symbols for reciprocity sheaves, which are a fundamental tool to prove Theorem 4.2, one of the main theorems of this paper. First we introduce some basic notation. In this section $X$ is a reduced noetherian separated scheme of dimension $d<\infty$ such that $X_{(0)}=X^{(d)}$.

Let $K$ be a field. For an integer $r\geq 0$, let $K^{\rm M}_r(K)$ be the Milnor $K$-group of $K$. Let $A$ be a local domain with the function field $K$. For an ideal $I\subset A$, let $\overline {K}^{\rm M}_r(A,I)\subset K^{\rm M}_r(K)$ denote the subgroup generated by symbols

\[ \{1+a,b_1,\dots,b_{r-1}\} \quad {\rm with}\ a\in I,\, b_i\in A^\times. \]

Let $A$ be a noetherian excellent one-dimensional local domain with function field $K$ and residue field $F$. Let $\tilde {A}$ be the normalization of $A$ and $S$ be the set of the maximal ideals of $\tilde {A}$. For $\mathfrak {m}\in S$, denote $\kappa (\mathfrak {m})=\tilde {A}/\mathfrak {m}$. Then we define

(3.0.1)\begin{equation} \partial_{A}:= \sum_{\mathfrak{m}\in S} \operatorname{Nm}_{\kappa(\mathfrak{m})/F}\circ \partial_{\mathfrak{m}} : K^{\rm M}_r(K)\to K^{\rm M}_{r-1}(F), \end{equation}

where $\partial _{\mathfrak {m}} :K^{\rm M}_r(K)\to K^{\rm M}_{r-1}(\kappa (\mathfrak {m}))$ denotes the tame symbol for the discrete valuation ring $\tilde {A}_{\mathfrak {m}}$, the localization of $\tilde {A}$ at $\mathfrak {m}$, and $\operatorname {Nm}_{\kappa (\mathfrak {m})/F}$ is the norm map.

For $x, y\in X$, we write

\[ y< x:\Longleftrightarrow \overline{\{y\}}\subsetneq \overline{\{x\}},\quad \text{that is, } y\in \overline{\{x\}} \text{ and } y\neq x. \]

A chain on $X$ is a sequence

(3.0.2)\begin{equation} {\underline{x}=(x_0,\ldots, x_n)\quad \text{with } x_0< x_1<\cdots < x_n.} \end{equation}

The chain $\underline {x}$ is a maximal Paršin chain (or maximal chain) if $n=d$ and $x_i\in X_{(i)}$. Note that the assumptions on $X$ imply $x_{i}\in \overline {\{x_{i+1}\}}^{(1)}$. We denote

\[ \mathrm{mc}(X)=\{\text{maximal chains on } X\}. \]

A maximal chain with break at $r\in \{0,\ldots, d\}$ is a chain (3.0.2) with $n=d-1$ and $x_i\in X_{(i)}$, for $i< r$, and $x_i\in X_{(i+1)}$, for $i\ge r$. We denote

\[ \mathrm{mc}_r(X)=\{\text{maximal chain with break at $r$ on $X$}\}. \]

For $\underline {x}=(x_0,\ldots, x_{d-1})\in \mathrm {mc}_r(X)$, we denote by $b(\underline {x})$ the set of $y\in X_{(r)}$ such that

(3.0.3)\begin{equation} {\underline{x}(y):=(x_0,\ldots, x_{r-1}, y, x_{r},\ldots, x_{d-1})\in \mathrm{mc}(X).} \end{equation}

In the rest of this section we fix $F=\underline {\omega }^{\operatorname {\mathbf {CI}}} G\in \operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{\operatorname {Nis}}$ with $G\in {\operatorname {\mathbf {RSC}}}_{\operatorname {Nis}}$ (cf. (1.0.4)). We also fix a function field $K$ over the base field $k$. Let $X$ be an integral scheme of finite type over $K$ and assume $d=\dim (X)\geq 1$. Recall from [Reference Rülling and SaitoRS21c, § 5] that we have a collection of bilinear pairings (cf. the convention from § 1(9))

(3.0.4)\begin{equation} {\big\{ (-,-)_{X/K,\underline{x}}: F(K(X))\otimes K^{\rm M}_d(K(X))\to F(K)\big\}_{\underline{x}\in \mathrm{mc}(X)}.} \end{equation}

The following properties hold for all $a\in F(K(X))$ (see Remark 3.1 below).

1. (HS1) Let $X\hookrightarrow X'$ be an open immersion where $X'$ is an integral $K$-scheme of dimension $d$. Then, for all $\beta \in K^{\rm M}_d(K(X))$,

   \[ (a,\beta)_{X/K,\underline{x}}=(a,\beta)_{X'/K,\underline{x}}. \]

2. (HS2) Let $\underline {x}=(x_0,\ldots,x_{d-1}, x_d)\in \mathrm {mc}(X)$ and $Z\subset X$ be the closure of $z=x_{d-1}$, and set $\underline {x}'=(x_0,\ldots, x_{d-1})\in \mathrm {mc}(Z)$. Assume $a\in F(\mathcal {O}_{X, z})$ and let $a(z)\in F(K(Z))$ be the restriction of $a$. Then

   \[ (a, \beta)_{X/K, \underline{x}} = (a(z), \partial_{z}\beta)_{Z/K, \underline{x}'}\quad \text{for $\beta\in K^{\rm M}_d(K(X))$}, \]
   where $\partial _{z}: K^{\rm M}_d(K(X)) \to K^{\rm M}_{d-1}(K(Z))$ is the map (3.0.1) for $A=\mathcal {O}_{X, z}$.

