Proof. Let C be the cokernel of $\varphi $ in $\operatorname {\mathbf {\underline {M}PST}}$. We want to show ${\underline {a}}_{\operatorname {Nis}}(C)=0$. For $\mathcal {X}\in \operatorname {\mathbf {\underline {M}Cor}}$, set $C_{\mathcal {X}}= \operatorname {Coker}(\varphi _{\mathcal {X}}: F_{\mathcal {X}}\to G_{\mathcal {X}})$ in the category of presheaves on $({\operatorname {\acute {e}t}}/\overline {X})$; denote by $C_{\mathcal {X},{\operatorname {Nis}}}$ its Nisnevich sheafification. By (1.2.4), it suffices to show $C_{\mathcal {X},{\operatorname {Nis}}}=0$, if $\overline {X}$ is normal. The latter is equivalent to the surjectivity of $\varphi _{\mathcal {X}}$ in the category of Nisnevich sheaves on $\overline {X}$, which is equivalent to the statement.

Proof. (1). First assume $G={\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}(\mathcal {X})$, for some $\mathcal {X}\in \operatorname {\mathbf {\underline {M}Cor}}$. In this case, $\operatorname {\underline {Hom}}(G, F)(\mathcal {Y})=F(\mathcal {X}\otimes \mathcal {Y})$. Clearly, this defines a cube invariant Nisnevich sheaf. It has M-reciprocity by [Reference SaitoSai20a, Lemma 1.27(2)] and has semipurity by [Reference SaitoSai20a, Lemma 1.29(2)]. Hence, $\operatorname {\underline {Hom}}(G,F)\in \operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$ in this case. In the general, case consider a resolution:

$$\begin{align*}\bigoplus_j {\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathcal{Y}_j)\to \bigoplus_i {\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathcal{X}_i)\to G\to 0.\end{align*}$$

We obtain an exact sequence:

(1.5.1)$$ \begin{align}0\to \operatorname{\underline{Hom}}(G, F)\to \prod_i \operatorname{\underline{Hom}}({\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathcal{X}_i), F)\to \prod_j \operatorname{\underline{Hom}}({\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathcal{Y}_j), F). \end{align} $$

This directly implies cube invariance and semipurity. The sheaf property holds since ${\underline {i}}_{\operatorname {Nis}}{\underline {a}}_{\operatorname {Nis}}: \operatorname {\mathbf {\underline {M}PST}}\to \operatorname {\mathbf {\underline {M}PST}}$ is left exact, where ${\underline {i}}_{\operatorname {Nis}}$ is the forgetful functor. In general, M-reciprocity won’t hold since $\tau _!\tau ^*$ does not commute with infinite products; however, it clearly holds if the first product in (1.5.1) is finite and by assumption we find such a resolution. (2) follows from (1) and adjunction.

Proof. The map $\theta _{m,n}$ is the composition of the two isomorphisms:

$$ \begin{align*} h_0({\overline{\square}}^{(m,n)})(R)& \xrightarrow{\simeq\, (*)} \operatorname{Pic}(\mathbf{P}^1_R, m\cdot 0+ n\cdot \infty)\\ & \xrightarrow{\simeq \, (**)} ((R[t]/t^m)^{\times}\oplus (R[z]/z^n)^{\times})/R^{\times}\oplus \mathbb{Z}, \end{align*} $$

which are defined as follows. We denote by $F_R:=m\cdot 0_R+ n\cdot \infty _R\subset \mathbf {P}^1_R$ the closed subscheme: (*) is induced by the classical map from Weil to Cartier divisors:

$$\begin{align*}{\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}({\overline{\square}}^{(m,n)})(R)\ni D\mapsto (\mathcal{O}(D), {\operatorname{id}}_{\mathcal{O}_{F_R}})\in \operatorname{Pic}(\mathbf{P}^1_R, F_R),\end{align*}$$

where $\mathcal {O}(D)$ is the line bundle on $\mathbf {P}^1_R$ given by $\mathcal {O}(D)(U)=\{f\in R(t)^{\times }\mid \operatorname {div}_U(f)\ge D \}$; it is an isomorphism by [Reference Rülling and YamazakiRY16, Theorem 1.1]. For (**), consider the exact sequence:

$$\begin{align*}H^0(\mathbf{P}^1_R, \mathcal{O}^{\times})\to H^0(F_R, \mathcal{O}^{\times})\to \operatorname{Pic}(\mathbf{P}^1_R, F_R) \to \operatorname{Pic}(\mathbf{P}_R^1)\to \operatorname{Pic}(F_R).\end{align*}$$

The last map decomposes as $\operatorname {Pic}(R)\oplus \operatorname {Pic}(\mathbf {P}^1) \to \operatorname {Pic}(F_R)$ given by:

$$\begin{align*}(M,\mathcal{O}(\{1\})^{\otimes n})\mapsto (M_{|\mathbf{P}^1_R}\otimes (\mathcal{O}(\{1\})^{\otimes n})_{|F_R}=M_{|F_R}.\end{align*}$$