3. (HS3) Let $D\subset X$ be an effective Cartier divisor with $I_D\subset \mathcal {O}_X$ its ideal sheaf. Assume that $X{\setminus} D$ is regular so that $(X,D)\in \operatorname {\mathbf {\underline {M}Cor}}^{\rm pro}$ and that $a\in F(X,D)$. For $\underline {x}=(x_0,\ldots, x_{d-1},x_d)\in \mathrm {mc}(X)$, we have

   \[ (a, \beta)_{X/K,\underline{x}}=0\quad \text{for $\beta\in \overline{K}^{\rm M}_d(\mathcal{O}_{X, x_{d-1}},I_D \mathcal{O}_{X, x_{d-1}})$.} \]

4. (HS4) Let $\underline {x}'\in \mathrm {mc}_r(X)$ with $0\leq r\leq d-1$. For $\beta \in K^{\rm M}_d(K(X))$,

   \[ (a, \beta)_{X/K, \underline{x}'(y)}=0\quad \text{for almost all } y\in _{-}(\underline{x}'). \]
   Assume that either $r\ge 1$ or $r=0$, $X$ is quasi-projective, and the closure of $x_1$ in $X$ is projective over $K$, where $\underline {x}'=(x_1,\ldots, x_d)$. Then
   \[ \sum_{y\in b(\underline{x}')} (a, \beta)_{X/K, \underline{x}'(y)}=0. \]

Remark 3.1 Properties (HS1)–(HS4) are slight variants of the (stronger) properties (HS1)–(HS4) in [Reference Rülling and SaitoRS21c, Proposition 5.3], where the Milnor $K$-group $K^{\rm M}_{d}(K^h_{X,\underline {x}})$ of the iterated henselization $K^h_{X,\underline {x}}$ of $K(X)$ along the chain $\underline {x}$ is used instead of $K^{\rm M}_{d}(K(X))$. The version stated here follows easily using the natural maps $\iota _{\underline {x}}: K(X) \to K^h_{X,\underline {x}}$ and the commutative diagram in the situation of (HS2),

and the commutative diagram in the situation of (HS4),

where $\partial _{\underline {x}}$ (respectively, $\iota _y$) is defined in [Reference Rülling and SaitoRS21c, (4.1.1)] (respectively, [Reference Rülling and SaitoRS21c, (3.2.3)]). We also note that $\overline {K}^{\rm M}_d(\mathcal {O}_{X, x_{d-1}},I_D \mathcal {O}_{X, x_{d-1}})$ in (HS2) coincides with the Zariski stalk at $x_{d-1}$ of the sheaf $\overline {V}_{d,X|D}$ defined in [Reference Rülling and SaitoRS21c, 4.4].

For a scheme $Z$ over $k$, write $Z_K=Z\otimes _k K$. If $Z_K$ is integral, we denote by $K(Z)$ the function field of $Z_K$. We quote the following result from [Reference Rülling and SaitoRS21c, Proposition 7.3]. It is a key tool in the proof of Theorem 4.2.

Proposition 3.2 Let $X\in \operatorname {\mathbf {Sm}}$ and assume $D$ is a reduced simple normal crossing divisor on $X$ with $I_D\subset \mathcal {O}_X$ its ideal sheaf. Let $U\subset X$ be an open subset containing all the generic points of $D$. Let $a\in F(X{\setminus} D)$. Assume that, for all function fields $K/k$ and for all $\underline {x}=(x_0,\ldots,x_{d-1}, x_d)\in \mathrm {mc}(U_K)$ with $x_{d-1}\in D_K^{(0)}$, we have

\[ (a,\beta)_{X_K/K, \underline{x}}=0\quad \text{for all } \beta\in \overline{K}^{\rm M}(\mathcal{O}_{X, x_{d-1}},I_D \mathcal{O}_{X, x_{d-1}}). \]

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

4. Logarithmic cohomology of reciprocity sheaves

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

We need a preliminary lemma for the proof of the theorem.

Lemma 4.3 Let $F\in \operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{{\operatorname {Nis}}}$. 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 $V=\operatorname {Spec} K\{x_1,\dots,x_n\}$ be the henselization of $\mathbf {A}^n_K$ at the origin and $\mathcal {L}_i=\{x_i=0\}\subset V$ 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 $\operatorname {\mathbf {\underline {M}Cor}}^{\rm pro}$ (cf. § 1(9)):

\[ \rho_r: (V,\mathcal{L}_1+\cdots +\mathcal{L}_r) \to (\mathbf{A}^r_S,\{x_1\cdots x_r=0\})\simeq (\mathbf{A}^1,0)^{\otimes r}\otimes (S,\emptyset) , \]

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

(4.3.1)\begin{equation} \rho_r^*: F(\mathbf{A}^r_S,\{x_1\cdots x_r=0\} ) \to F(V,\mathcal{L}_1+\cdots + \mathcal{L}_r). \end{equation}

Then $F(V,\mathcal {L}_1+\cdots + \mathcal {L}_r)$ is generated by the image of $\rho _r^*$ and

\[ F(V,\mathcal{L}_1+\cdots+\overset{\vee}{\mathcal{L}_i}+\cdots \mathcal{L}_{r}) \quad\text{for } i=1,\dots, r. \]

Proof. For $\mathcal {Y}\in \operatorname {\mathbf {\underline {M}Cor}}$, let $F^\mathcal {Y}\in \operatorname {\mathbf {\underline {M}PST}}$ be defined by $F^\mathcal {Y}(\mathcal {Z})=F(\mathcal {Y}\otimes \mathcal {Z})$. Clearly, we have $F^\mathcal {Y}\in \operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{{\operatorname {Nis}}}$ for $F\in \operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{{\operatorname {Nis}}}$. We prove the lemma by induction on $r$. The case $r=1$ holds since by [Reference SaitoSai20, Lemmas 7.1 and 5.9], $\rho _1$ induces an isomorphism

\[ F^{(\mathbf{A}^1,0)}(S)/ F^{(\mathbf{A}^1,\emptyset)}(S) \overset{\simeq}\longrightarrow F(V,\mathcal{L}_1)/F(V). \]

By definition $\mathcal {L}_1=\operatorname {Spec} K\{x_2,\dots,x_n\}$ and we have a map in $\operatorname {\mathbf {\underline {M}Cor}}^{\rm pro}$,

\[ (V,\mathcal{L}_1+\cdots +\mathcal{L}_r) \to (\mathbf{A}^1,0)\otimes(\mathcal{L}_1,\mathcal{L}_1\cap (\mathcal{L}_2+\cdots +\mathcal{L}_r)), \]

induced by the natural map $K\{x_2,\dots,x_{n}\} [x_1] \to K\{x_1,\dots,x_n\}$. By [Reference SaitoSai20, Lemmas 7.1 and 5.9], it induces an isomorphism

\[ F^{(\mathbf{A}^1,0)}(\mathcal{L}_1,E)/ F^{(\mathbf{A}^1,\emptyset)}(\mathcal{L}_1,E) \overset{\simeq}\longrightarrow F(V,\mathcal{L}_1+\cdots +\mathcal{L}_r)/F(V,\mathcal{L}_2+\cdots +\mathcal{L}_r) \]

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 $\mathcal {X}=(X,D),\mathcal {Y}=(Y,E)\in \operatorname {\mathbf {\underline {M}Cor}}_{\rm ls}$ 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 $(X,D)=(V,\mathcal {L}_1+\cdots +\mathcal {L}_r)\in \operatorname {\mathbf {\underline {M}Cor}}^{\rm pro}$ in the notation of 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 induction on $r$. By Lemma 4.3, we may then assume

\[ (X,D)= \mathcal{M}:=(\mathbf{A}^1,0)^{\otimes r}\otimes(S,\emptyset) \quad {\rm for}\ S\in \operatorname{\mathbf{Sm}}^{\rm pro}. \]

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 subscheme of $(Y-E)\times (X-D)$ finite surjective over $Y-E$, and its closure $\overline {\alpha }$ in $Y\times X$ is finite surjective over $Y$.

• $(\spadesuit )$ Let $Y'$ be the normalization of $\overline {\alpha }$ and $E':=E\times _Y Y'$. Then $X$, $Y$, $E$ and $E'$ are irreducible, and $\alpha$, $Y'$, $E_{{\operatorname {red}}}$ and $E'_{{\operatorname {red}}}$ are essentially smooth over $k$.

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

\[ F(X,D)\overset{f^*}{\longrightarrow} F(Y',E') \overset{g_*}{\longrightarrow} F(Y,E). \]

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 $^t\Gamma _g\in \operatorname {\mathbf {Cor}}(Y-E,Y'-E')$ is the transpose of the graph of $g$ and $\Gamma _f\in \operatorname {\mathbf {Cor}}(Y'-E', X-D)$ is the graph of $f$. By definition this follows from the equality

\[ ^t\Gamma_g\times_{Y'-E'} \Gamma_f =\alpha\subset (Y-E)\times (X-D) \]

which one can check easily, noting that $Y'\to \overline {\alpha }$ is an isomorphism over $\alpha$ since $\alpha$ is regular by $(\spadesuit )$. Then we get a commutative diagram

where the top inclusion comes from the inequality $E_{{\operatorname {red}}} \times _Y Y'\geq E'_{{\operatorname {red}}}$ as Cartier divisors on $Y'$ thanks to the semipurity of $F$ (cf. § 1(16)). Hence, it suffices to show $f^*(F(X,D))\subset F(Y',E'_{{\operatorname {red}}})$. By replacing $(Y,E)$ with $(Y',E')$, 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

\[ (Y,E) \overset{i}{\longrightarrow} (\mathbf{A}^1,0)^{\otimes r}\otimes (Y,\emptyset) \to (\mathbf{A}^1,0)^{\otimes r}\otimes(S,\emptyset), \]

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 is 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.

Claim 4.3.1 Take $a\in F((\mathbf {A}^1,0)^{\otimes r}\otimes (Y,\emptyset ))$. There exists an open neighborhood $U\subset Y$ of the generic point of $E$ such that for every function field $K$ over $k$ and every $\delta =(\delta _0,\dots,\delta _{e-1},\delta _{e})\in \mathrm {mc}(U_K)$ with $\xi :=\delta _{e-1}\in E_K^{(0)}$ and $e=\dim (Y)$, we have

\[ (i^*(a)_K,\gamma)_{Y_K/K,\delta}=0\quad\forall \gamma\in \overline{K}^{\rm M}_e(\mathcal{O}_{Y_K,\xi},\mathfrak{m}_\xi) \]

for the pairing from (3.0.4):

\[ (-,-)_{Y_K/K,\delta}: F(K(Y))\otimes K^{\rm M}_d(K(Y))\to F(K). \]

Proof. After replacing $Y$ by an open neighborhood of the generic point of $E$, we may assume that $Y=\operatorname {Spec}(A)$ is affine and $E_{{\operatorname {red}}}=\operatorname {Spec}(A/(\pi ))$ for $\pi \in A$ and, moreover, that writing

\[ \mathbf{A}^r\times Y=\operatorname{Spec} A[x_1,\dots,x_r],\quad (\mathbf{A}^1,0)^{\otimes r}\otimes(Y,\emptyset) =(\mathbf{A}^r_Y,\{x_1\cdots x_r=0\}), \]

we have

\[ i(Y) =\underset{1\leq i\leq r}{\bigcap} \{x_i-u_i\pi^{m_i}=0\} \quad {\rm with}\ m_i\in \mathbb{Z}_{\geq 0},\; u_i\in A^\times. \]

Let $\delta =(\delta _0,\dots,\delta _e)$ be as in the claim and put $\delta '=(\delta _0,\dots,\delta _{e-1})\in \mathrm {mc}((E_{{\operatorname {red}}})_K)$. Put $\tilde {X}_K=\mathbf {A}^r\times Y_K$ and $F=\{\pi =0\}\subset \tilde {X}_K$. Note $d:=\dim (\tilde {X}_K)=e+r$. Let $z_j$ for $e\leq j\leq d-1$ be the generic point of

\[ Z_j=\underset{1\leq i\leq d-j}{\bigcap} \{x_i-u_i\pi^{m_i}=0\} \subset \tilde{X}_K \]

which lies over $\delta _e$,Footnote ⁵ and $w_j$ for $e-1\leq j\leq d-2$ be the generic point of

\[ W_j=F\cap Z_{j+1}=\{\pi=x_1=\cdots=x_{d-j-1}=0\} \]

which is contained in the closure of $z_{j+1}$. Note $\dim (Z_j)=\dim (W_j)=j$ and the section $i$ induces isomorphisms

(4.3.2)\begin{equation} {Y_K\simeq Z_e \quad\text{and}\quad (E_{{\operatorname{red}}})_K\simeq W_{e-1}.} \end{equation}

Let $\sigma =(i(\delta '),w_e,\dots,w_{d-2},\eta _1,\nu )\in \mathrm {mc}(\tilde {X}_K)$, where $\nu$ is the generic point of $\tilde {X}_K$ lying over $\delta _e$, $\eta _1$ is the generic point of $D_1=\{x_1=0\}\subset \tilde {X}_K$ contained in the closure of $\nu$, and $i(\delta ')\in \mathrm {mc}(W_{e-1})$ is the image of $\delta '$ under (4.3.2). Take any $\gamma \in \overline {K}^{\rm M}_e(\mathcal {O}_{Y_K,\xi },\mathfrak {m}_\xi )$ and put