Since $F_R\to \operatorname {Spec} R$ has a section, the map $\operatorname {Pic}(R)\to \operatorname {Pic}(F_R)$ is injective. Hence, the above sequence yields an exact sequence:

$$\begin{align*}H^0(\mathbf{P}^1_R, \mathcal{O}^{\times})\to H^0(F_R, \mathcal{O}^{\times})\to \operatorname{Pic}(\mathbf{P}^1_R, F_R) \xrightarrow{d} \mathbb{Z} \to 0,\end{align*}$$

where $d(L,\alpha ):=d(L):=\deg (L_{|\mathbf {P}^1_{{\operatorname {Frac}}(R)}})$; we can choose a splitting of d by $r\mapsto (\mathcal {O}_{\mathbf {P}^1_R}(\{1\})^{\otimes r}, {\operatorname {id}}_{\mathcal {O}_{F_R}})$; the map in the middle sends $u\in H^0(F_R, \mathcal {O}^{\times })$ to $(\mathcal {O}_{\mathbf {P}^1_R}, u\cdot : \mathcal {O}_{F_R}\xrightarrow {\simeq } \mathcal {O}_{F_R})$, where $u\cdot $ is the isomorphism given by multiplication by u. Let $(L,\alpha )$ be a pair with L a line bundle on $\mathbf {P}^1_R$ with $d(L)=r$ and $\alpha : \mathcal {O}_{F_R}\xrightarrow {\simeq } L_{|F_R}$ an isomorphism; we find an isomorphism $\varphi :L\otimes \mathcal {O}(\{1\})^{\otimes -r}\xrightarrow {\simeq } \mathcal {O}_{\mathbf {P}_R^1}$ and define the isomorphism $\alpha '$ as the composition:

$$\begin{align*}\alpha'=(\alpha^{\prime}_{m\cdot 0},\alpha^{\prime}_{n\cdot \infty}): \mathcal{O}_{F_R}\xrightarrow{\simeq\, \alpha}L_{|F_R} = (L\otimes\mathcal{O}(\{1\})^{\otimes -r})_{|F_R} \xrightarrow{\varphi_{|F_R}} (\mathcal{O}_{\mathbf{P}^1})_{|F_R}, \end{align*}$$

where the equality follows from the fact that we have a canonical identification $\mathcal {O}(\{1\})_{|F_R} = \mathcal {O}_{|F_R}$. Hence, $\varphi $ induces an isomorphism $(L\otimes \mathcal {O}(\{1\})^{\otimes -r}, \alpha )\cong (\mathcal {O}_{\mathbf {P}^1_R}, \alpha ')$; the isomorphism (**) is given by:

$$\begin{align*}(L,\alpha)\mapsto (\alpha^{\prime}_{m\cdot0}(1), \alpha^{\prime}_{n\cdot \infty}(1), d(L)).\end{align*}$$

Let $Z=V(g)\in {\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}({\overline {\square }}^{(m,n)})(R)$ be a prime correspondence as in the statement. Write $t=T_0/T_1$, and let $G\in R[T_0, T_1]$ be the homogenisation of g. We have an isomorphism:

$$\begin{align*}\mathcal{O}(Z)\otimes \mathcal{O}(\{1\})^{\otimes -r}= \mathcal{O}_{\mathbf{P}^1_R}\cdot \tfrac{(T_0-T_1)^r}{G} \xrightarrow{\simeq}\mathcal{O}_{\mathbf{P}^1_R},\end{align*}$$

where the second isomorphism is given by multiplication with $G/(T_0-T_1)^r$. Thus, $\theta _{m,n}$ admits the description from the statement, where $z=1/t$. The commutativity of the diagram follows directly from this.

Proof. The second isomorphism in (2.4.1) holds by Lemma 2.2 and Remark 2.3; the unit map is injective by semipurity. We show the surjectivity of the composite map:

(2.4.4)$$ \begin{align}h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{sp}}({\overline{\square}}^{(m,n)})\to h_{0,{\operatorname{Nis}}}^{{\overline{\square}},\mathrm{sp}}(\overline{\square}^{(1)}) \to \underline{\omega}^*(\mathbf{G}_m\oplus \mathbb{Z}) \end{align} $$

for $m,n\ge 1$. By Lemma 1.3, it suffices to show the surjectivity on $(\operatorname {Spec} R, (f))$, where R is an integral normal local k-algebra and $f\in R\setminus \{0\}$, such that $R_f$ is regular. Denote by:

$$\begin{align*}\psi: {\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}({\overline{\square}}^{(m,n)})(R,f)\to R_f^{\times}\oplus\mathbb{Z}\end{align*}$$

the precomposition of (2.4.4) evaluated at $(R,f)$ with the quotient map:

$$\begin{align*}{\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}({\overline{\square}}^{(m,n)})(R,f)\to h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{ sp}}({\overline{\square}}^{(m,n)})(R,f).\end{align*}$$

By Lemma 2.2:

(2.4.5)$$ \begin{align}\psi(V(a_0+ a_1t+\ldots +a_r t^r))= ((-1)^r a_0/a_r, r), \end{align} $$

provided that $Z=V(a_0+ a_1t+\ldots +a_r t^r)$ is an admissible prime correspondence and $a_i\in R_f$. We claim that $\psi $ is surjective. To this end, observe that for $a\in R_f^{\times }$, we find $N\ge 0$ and $b\in R$, such that:

(2.4.6)$$ \begin{align}ab=f^{nN}, \quad \text{and}\quad af^{mN}\in R. \end{align} $$