(4.3.3)\begin{equation} \beta=\bigg\{\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}}\bigg\}\in K^{\rm M}_d(\mathcal{O}_{\tilde{X}_K,\nu}), \end{equation}

where $\iota : K^{\rm M}_e(\mathcal {O}_{Y_K,\delta _e}) \to K^{\rm M}_e(\mathcal {O}_{\tilde {X}_K,\nu })$ is induced by the projection $\tilde {X}_K \to Y_K$. 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 have

\begin{align*} 0=(a_K,\beta)_{\tilde{X}_K/K,\sigma} &=-\underset{\substack{\tau\in \tilde{X}_K^{(1)}-\{\eta_1\}\\\tau>w_{d-2}}}{\sum} (a_K,\beta)_{\tilde{X}_K/K,(i(\delta'),w_e,\dots,w_{d-2},\tau,\nu)} \\ &=-(a_K,\beta)_{\tilde{X}_K/K,(i(\delta'),w_e,\dots,w_{d-2},z_{d-1},\nu)} \\ &=\pm ((a_K)_{|Z_{d-1}},\beta_1)_{Z_{d-1}/K,(i(\delta'),w_e,\dots,w_{d-2},z_{d-1})}, \end{align*}

\[ \beta_1=\bigg\{\iota_1(\gamma),\frac{u_2\pi^{m_2}-x_2}{u_2\pi^{m_2}},\dots,\frac{u_r\pi^{m_r}-x_r}{u_r\pi^{m_r}}\bigg\}\in K^{\rm M}_{d-1}(\mathcal{O}_{Z_{d-1},z_{d-1}}), \]

where $\iota _1: K^{\rm M}_e(\mathcal {O}_{Y_K,\delta _e}) \to K^{\rm M}_e(\mathcal {O}_{Z_{d-1},z_{d-1}})$ is induced by the dominant map $Z_{d-1}\to Y_K$ induced by the projection $\tilde {X}_K \to Y_K$. The first equality follows from § 3 (HS3) applied to $D_1\subset \tilde {X}_K$, noting that $\beta$ lies in $\overline {K}^{\rm M}_d(\mathcal {O}_{\tilde {X}_K, \eta _1},\mathfrak {m}_{\eta _1})$ since $(u_1\pi ^{m_1}-x_1)/u_1\pi ^{m_1} \in 1+ x_1 \mathcal {O}_{\tilde {X}_K,\eta _1}$. The second follows from (HS4). The third equality holds since $z_{d-1}$ is the unique $\tau \in \tilde {X}_K^{(1)}-\{\eta _1\}$ such that $\tau >w_{d-2}$ and $(a_K,\beta )_{\tilde {X}_K/K,(i(\delta '),w_e,\dots,w_{d-2},\tau,\nu )}$ may not vanish, which follows from (HS2), noting that $\iota (\gamma )_{|F}=0$. The final equality follows from (HS2). When $r=1$, the last term in the above formula is equal to $((a_K)_{|Y_K},\gamma )_{Y_K/K,\delta }$ by (4.3.2), so that the proof is complete. When $r>1$, we further get

\begin{align*} 0&=((a_K)_{|Z_{d-1}},\beta_1)_{Z_{d-1}/K,(i(\delta'),w_e,\dots,w_{d-2},z_{d-1})}\\ &=-\underset{\substack{\tau\in Z_{d-1}^{(1)}-\{w_{d-2}\}\\\tau>w_{d-3}}}{\sum} ((a_K)_{|Z_{d-1}},\beta_1)_{Z_{d-1}/K,(i(\delta'),w_e,\dots,w_{d-3},\tau,z_{d-1})} \\ &=-((a_K)_{|Z_{d-1}},\beta_1)_{Z_{d-1}/K,(i(\delta'),w_e,\dots,w_{d-3},z_{d-2},z_{d-1})} \\ &=\pm ((a_K)_{|Z_{d-2}},\beta_2)_{Z_{d-2}/K,(i(\delta'),w_e,\dots,w_{d-3},z_{d-2})} , \end{align*}

\[ \beta_2=\bigg\{\iota_2(\gamma),\frac{u_3\pi^{m_3}-x_3}{u_3\pi^{m_3}},\dots,\frac{u_r\pi^{m_r}-x_r}{u_r\pi^{m_r}}\bigg\}\in K^{\rm M}_{d-1}(\mathcal{O}_{Z_{d-2},z_{d-2}}), \]

where $\iota _2: K^{\rm M}_e(\mathcal {O}_{Y_K,\delta _e}) \to K^{\rm M}_e(\mathcal {O}_{Z_{d-2},z_{d-2}})$ is induced by the dominant map $Z_{d-2}\to Y_K$ induced by the projection $\tilde {X}_K \to Y_K$. The above equalities hold by the same arguments as above, except that for the third equality there are a priori two $\tau \in Z_{d-1}^{(1)}-\{w_{d-2}\}$ with $\tau >w_{d-3}$ for which $((a_K)_{|Z_{d-1}},\beta _1)_{Z_{d-1}/K,(i(\delta '),w_e,\dots,w_{d-3},\tau,z_{d-1})}$ may not vanish. One is $z_{d-2}$ and the other is the generic point $\eta _2$ of $Z_{d-1}\cap D_2$ with $D_2= \{x_2=0\}\subset \tilde {X}_K$ which is contained in the closure of $z_{d-1}$. But $((a_K)_{|Z_{d-1}},\beta _1)_{Z_{d-1}/K,(i(\delta '),w_e,\dots,w_{d-3},\eta _2,z_{d-1})}=0$. Indeed, $(a_K)_{|Z_{d-1}}\in F(\operatorname {Spec}(\mathcal {O}_{Z_{d-1},\eta _2}),\eta _2)$ since $Z_{d-1}$ and $D_2$ intersect transversally in $\tilde {X}_K$. Hence, the vanishing follows from (HS3) applied to $Z_{d-1}\cap D_2\subset Z_{d-1}$, noting that $\big ((u_2\pi ^{m_2}-x_2)/u_2\pi ^{m_2}\big )_{|Z_{d-1}} \in 1+ x_2 \mathcal {O}_{Z_{d-1},\eta _2}$ so that $\beta _1\in K^{\rm M}_d(\mathcal {O}_{Z_{d-1},\eta _2},\mathfrak {m}_{\eta _2})$. Repeating the same arguments, we finally get

\[ 0=((a_K)_{|Z_e},\iota_r(\gamma))_{Z_e/K,(i(\delta'),z_e)} =((a_K)_{|Y_K},\gamma)_{Y_K/K,\delta}, \]

where $\iota _r: K^{\rm M}_e(\mathcal {O}_{Y_K,\delta _e}) \to K^{\rm M}_e(\mathcal {O}_{Z_e,z_e})$ is induced by the isomorphism $Z_e\to Y_K$ induced by the projection $\tilde {X}_K \to Y_K$ 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

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

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

5. Invariance of logarithmic cohomology under blowups

Retain the notation of § 4.