Set $W:=V(t^{mnN} + (-1)^{mnN} a)\subset \operatorname {Spec} R_f[t, 1/t]$ and $K={\operatorname {Frac}}(R)$. Let $t^{mnN} + (-1)^{mnN} a=\prod _i h_i$ be the decomposition into monic irreducible factors in $K[t,1/t]$, and denote by $W_i\subset \operatorname {Spec} R_f[t, 1/t]$ the closure of $V(h_i)$ (note that $W_i=W_j$ for $i\neq j$ is allowed). The $W_i$ correspond to the components of W which are dominant over $R_f$; since W is finite (the polynomial defining W is monic) and surjective over $R_f$, so are the $W_i$. We claim:

(2.4.7)$$ \begin{align}W_i\in {\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}({\overline{\square}}^{(m,n)})(R,f). \end{align} $$

Indeed, let $I_i$ (respectively, $J_i$) be the ideal of the closure of $W_i$ in $\operatorname {Spec} R[t]$ (respectively, $\operatorname {Spec} R[z]$ with $z=1/t$). By (2.4.6):

$$\begin{align*}bt^{nmN} + (-1)^{mnN} f^{nN}\in I_i \quad \text{and} \quad f^{mN} + (-1)^{mnN} f^{mN}a z^{mnN}\in J_i.\end{align*}$$

Hence, $(f/t^m)^{nN}\in R[t]/I_i$ and $(f/z^n)^{mN}\in R[z]/J_i$. It follows that $f/t^m$ (respectively, $f/z^n$) is integral over $R[t]/I_i$ (respectively, $R[z]/J_i$); thus, (2.4.7) holds. Put:

$$\begin{align*}W_a= \sum_i W_i \in{\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}({\overline{\square}}^{(m,n)})(R,f).\end{align*}$$

We claim:

(2.4.8)$$ \begin{align}\psi(W_a)=(a, mnN)\in R_f^{\times}\oplus \mathbb{Z}. \end{align} $$

Indeed, it suffices to show this after restriction to the generic point of R, in which case, it follows directly from the definition of the $W_i$ and (2.4.5). This implies the surjectivity of $\psi $ and that of (2.4.4). Next, we show that (2.4.4) has a splitting. Let $\omega _a^{m,n}\in h_{0,{\operatorname {Nis}}}^{{\overline {\square }}, \mathrm {sp}}({\overline {\square }}^{(m,n)})(R,f)$ be the class of $W_a$ and $\lambda _a^{m,n}=\omega _a^{m,n}-\omega _1^{m,n}$, where $\omega _1^{m,n}$ is defined as $\omega _a^{m,n}$ replacing a by $1$ (and using the same N). By (2.4.8), the image of $\lambda _a^{m,n}$ under the map (2.4.4):

$$\begin{align*}h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{sp}}({\overline{\square}}^{(m,n)})(R,f) \to R_f^{\times}\oplus \mathbb{Z}\end{align*}$$

is $(a,0)$.

Claim 2.4.1. $\lambda _a^{m,n}$ is independent of the choice of N, and we have:

(2.4.9)$$ \begin{align}\lambda_{ab}^{m,n} =\lambda_a^{m,n} + \lambda_b^{m,n}\quad\text{ for } a,b \in R_f^{\times}. \end{align} $$

Moreover, for $m'\geq m$ and $n'\geq n$, the image of $\lambda ^{m',n'}_a$ under:

$$\begin{align*}h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{sp}}({\overline{\square}}^{(m',n')}) \to h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{sp}}({\overline{\square}}^{(m,n)})\end{align*}$$

coincides with $\lambda _a^{m,n}$.

By the semipurity of $h_{0,{\operatorname {Nis}}}^{{\overline {\square }}, \mathrm {sp}}({\overline {\square }}^{(m,n)})$ and [Reference SaitoSai20a, Theorem 3.1], we have an injective homomorphism:

(2.4.10)$$ \begin{align}h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{sp}}({\overline{\square}}^{(m,n)})(R,f) \hookrightarrow \underline{\omega}_! h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{ sp}}({\overline{\square}}^{(m,n)})(K)=h_0({\overline{\square}}^{(m,n)})(K). \end{align} $$

By Lemma 2.2, the isomorphism:

$$\begin{align*}\theta_{m,n}: h_0({\overline{\square}}^{(m,n)})(K)\xrightarrow{\simeq} ((K[t]/t^m)^{\times} \oplus (K[z]/z^n)^{\times})/K^{\times} \oplus \mathbb{Z}\end{align*}$$

sends $\omega ^{m,n}_a$ to:

$$\begin{align*}\theta_{m,n}(\omega^{m,n}_a) =\left (\frac{(-1)^{mnN}a}{(t-1)^{mnN}}, \frac{1}{(1-z)^{mnN}}, mnN\right).\end{align*}$$