Theorem 5.2 For $F\in \operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{{\operatorname {Nis}}}$ and $\rho :\mathcal {Y}\to \mathcal {X}$ in $\Lambda ^{{\operatorname {fin}}}_{\rm ls}$, we have

(5.2.1)\begin{equation} \rho^*: {H_{\log}^{i}}(\mathcal{X},F) \cong {H_{\log}^{i}}(\mathcal{Y},F) \quad\forall i\geq 0. \end{equation}

Proof. Write $\mathcal {Y}=(Y,E)$ and $\mathcal {X}=(X,D)$. First we prove the theorem for $i=0$. We may assume that $D$ is reduced and $E=\rho ^*D$. By [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a, Proposition 1.9.2 b)], $\rho$ is invertible in $\operatorname {\mathbf {\underline {M}Cor}}$, so that $\rho ^*:F(\mathcal {X})\cong F(\mathcal {Y})$. Since this factors through $F(Y,E_{{\operatorname {red}}})$ by Theorem 4.2, we get (5.2.1) for $i=0$.

To show (5.2.1) for $i>0$, it suffices to prove $R^i\rho _* F_{(Y,E_{{\operatorname {red}}})} =0$. The problem is Nisnevich local, so we may assume that $\rho$ is induced by a blowup $\rho :Y\to X$ in a smooth center $Z\subset D$ normal crossing to $D$. By [Reference Kelly and SaitoKS21, Corollary 9], Nisnevich locally around a point of $Z$, $(X,D)$ is isomorphic to

\[ (\mathbf{A}^c,L_1+\cdots +L_r)\otimes \mathcal{W} \quad {\rm with}\ \mathcal{W}=(W,W^\infty)\in \operatorname{\mathbf{\underline{M}Cor}}_{\rm ls}, \]

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 5.3 Let $F\in \operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{{\operatorname {Nis}}}$ and $\mathcal {W}=(W,W^\infty )\in \operatorname {\mathbf {\underline {M}Cor}}_{\rm ls}$. 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 blowup at the origin $0\in \mathbf {A}^n$ and $\tilde {L}_i\subset Y$ be the strict transforms of $L_i$ for $1\leq i\leq n$ and $E=\rho ^{-1}(0)\subset Y$. For any $1\leq r\leq n$, we have

(5.3.1)\begin{equation} R^i\rho_{W*} F_{(Y, \tilde{L}_1 +\cdots +\tilde{L}_r + E)\otimes \mathcal{W}}=0\quad \text{for } i\ge 1, \end{equation}

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

Proof. The case $r=1$ is proved in [Reference Binda, Rülling and SaitoBRS22, Lemma 2.13] and we show the case $r=2$.Footnote ⁶ Put $D= L_1+ L_2$. By the case $i=0$ of Theorem 5.2, we get

(5.4.1)\begin{equation} F_{(\mathbf{A}^2,D)\otimes \mathcal{W}} \cong \rho_{W*} F_{(Y, \tilde{L}_1+\tilde{L}_2 + E)\otimes \mathcal{W}}. \end{equation}

Set

\[ \mathcal{F}:=F_{(Y, \tilde{L}_1+\tilde{L}_2 + E)\otimes \mathcal{W}}, \]

and $\mathbf {A}^2_W=\mathbf {A}^2\times W$ with the projection $p:A^2_W \to W$. Since $R^i\rho _{W*}\mathcal {F}$ for $i\ge 1$ is supported in $0\times W$, we have

\begin{align*} R^i\rho_{W*}\mathcal{F}=0&\Longleftrightarrow p_*R^i\rho_{W*}\mathcal{F}=0\\ &\Longleftrightarrow (p_*R^i\rho_{W*}\mathcal{F})_w=0\quad\forall w\in W\\ &\Longleftrightarrow H^0(\mathbf{A}^2_{W_w}, R^i\rho_{W*}\mathcal{F})=0\quad\forall w\in W, \end{align*}

where $W_w$ is the henselization of $W$ at $w$. Hence, it suffices to show $H^0(\mathbf {A}^2_W, R^i\rho _{W*}\mathcal {F})=0$, assuming $W$ is henselian local. Then we have

\[ H^j(\mathbf{A}^2_W, R^i\rho_{W*}\mathcal{F})=0, \quad\forall i,j\ge 1. \]

By (5.4.1) and [Reference Binda, Rülling and SaitoBRS22, Lemma 2.10],

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

Thus, the Leray spectral sequence yields

\[ H^0(\mathbf{A}_W^2, R^i\rho_{W*}\mathcal{F})= H^i(Y\times W, \mathcal{F}), \quad i\ge 0, \]

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

\[ \mathbf{A}^2=\operatorname{Spec} k[x,y]\quad\text{and}\quad L_1=V(x),\ L_2=V(y) \subset \mathbf{A}^2. \]

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

(5.4.2)\begin{equation} \tilde{L}_1=\pi_0^{-1}(0),\quad \tilde{L}_2=\pi_0^{-1}(\infty). \end{equation}

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 $U:=\mathbf {A}^1_s \times W$, $V:=(\mathbf {P}^1{\setminus} \{0\})\times W$ 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 (5.4.2), we have

(5.4.3)\begin{equation} \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}}. \end{equation}

Thus, [Reference Binda, Rülling and SaitoBRS22, Lemma 2.10] yields

\[ R^j\pi_*\mathcal{F}=0 \quad {\rm for}\ j\ge 1, \]

and it remains to show

(5.4.4)\begin{equation} {H^i(\mathbf{P}^1_W, \pi_*\mathcal{F})=0 \quad {\rm for}\ i\ge 1,} \end{equation}

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 (5.4.2), 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

\[ F_1:=\operatorname{\underline{Hom}}({\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathbf{A}^1_x, 0), F )\in \operatorname{\mathbf{CI}}^{\tau,{\rm sp}}_{{\operatorname{Nis}}}, \]

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, (5.4.4) follows from

\[ H^i(\mathbf{P}^1_W, F_{1, (\mathbf{P}^1,\infty)\otimes \mathcal{W}})=0 \quad {\rm for}\ i\ge 1, \]

which follows from [Reference SaitoSai20, Theorem 0.6].

Lemma 5.5 Let $N>2$ be an integer and assume that Proposition 5.3 holds for $n< N$. Let $(X,D)\in \operatorname {\mathbf {\underline {M}Cor}}_{\rm ls}$ and $Z\subset X$ be a smooth integral closed subscheme with $2\le {\operatorname {codim}}(Z,X)=:c < N$. Assume

\[ D= D_1+\cdots + D_r + D' \quad {with}\ r\leq c , \]

where $D_1,\dots,D_r$ are distinct and reduced irreducible components of $D$ containing $Z$, and $D'$ is an effective divisor on $X$ such that none of the component of $D'$ contains $Z$ and $Z$ is transversal to $|D'|$. Let $\rho : Y\to X$ be the blowup of $X$ in $Z$, let $\tilde {D}_i$ and $\tilde {D}'\subset Y$ be the strict transforms of $D_i$ and $D'$ respectively, and let $E_Z=\rho ^{-1}(Z)$. Then, for all $\mathcal {W}=(W,W^\infty )\in \operatorname {\mathbf {\underline {M}Cor}}_{\rm ls}$,

\[ R^i\rho_{W*} F_{(Y, \tilde{D}_1+\cdots + \tilde{D}_r +E_Z + \tilde{D}' )\otimes \mathcal{W}}=0 \quad\text{for } i\ge 1, \]

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

Proof. This proof is adapted from [Reference Binda, Rülling and SaitoBRS22, Lemma 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 $V_{(z)}:= V\times _{X\times W} \operatorname {Spec} A$. By assumption we find a regular system of local parameters $t_1,\ldots, t_m$ of $A$, such that

\[ (D_i\times W)_{(z)}=V(t_i) \quad {\rm for}\ 1\leq i\leq r,\quad (Z\times W)_{(z)}=V(t_1,\ldots, t_c), \]

\[ (D'\times W)_{(z)}=V(t_{c+1}^{e_{c+1}}\cdots t_{m_0}^{e_{m_0}}) \quad {\rm with}\ c+1\leq m_0\le m, \]

\[ (X\times W^\infty)_{(z)}=V(t_{m_0+1}^{e_{m_0+1}}\cdots t_{m_1}^{e_{m_1}}) \quad {\rm with}\ m_0\le m_1\leq m. \]

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 blowup in $0$. By the above,

\[ \rho_W: (Y,\tilde{D}_1+\cdots + \tilde{D}_r +E_Z + \tilde{D}' )\otimes \mathcal{W}\to (X, D)\otimes \mathcal{W} \]

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

\[ (\widetilde{\mathbf{A}^c},\tilde{L}_1+\cdots+\tilde{L}_r + E)\otimes \mathcal{W}' \to (\mathbf{A}^c, L_1+\cdots+ L_r)\otimes \mathcal{W}', \]

\[ \bigg(\mathcal{W}'=\bigg(\mathbf{A}_K^{m-c}, \bigg(\prod_{i=c+1}^{m_1} t_{i}^{e_{i}}\bigg)\bigg)\bigg) \]

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 5.3. Hence, the statement follows from the proposition for $n=c< N$.

Proof of Proposition 5.3 The proof is by induction on $n\geq 2$. The case $n=2$ follows from Lemma 5.4. Assume that $n>2$ and that the proposition is proven for $\mathbf {A}^m$ with $m< n$. For $r=1$, Proposition 5.3 is proved in [Reference Binda, Rülling and SaitoBRS22, Theorem 2.12]. Assume that $r\geq 2$. Let $Z:=L_1\cap L_2\subset \mathbf {A}^n$ and $\tilde {Z}\subset Y$ be the strict transform of $Z$. Denote by $\rho ': Y'\to Y$ the blowup of $Y$ in $\tilde {Z}$, let $\tilde {L}_i', E'\subset Y'$ be the strict transforms of $\tilde {L}, E$ respectively, and let $E''=(\rho ')^{-1}(\tilde {Z})$. 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 5.5,

\[ R^i\rho'_{W*} F_{(Y', \tilde{L}_1'+\cdots+\tilde{L}_r' + E' + E'')\otimes \mathcal{W}}=0\quad {\rm for}\ i\ge 1. \]

Since Theorem 5.2 has been proved for $i=0$, we have

\[ \rho'_* F_{(Y', \tilde{L}_1'+\cdots+\tilde{L}_r' + E' + E'')\otimes \mathcal{W}} = F_{(Y, \tilde{L}_1+\cdots+\tilde{L}_r + E)\otimes \mathcal{W}}. \]

Hence, we obtain

(5.5.1)\begin{equation} {R^i\rho_{W*} F_{(Y, \tilde{L}_1+\cdots+\tilde{L}_r + E)\otimes \mathcal{W}}= R^i(\rho\rho')_{W*} F_{(Y', \tilde{L}_1'+\cdots+\tilde{L}_r' + E'+ E'')\otimes \mathcal{W}}. } \end{equation}

Denote by $\sigma :\hat {Y} \to \mathbf {A}^n$ the blowup in $Z$, let $\hat {L}_i\subset \hat {Y}$ be the strict transform of $L_i$, and let $\Xi =\sigma ^{-1}(Z)$. By Lemma 5.5 we get

(5.5.2)\begin{equation} {R^i\sigma_{W*} F_{(\hat{Y}, \hat{L}_1+\cdots+\hat{L}_r+ \Xi)\otimes \mathcal{W}}=0 \quad {\rm for}\ i\ge 1.} \end{equation}

Denote by $\sigma ':\hat {Y}'\to \hat {Y}$ the blowup in $\hat {Z}=\sigma ^{-1}(0)\subset \Xi$, let $\hat {L}'_i,\; \Xi '\subset \hat {Y}'$ be the strict transforms of $\hat {L}_i,\; \Xi$ respectively, and let $\Xi ''=\sigma '^{-1}(\hat {Z})$. Note that $\hat {Z}\subset \hat {L}_3\cap \cdots \cap \hat {L}_n\cap \Xi$ and ${\operatorname {codim}}(\hat {Z}, \hat {Y})= n-1$ and $\hat {Z}$ intersects transversally with $\hat {L}_1+ \hat {L}_2$. Thus, by Lemma 5.5 and the case $i=0$ of Theorem 5.2, we obtain