Thus, $\theta _{m,n}(\lambda ^{m,n}_a)= (a,1,0)$, which is independent of N. By the injectivity of (2.4.10), this implies the first two assertions of the claim; similarly, the final assertion of the claim follows from the commutative diagram in Lemma 2.2. Since $\lambda _a^{m,n}$ does not change if we replace f by $uf$ with $u\in R^{\times }$, the map $a\to \lambda _a^{m,n}$ glues to give a global morphism of Nisnevich sheaves which induces the splitting $s_{m,n}$ from the statement. It remains to check that $s_{m,n}$ is compatible with transfers. To this end, it suffices to check that ${\underline {\omega }}_!(s_{m,n})$ is compatible with transfers, and since the transition maps are surjective, it further suffices to show that:

$$\begin{align*}\varprojlim_{m,n} {\underline{\omega}}_!(s_{m,n}): \mathbf{G}_m\oplus \mathbb{Z}\to \varprojlim_{m,n} {\underline{\omega_!}}h^{{\overline{\square}}, \mathrm{sp}}_{0,{\operatorname{Nis}}}({\overline{\square}}^{(m,n)})\end{align*}$$

is compatible with transfers. Since we can identify the target with $\mathbb {W}\oplus \mathbb {W}\oplus \mathbf {G}_m\oplus \mathbb {Z}$ by Remark 2.3, the compatibility holds automatically by [Reference Merici and SaitoMS20, Lemma 1.1].

Proof. It is direct to check that $\psi $ induces a morphism (2.5.1). To check the factorisation statement, we may replace F by $\operatorname {\underline {Hom}}({\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}(\mathcal {X}), F)$ to reduce to the case $\mathcal {X}=(\operatorname {Spec} k,\emptyset )$ (see Lemma 1.5(1)). By Yoneda and (1.4.2), we are reduced to show that we have a factorisation as follows:

(2.5.2)

By [Reference Merici and SaitoMS20, Lemma 1.14(iii)] and Lemma 2.4, the map a is surjective. Thus, we have to show $\psi (\operatorname {Ker} a)=0$. By semipurity, it suffices to show that we have a factorisation as in (2.5.2) after applying $\underline {\omega }_!$. By [Reference Rülling, Sugiyama and YamazakiRSY22, Proposition 5.6], we have:

$$\begin{align*}H:=\underline{\omega}_!(h_{0,{\operatorname{Nis}}}^{{\overline{\square}}, \mathrm{ sp}}(\overline{\square}^{(1)}_x\otimes\overline{\square}^{(1)}_s))=h_{0,{\operatorname{Nis}}}(\overline{\square}^{(1)}_x\otimes\overline{\square}^{(1)}_s)=\mathcal{K}^M_2\oplus \mathbf{G}_m\oplus\mathbf{G}_m\oplus \mathbb{Z},\end{align*}$$

where $\mathcal {K}^M_2$ is the (improved) Milnor K-theory sheaf; in particular, H is $\mathbf {A}^1$-invariant. Thus, $\underline {\omega }_!(\psi )$ and $\underline {\omega }_!(a)$ factor via $h^{\mathbf {A}^1}_{0,{\operatorname {Nis}}}(h_{0,{\operatorname {Nis}}}(\overline {\square }^{(1)}_y\otimes {\overline {\square }}^{(2)}_s))$ (cf. (1.6.2)). Thus, we obtain solid arrows in $\operatorname {\mathbf {NST}}$:

(2.5.3)

Since $\bar {a}$ is the composition of the natural isomorphisms (cf. (1.6.3)):

$$\begin{align*}h^{\mathbf{A}^1}_{0,{\operatorname{Nis}}}(h_{0,{\operatorname{Nis}}}(\overline{\square}^{(1)}_y\otimes{\overline{\square}}^{(2)}_s)) \cong h^{\mathbf{A}^1}_{0,{\operatorname{Nis}}}({\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathbf{A}^1_y\setminus\{0\})\otimes_{{\operatorname{\mathbf{PST}}}} {\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathbf{A}^1_s\setminus\{0\})) \cong H\end{align*}$$

the dotted arrow exists, which completes the proof.

Proof. This is an immediate consequence of [Reference SaitoSai20a, Corollary 8.6(3)].

Proof. First consider the case $n=1$. Set $E:=E_1$ and ${\overline {\square }}^{(E,r)}:=(\mathbf {P}^1, E+ r\cdot \infty )$, for $r\ge 1$. The natural morphism ${\overline {\square }}^{(E,r)}\to {\overline {\square }}$ induces a map $F_{\mathcal {X}\otimes {\overline {\square }}} \to F_{\mathcal {X}\otimes {\overline {\square }}^{(E,r)}}$. The cohomology sheaves of the cone C of this map are supported in $X\times |E+\infty |$, whence $R^i\overline {\pi }_*C=0$, $i\ge 1$, where $\overline {\pi }:\mathbf {P}^1_X\to X$ is the projection. We obtain surjections:

$$\begin{align*}R^i\overline{\pi}_*F_{\mathcal{X}\otimes{\overline{\square}}}\to R^i\overline{\pi}_*F_{\mathcal{X}\otimes{\overline{\square}}^{(E,r)}}\to 0,\quad \text{for all } i\ge 1.\end{align*}$$

By the cube invariance of cohomology (see [Reference SaitoSai20a, Theorem 9.3]), the left term vanishes. Thus, M-reciprocity (see [Reference SaitoSai20a, Lemma 1.27(1)]) yields:

$$\begin{align*}0=\varinjlim_r R^i\overline{\pi}_*F_{\mathcal{X}\otimes{\overline{\square}}^{(E,r)}}= R^i\overline{\pi}_* j_*F_{(\mathbf{A}^1, E)\otimes\mathcal{X}},\end{align*}$$

where $j:\mathbf {A}^1_X\hookrightarrow \mathbf {P}^1_X$ is the open immersion. Together with Lemma 2.9, we obtain:

$$\begin{align*}R^i\overline{\pi}_* Rj^k_*F_{(\mathbf{A}^1,E)\otimes\mathcal{X}}=0, \quad \text{for all } i\geq 1, k\geq 0.\end{align*}$$

Thus, the vanishing $R^i\pi _*F_{(\mathbf {A}^1,E)\otimes \mathcal {X}}=0$ follows from the Leray spectral sequence.