(5.5.3)\begin{equation} {R\sigma'_{W*}F_{(\hat{Y}',\hat{L}_1'+\cdots+\hat{L}_r'+ \Xi'+\Xi'')\otimes \mathcal{W}}= F_{(\hat{Y}, \hat{L}_1+\cdots+\hat{L}_r+ \Xi)\otimes \mathcal{W}}.} \end{equation}

Finally, by [Reference Binda, Rülling and SaitoBRS22, Lemma 2.15], there is an isomorphism of $\mathbf {A}^n\times W$-schemes

(5.5.4)\begin{equation} { (\hat{Y}',\hat{L}_1',\dots,\hat{L}_r,\Xi',\Xi'')\cong (Y', \tilde{L}_1',\dots,\tilde{L}_r',E',E'').} \end{equation}

Altogether we obtain, for $i\ge 1$,

\begin{align*} R^i\rho_{W*} F_{(Y, \tilde{L}_1+\cdots+\tilde{L}_r + E)\otimes \mathcal{W}} &= R^i(\rho\rho')_{W*} F_{(Y', \tilde{L}_1'+\cdots+\tilde{L}_r' + E' + E'')\otimes \mathcal{W}}, & & \text{by (5.5.1)},\\ & = R^i(\sigma\sigma')_{W*} F_{(\hat{Y}',\hat{L}_1'+\cdots+\hat{L}_r'+ \Xi'+\Xi'')\otimes \mathcal{W}}, & & \text{by (5.5.4)},\\ & = R^i\sigma_{W*} F_{(\hat{Y}, \hat{L}_1+\cdots+\hat{L}_r+ \Xi)\otimes \mathcal{W}}, & & \text{by (5.5.3)},\\ & = 0, & & \text{by (5.5.2)}. \end{align*}

This completes the proof of the proposition.

Remark 5.6 For simplicity, we write

\[ {H_{\log}^{i}}(-,F)={H_{\log}^{i}}(-,\underline{\omega}^{\operatorname{\mathbf{CI}}} F)\quad {\rm for}\ F\in {\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}}. \]

By [Reference Rülling and SaitoRS21a, Corollary 6.8], if ${\operatorname {ch}}(k)=0$ and $F=\Omega ^i$, we have

\[ {H_{\log}^{i}}(-,\Omega^i)=H^i(X,\Omega^i(\log |D|) \quad {\rm for}\ (X,D)\in \operatorname{\mathbf{\underline{M}Cor}}_{\rm ls}\!. \]

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 notation as [Reference Binda, Park and ØstværBPØ22].

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 [Reference Binda, Park and ØstværBPØ22, Definition 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 ${\operatorname {\mathbf {PSh}}^{\rm ltr}}$ be the category of additive presheaves of abelian groups on ${\operatorname {l\mathbf {Cor}}}$ and ${\operatorname {\mathbf {Shv}}^{\rm ltr}_{\rm dNis}}\subset {\operatorname {\mathbf {PSh}}^{\rm ltr}}$ be the full subcategory consisting of those $\mathcal {F}$ whose restrictions to ${\operatorname {l\mathbf {Sm}}}$ are dividing Nisnevich sheaves (see [Reference Binda, Park and ØstværBPØ22, Definition 3.1.4]). It is shown in [Reference Binda, Park and ØstværBPØ22, Theorem 1.2.1 and Proposition 4.7.5] that ${\operatorname {\mathbf {Shv}}^{\rm ltr}_{\rm dNis}}$ is a Grothendieck abelian category and there is an equivalence of categories

(6.0.1)\begin{equation} {{\operatorname{\mathbf{Shv}}^{\rm ltr}_{\rm dNis}} \simeq {\operatorname{\mathbf{Shv}}^{\rm ltr}_{\rm dNis}}({\operatorname{\mathbf{Sm}l\mathbf{Sm}}}),} \end{equation}

where the right-hand side denotes the full subcategory of the category ${\operatorname {\mathbf {PSh}}^{\rm ltr}}({\operatorname {\mathbf {Sm}l\mathbf {Sm}}})$ 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.

We now construct a functor

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

For $\mathfrak {X}=(X,\mathcal {M})\in {\operatorname {\mathbf {Sm}l\mathbf {Sm}}}$, we put $\mathfrak {X}^{\rm MP}=(X, \partial \mathfrak {X})$, where $\partial \mathfrak {X}\subset X$ is the closed subscheme consisting of the points where the log structure $\mathcal {M}$ is not trivial. By [Reference Binda, Park and ØstværBPØ22, Lemma A.5.10], $\partial \mathfrak {X}$ with reduced structure is a normal crossing divisor on $X$, so that we can view $\mathfrak {X}^{\rm MP}$ as an object of $\operatorname {\mathbf {\underline {M}Cor}}_{\rm ls}$. For $F\in \operatorname {\mathbf {\underline {M}PST}}_{\log }$ and $\mathfrak {X}\in {\operatorname {\mathbf {Sm}l\mathbf {Sm}}}$, we put

(6.0.3)\begin{equation} {F^{\log}(\mathfrak{X})= F(\mathfrak{X}^{\rm MP}).} \end{equation}

Take $\mathfrak {Y} \in {\operatorname {\mathbf {Sm}l\mathbf {Sm}}}$ and $\alpha \in {\operatorname {l\mathbf {Cor}}}(\mathfrak {Y},\mathfrak {X})$. By [Reference Binda, Park and ØstværBPØ22, Definition 2.1.1 and Remark 2.1.2(iii)], we have

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

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

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

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

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

(6.0.4)\begin{equation} { \mathcal{L}og : \operatorname{\mathbf{\underline{M}PST}}_{\log} \to {\operatorname{\mathbf{PSh}}^{\rm ltr}}({\operatorname{\mathbf{Sm}l\mathbf{Sm}}}),\quad F \to F^{\log}.} \end{equation}

By the definitions of sheaves ([Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a, Definition 1], [Reference Binda, Park and ØstværBPØ22, Definition 3.1.4] and [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a, Proposition 1.9.2]), this induces a functor

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

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 $\mathcal {X}=\mathfrak {X}^{\rm MP}\in \operatorname {\mathbf {\underline {M}Cor}}_{\rm ls}$, we have

(6.0.5)\begin{equation} {H^i_{\operatorname{Nis}}(X,F_{\mathcal{X}}) = H^i_{s\mathrm{Nis}}(\mathfrak{X},F^{\log})\; (F^{\log}=\mathcal{L}og(F)),} \end{equation}

where the right-hand side is the cohomology for the strict Nisnevich topology (see [Reference Binda, Park and ØstværBPØ22, Definition 4.3.1]).