The general case follows by induction (by factoring $\pi $ as $\mathbf {A}^n_X\xrightarrow {\pi _1} \mathbf {A}^{n-1}_X\xrightarrow {\pi _{n-1}} X$ and observing):

$$\begin{align*}\pi_{1*}(F_{(\mathbf{A}^1, E_1)\otimes \ldots\otimes(\mathbf{A}^1, E_n)\otimes\mathcal{X}})= F_{1, (\mathbf{A}^1, E_2)\otimes\ldots\otimes(\mathbf{A}^1, E_n)\otimes\mathcal{X}},\end{align*}$$

where $F_1:=\operatorname {\underline {Hom}}_{\operatorname {\mathbf {\underline {M}PST}}}({\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}(\mathbf {A}^1,E_1), F)$ lies in $\operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$ by Lemma 1.5(1).

Proof. We can assume X is henselian local and:

$$\begin{align*}L=V(x)\subset \mathbf{A}^2=\operatorname{Spec} k[x,y].\end{align*}$$

Set:

$$\begin{align*}\mathcal{F}:=F_{(Y, \rho^*L)\otimes \mathcal{X}};\end{align*}$$

it is a Nisnevich sheaf on $Y\times X$. For $i\ge 1$, the higher direct images $R^i\rho _{X*}\mathcal {F}$ are supported in $0\times X$, whence:

$$\begin{align*}H^j(\mathbf{A}^2_X, R^i\rho_{X*}\mathcal{F})=0, \quad \text{for all }i,j\ge 1,\end{align*}$$

and:

$$\begin{align*}R^i\rho_{X*}\mathcal{F}=0\Longleftrightarrow H^0(\mathbf{A}^2_X, R^i\rho_{X*}\mathcal{F})=0.\end{align*}$$

Furthermore, $\rho _{X*} \mathcal {F}= F_{(\mathbf {A}^2, L)\otimes \mathcal {X}}$, since $(Y, \rho ^*L)\otimes \mathcal {X}\cong (\mathbf {A}^2, L)\otimes \mathcal {X}$ in $\operatorname {\mathbf {\underline {M}Cor}}$ (see [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a, Section 1.7]). Hence, by Lemma 2.10:

$$\begin{align*}H^i(\mathbf{A}^2_X, \rho_{X*}\mathcal{F})=H^i(\mathbf{A}^2_X, F_{(\mathbf{A}^2, L)\otimes \mathcal{X}})=0.\end{align*}$$

Thus, the Leray spectral sequence yields:

(2.13.1)$$ \begin{align}H^0(\mathbf{A}_X^2, R^i\rho_{X*}\mathcal{F})= H^i(Y\times X, \mathcal{F}), \quad i\ge 0,\end{align} $$

and we have to show that this group vanishes for $i\ge 1$. Write:

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

and denote by:

$$\begin{align*}\pi: Y\times X\hookrightarrow \mathbf{A}^2\times \mathbf{P}^1_X \to \mathbf{P}^1_X=\operatorname{Proj} \mathcal{O}_X[S,T]\end{align*}$$

the morphism induced by projection. In order to show that (2.13.1) vanishes, we can project along $\pi $ and use the Leray spectral sequence:

$$\begin{align*}H^{i-j}(\mathbf{P}^1_X, R\pi^j_*\mathcal{F})\Rightarrow H^i(Y\times X, \mathcal{F})\end{align*}$$

to reduce the problem to showing that:

(2.13.2)$$ \begin{align}H^i(\mathbf{P}^1_X, R^j\pi_*\mathcal{F})=0, \quad i\ge 1, j\geq 0. \end{align} $$

The terms $R^j\pi _* \mathcal {F}$ for $j\geq 1$ are easy to handle using Lemma 2.10. Indeed, set $s=S/T$, and write:

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

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

$$\begin{align*}\mathcal{U}:=(\mathbf{A}^1_s,0)\otimes\mathcal{X}, \quad \mathcal{V}:=(\mathbf{P}^1\setminus\{0\}) \otimes \mathcal{X}.\end{align*}$$

We have:

$$\begin{align*}\pi^{-1}(U)= \mathbf{A}^1_y\times U, \quad \pi^{-1}(V)=\mathbf{A}^1_x\times V,\end{align*}$$

and the restriction of $\pi $ to these open subsets is given by projection. Furthermore by construction,

(2.13.3)$$ \begin{align} \mathcal{F}_{|\pi^{-1}(U)}=F_{(\mathbf{A}^1_y,0)\otimes \mathcal{U}},\quad \mathcal{F}_{|\pi^{-1}(V)}= F_{(\mathbf{A}^1_x, 0)\otimes \mathcal{V}}. \end{align} $$