Theorem 6.1 For $F\in \operatorname {\mathbf {CI}}^{\tau,{\rm sp}}_{{\operatorname {Nis}}}$, $F^{\log }=\mathcal {L}og(F)\in {\operatorname {\mathbf {Shv}}^{\rm ltr}_{\rm dNis}}$ is strictly ${\overline {\square }}$-invariant in the sense of [Reference Binda, Park and ØstværBPØ22, Definition 5.2.2]. For $\mathfrak {X}\in {\operatorname {\mathbf {Sm}l\mathbf {Sm}}}$ with $\mathcal {X}=\mathfrak {X}^{\rm MP}\in \operatorname {\mathbf {\underline {M}Cor}}_{\rm ls}$, we have a natural isomorphism

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

where $\operatorname {\mathbf {logDM}}^{{\operatorname {eff}}}$ is the triangulated category of logarithmic motives defined in [Reference Binda, Park and ØstværBPØ22, Definition 5.2.1].

Proof. Let $\mathfrak {X}_{\rm div}^{\rm Sm}$ be the category of log modifications $\mathfrak {Y} \to \mathfrak {X}$ such that $\mathfrak {Y}\in {\operatorname {\mathbf {Sm}l\mathbf {Sm}}}$ (see [Reference Binda, Park and ØstværBPØ22, Definition A.11.12]) and $\mathfrak {X}_{\rm divsc}^{\rm Sm}\subset \mathfrak {X}_{\rm div}^{\rm Sm}$ be the full subcategory given by those maps $\mathfrak {Y}\to \mathfrak {X}$ that are isomorphic to compositions of log modifications along smooth centers (see [Reference Binda, Park and ØstværBPØ22, Definitions 4.4.4 and A.14.10]). We have isomorphisms

\begin{align*} H^i_{\operatorname{Nis}}(X,F_{\mathcal{X}}) &\overset{(6.0.5)}{\simeq} H^i_{s\mathrm{Nis}}(\mathfrak{X},F^{\log}) \overset{(*1)}{\simeq} \mathop {\varinjlim}_{\mathfrak{Y}\in \mathfrak{X}_{\rm divsc}^{\rm Sm}} H^i_{s\mathrm{Nis}}(\mathfrak{Y},F^{\log}) \\ &\overset{(*2)}{\simeq} \mathop {\varinjlim}_{\mathfrak{Y}\in \mathfrak{X}_{\rm div}^{\rm Sm}} H^i_{s\mathrm{Nis}}(\mathfrak{Y},F^{\log}) \overset{(*3)}{\simeq} H^i_{d\mathrm{Nis}}(\mathfrak{X},F^{\log}), \end{align*}

where $(*2)$ follows from [Reference Binda, Park and ØstværBPØ22, Corollary 4.4.5] and $(*3)$ from [Reference Binda, Park and ØstværBPØ22, Theorem 5.1.8], and $(*1)$ is a consequence of Theorem 5.2 in view of (6.0.5) and the fact that a log modification of $\mathfrak {X}=(X,\mathcal {M})\in {\operatorname {\mathbf {Sm}l\mathbf {Sm}}}$ along smooth center is induced Zariski locally by a blowup of $X$ in an intersection of irreducible components of $\partial \mathfrak {X}$ so that it corresponds to a morphism in $\Lambda ^{{\operatorname {fin}}}_{\rm ls}$ from Definition 5.1.

Hence, the strict ${\overline {\square }}$-invariance of $F^{\log }$ follows from [Reference SaitoSai20, Theorem 0.6]. Finally, (6.1.1) follows from [Reference Binda, Park and ØstværBPØ22, Proposition 5.2.3].

We now consider the composite functor

\[ \mathcal{L}og': {\operatorname{\mathbf{RSC}}}_{\operatorname{Nis}} \overset{\underline{\omega}^{\operatorname{\mathbf{CI}}}}{\longrightarrow} \operatorname{\mathbf{CI}}^{\tau,{\rm sp}}_{{\operatorname{Nis}}} \overset{\mathcal{L}og}{\longrightarrow} \operatorname{\mathbf{CI}}^{\rm ltr}_{\rm dNis}, \]

where $\operatorname {\mathbf {CI}}^{\rm ltr}_{\rm dNis}\subset {\operatorname {\mathbf {Shv}}^{\rm ltr}_{\rm dNis}}$ is the full subcategory consisting of strictly ${\overline {\square }}$-invariant objects. By [Reference Binda and MericiBM12, Theorem 5.7], $\operatorname {\mathbf {CI}}^{\rm ltr}_{\rm dNis}$ is a Grothendieck abelian category.

In what follows, we let

(6.2.1)\begin{equation} { \mathcal{L}og : {\operatorname{\mathbf{RSC}}}_{\operatorname{Nis}} \to \operatorname{\mathbf{CI}}^{\rm ltr}_{\rm dNis}\;:\; F\to F^{\log} } \end{equation}

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

(6.2.2)\begin{equation} { F^{\log}(X,\mathrm{triv})= F(X)\quad {\rm for}\ F\in {\operatorname{\mathbf{RSC}}}_{{\operatorname{Nis}}},\; X\in \operatorname{\mathbf{Sm}},} \end{equation}

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

Proof. First we prove the full faithfulness. 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 ${\operatorname {\mathbf {Shv}}^{\rm ltr}_{\rm dNis}}$. 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 [Reference Binda, Park and ØstværBPØ22, Example 2.1.3(3)],

\[ \operatorname{\mathbf{Cor}}(Y,X)= {\operatorname{l\mathbf{Cor}}}(Y,\mathrm{triv}),(X,\mathrm{triv}))\quad {\rm for}\ X,Y\in \operatorname{\mathbf{Sm}}\!. \]

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

where $j^*$ are induced by the natural map $(X\backslash \partial \mathfrak {X},\mathrm {triv})\to \mathfrak {X}$ of log schemes and are 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.

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

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

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