Thus, Lemma 2.10 (in the case $n=1$) yields:

$$\begin{align*}R^j\pi_*\mathcal{F}=0, \quad j\ge 1.\end{align*}$$

It remains to show:

(2.13.4)$$ \begin{align}H^i(\mathbf{P}^1_X, \pi_*\mathcal{F})=0, \quad i\ge 1. \end{align} $$

Set:

(2.13.5)$$ \begin{align}F_1:=\operatorname{\underline{Hom}}({\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathbf{A}^1_x, 0), F ).\end{align} $$

Note that $F_1\in \operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$ by Lemma 1.5(1). Let $j: V\hookrightarrow \mathbf {P}^1_X$ be the open immersion. Its base change along $\pi $ induces a morphism:

(2.13.6)$$ \begin{align}\iota: (\mathbf{A}^1_x,0)\otimes \mathcal{V}\to (Y,\rho^*L)\otimes \mathcal{X} \quad \text{in } \operatorname{\mathbf{\underline{M}Cor}}. \end{align} $$

This yields an exact sequence of Nisnevich sheaves on $\mathbf {P}^1_X$:

$$\begin{align*}0\to \pi_*\mathcal{F}\xrightarrow{\pi_*(\iota^*)} j_*F_{1,\mathcal{V}}\to \Gamma\to 0,\end{align*}$$

defining $\Gamma $; here, the first map is injective by the semipurity of F. Since $\Gamma $ is supported on $0\times X$, we obtain for $i\ge 2$:

$$ \begin{align*} H^i(\mathbf{P}^1_X, \pi_*\mathcal{F}) & = H^i(\mathbf{P}^1_X, j_*F_{1,\mathcal{V}})\\ &= H^i(V, F_{1,\mathcal{V}}), & & \text{by Lemma}\ {2.9},\\ &=0, &&\text{by Lemma}\ {2.10}. \end{align*} $$

It remains to prove the vanishing (2.13.4) for $i=1$. This will occupy the rest of the proof. Let:

$$\begin{align*}a\colon Y\times X\to \mathbf{A}^1_x\times \mathbf{P}^1\times X\end{align*}$$

be induced by the base change of the closed immersion $Y\hookrightarrow \mathbf {A}^2\times \mathbf {P}^1$ followed by the base change of the projection $\mathbf {A}^2\to \mathbf {A}^1_x$. The map a induces a morphism:

(2.13.7)$$ \begin{align}\alpha: (Y,\rho^*L)\otimes \mathcal{X}\to (\mathbf{A}^1_x, 0)\otimes \mathbf{P}^1_{\mathcal{X}}\quad \text{in }\operatorname{\mathbf{\underline{M}Cor}}, \end{align} $$

where $\mathbf {P}^1_{\mathcal {X}}:=\mathbf {P}^1\otimes \mathcal {X}$ and which precomposed with $\iota $ from (2.13.6) yields the morphism:

(2.13.8)$$ \begin{align}\alpha\iota: (\mathbf{A}^1_x,0)\otimes \mathcal{V}\to (\mathbf{A}^1_x, 0)\otimes \mathbf{P}^1_{\mathcal{X}} \end{align} $$

induced by the open immersion $\mathbf {A}^1_x\times V\hookrightarrow \mathbf {A}^1_x\times \mathbf {P}^1_X$. This gives a factorisation:

where the diagonal morphism is injective by [Reference SaitoSai20a, Theorem 3.1(2)] and the semipurity of $F_1$. This implies that the morphism labeled $\pi _*(\alpha ^*)$ is injective too. Similarly, the embedding $\mathcal {V} \to (\mathbf {P}^1, 0)\otimes \mathcal {X}$ induces another injective morphism $F_{1, (\mathbf {P}^1, 0)\otimes \mathcal {X}} \to j_*F_{1,\mathcal {V}}$. In total, we obtain the following commutative diagram:

(2.13.9)

with exact rows, defining the cokernels ${\underline {\Sigma }}$, ${\underline {\Lambda }}$ and ${\underline {\Lambda }}(0)$, as well as the map $\varphi $. Applying $R\Gamma (\mathbf {P}^1_X,-)$ yields:

with exact rows and in which the $\partial _i$ are the connecting homomorphisms and where:

$$\begin{align*}\Sigma:= H^0(\mathbf{P}^1_X, {\underline{\Sigma}}), \quad \Lambda:= H^0(\mathbf{P}^1_X, {\underline{\Lambda}}), \quad \Lambda(0):=H^0(\mathbf{P}^1_X, {\underline{\Lambda}}(0)).\end{align*}$$

The group $H^1(\mathbf {P}^1, F_{1, (\mathbf {P}^1, 0)\otimes \mathcal {X}})$ vanishes by the cube invariance of cohomology (see [Reference SaitoSai20a, Theorem 9.3]), thus, $\partial _{2|\Lambda (0)}$ is surjective, the vanishing (2.13.4) for $i=1$ will follow, if we can show:

(2.13.10)$$ \begin{align}\Lambda(0)\subset \varphi(\Sigma). \end{align} $$

Note that ${\underline {\Sigma }}$, ${\underline {\Lambda }}$ and ${\underline {\Lambda }}(0)$ have support in $0\times X\subset U$, so we can compute the global sections on U instead of $\mathbf {P}^1$ to show (2.13.10). Now, since $H^1(U, F_{1,\mathcal {U}})=0$, by Lemma 2.10, unravelling the definitions, we obtain from (2.13.3) and (2.13.9) with $G:=F(-\otimes \mathcal {X})$ and $\mathbf {A}^1_s=\mathbf {P}^1\setminus \{\infty \}$ the following descriptions:

$$\begin{align*}\Sigma= \frac{G((\mathbf{A}^1_y,0)\otimes (\mathbf{A}^1_s,0))} {\alpha^* G((\mathbf{A}^1_x,0)\otimes \mathbf{A}^1_s)}, \end{align*}$$

$$\begin{align*}\Lambda= \frac{G((\mathbf{A}^1_x,0)\otimes (\mathbf{A}^1_s\setminus\{0\},\emptyset))} { G((\mathbf{A}^1_x,0)\otimes \mathbf{A}^1_s)}, \end{align*}$$

(2.13.11)$$ \begin{align}\Lambda(0)= \frac{G((\mathbf{A}^1_x,0)\otimes (\mathbf{A}^1_s,0))} { G((\mathbf{A}^1_x,0)\otimes \mathbf{A}^1_s)}. \end{align} $$

By [Reference SaitoSai20a, Lemma 5.9], we have isomorphisms (see Notation 2.1):

(2.13.12)$$ \begin{align}\frac{G(\overline{\square}^{(1)}_x\otimes (\mathbf{A}^1_s,0))}{G((\mathbf{P}^1_x,\infty)\otimes(\mathbf{A}^1_s,0))} \xrightarrow{\simeq} \frac{G((\mathbf{A}^1_x,0)\otimes (\mathbf{A}^1_s,0))}{G(\mathbf{A}^1_x\otimes(\mathbf{A}^1_s,0))},\end{align} $$

(2.13.13)$$ \begin{align}\frac{G(\overline{\square}^{(1)}_x\otimes \overline{\square}^{(1)}_s)}{G(\overline{\square}^{(1)}_x\otimes(\mathbf{P}^1_s,\infty))} \xrightarrow{\simeq} \frac{G(\overline{\square}^{(1)}_x\otimes (\mathbf{A}^1_s,0))}{G(\overline{\square}^{(1)}_x\otimes \mathbf{A}^1_s)}.\end{align} $$

Write j for the open immersion $(\mathbf {A}^1_x,0) \hookrightarrow \overline {\square }^{(1)}_x$. The base change of $j^*$ induces a commutative diagram:

(2.13.14)

The horizontal composite morphism is zero by (2.13.11), hence, the kernel of the diagonal arrow contains $G(\overline {\square }^{(1)}_x\otimes \mathbf {A}^1_s)$. Next, note that from (2.13.12), we get the surjective morphism:

(2.13.15)$$ \begin{align} G(\overline{\square}^{(1)}_x\otimes (\mathbf{A}^1_s,0)) \oplus G(\mathbf{A}^1_x \otimes (\mathbf{A}^1_s, 0)) \to G( (\mathbf{A}^1_x,0) \otimes (\mathbf{A}^1_s, 0))\to 0. \end{align} $$

Combining (2.13.15), (2.13.13) and (2.13.14), we get a surjection:

(2.13.16)$$ \begin{align} G(\mathbf{A}^1_x\otimes(\mathbf{A}^1_s,0))\oplus G(\overline{\square}^{(1)}_x\otimes \overline{\square}^{(1)}_s)\rightarrow\!\!\!\!\!\rightarrow \Lambda(0). \end{align} $$

Note that the pullback of the open immersion $\pi ^{-1}(V)\hookrightarrow Y\times X$ along $\pi ^{-1}(U)\hookrightarrow Y\times X$ induces the open immersion:

$$\begin{align*}\mathbf{A}^1_x\times (\mathbf{A}^1_s\setminus\{0\})\times X\to \mathbf{A}^1_y\times \mathbf{A}^1_s \times X,\end{align*}$$

which is induced by base change from the $k[s]$-linear map:

$$\begin{align*}k[y, s]\mapsto k[x, s, 1/s], \quad y\mapsto x/s.\end{align*}$$

It gives the following two morphisms in $\operatorname {\mathbf {\underline {M}Cor}}$:

$$\begin{align*}\iota_1: (\mathbf{A}^1_x,0)\otimes (\mathbf{A}^1_s\setminus\{0\})\otimes\mathcal{X}\to (\mathbf{A}^1_y,0)\otimes(\mathbf{A}^1_s,0)\otimes \mathcal{X}.\end{align*}$$

$$\begin{align*}\iota_2: (\mathbf{A}^1_x,0)\otimes (\mathbf{A}^1_s\setminus \{0\})\otimes \mathcal{X}\to \overline{\square}^{(1)}_y\otimes{\overline{\square}}^{(2)}_s\otimes \mathcal{X}.\end{align*}$$

Furthermore, consider the base change of the map (2.5.1):

$$\begin{align*}\psi: \overline{\square}^{(1)}_y\otimes{\overline{\square}}^{(2)}_s\otimes \mathcal{X}\to \overline{\square}^{(1)}_x\otimes\overline{\square}^{(1)}_s\otimes\mathcal{X},\end{align*}$$

which is induced by $x\mapsto ys$; it restricts to:

$$\begin{align*}\psi_1: (\mathbf{A}^1_y,0)\otimes(\mathbf{A}^1_s,0)\to\mathbf{A}^1_x\otimes (\mathbf{A}^1_s,0).\end{align*}$$

In particular, $\psi _1\circ \iota _1$ is induced by the open immersion $\mathbf {A}^1_x\setminus \{0\}\times \mathbf {A}^1_s\setminus \{0\}\hookrightarrow \mathbf {A}^1_x\times \mathbf {A}^1_s\setminus \{0\}$ and $\psi \circ \iota _2$ is induced by the identity on $\mathbf {A}^1_x\setminus \{0\}\times \mathbf {A}^1_s\setminus \{0\}$. Consider the following diagram:

Here, the maps $r_i$ are the natural maps into the quotients; the diagram commutes by definition of the morphisms involved. Hence:

(2.13.17)$$ \begin{align}\operatorname{Im}(G(\mathbf{A}^1_x\otimes (\mathbf{A}^1_s,0))\to \Lambda(0))\subset \varphi(\Sigma). \end{align} $$

Consider now the following diagram:

Here, the maps $r_1$ and $r_3$ are induced by restriction followed by the quotient map using (2.13.12) and (2.13.13); the two squares and the triangle on the lower left commute by definition of the morphisms involved; the map $\psi ^*$ factors via the dotted arrow in the diagram (by Proposition 2.5). This shows:

$$\begin{align*}\operatorname{Im}(G(\overline{\square}^{(1)}_x\otimes \overline{\square}^{(1)}_s)\to \Lambda(0))\subset \varphi(\Sigma),\end{align*}$$

which together with (2.13.17) and (2.13.16) implies (2.13.10). This completes the proof of the lemma.

Proof. There is nothing to prove for ${\operatorname {codim}}(Z,X)=1$, we therefore consider the case ${\operatorname {codim}}(Z,X)=2$. Since $(Y,\rho ^*D)\cong (X,D)$ in $\operatorname {\mathbf {\underline {M}Cor}}$ we have $\rho _*F_{(Y,\rho ^*D)}\cong F_{(X,D)}$. Thus, it remains to show the vanishing:

(2.14.1)$$ \begin{align} R^i\rho_*F_{(Y,\rho^*D)}=0,\quad \text{for all } i\ge 1. \end{align} $$

The question is Nisnevich local around the points in Z. Let $z\in Z$ be a point, and consider the regular henselian local ring $A=\mathcal {O}_{X,z}^h$. For $V\subset X$, set $V_{(z)}:= V\times _{ X} \operatorname {Spec} A$. Denote by $D'\subset X$ the closed subscheme defined by $D-D_0$. By assumption, we find a regular system of local parameters $x,y,t_1\ldots , t_s$ of A, such that $Z_{(z)}=V(x,y)$, $D_{0, (z)}=V(x)$ and $D^{\prime }_{(z)}=V(t_{1}^{n_{1}}\cdots t_{r}^{n_r})$, for some $r\le s$ and $n_i\ge 1$. Let $K\hookrightarrow A$ be a coefficient field over k; we obtain an isomorphism:

$$\begin{align*}K\{x,y, t_1,\ldots, t_s\}\xrightarrow{\simeq} A.\end{align*}$$

Let $\rho _1:\widetilde {\mathbf {A}^2}\to \mathbf {A}^2$ be the blow-up in $0$. By the above, the blow-up in Z:

$$\begin{align*}\rho: (Y,\rho^*D)\to (X, D)\end{align*}$$

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

$$\begin{align*}(\widetilde{\mathbf{A}^2},\rho_1^*(x))\otimes (\mathbf{A}_K^{s}, (\prod_{i=1}^{r} t_i^{n_i})) \to (\mathbf{A}^2, (x))\otimes (\mathbf{A}_K^{s}, (\prod_{i=1}^{r} t_i^{n_i})), \end{align*}$$

which is induced by base change from $\rho _1$. Hence, the vanishing (2.14.1) follows from Lemma 2.13.

Proof. Recall the following general fact: Let $\mathcal {I}, \mathcal {J}\subset \mathcal {O}_X$ be two coherent ideal sheaves. Then the blow-up $\tilde {X}\to X$ of X in $\mathcal {I}\cdot \mathcal {J}$ is equal to the composition $X_2\xrightarrow {\pi _2} X_1\xrightarrow {\pi _1}X$, where $\pi _1$ is the blow-up in $\mathcal {I}$ and $\pi _2$ is the blow-up in $\pi _1^{-1}\mathcal {J}\cdot \mathcal {O}_{X_1}$. This is proven using the universal property of blow-ups (see, e.g. [Sta19, Tag 080A]. Here, denote by $\mathcal {I}_i\subset \mathcal {O}_X$ the ideal sheaves of $Z_i$. We have $\mathcal {I}_1\subset \mathcal {I}_0$. Let $\pi :\tilde {X}\to X$ be the blow-up of X in $\mathcal {I}_1\cdot \mathcal {I}_0$. By the remark above, $\pi $ is isomorphic as X-scheme to $\sigma \sigma '$. Furthermore, note that $\rho '$ is also equal to the blow-up of $X'$ in $\rho ^{-1}(Z_1)$. Indeed, the ideal sheaf of $\rho ^{-1}(Z_1)$ is equal to $\rho ^{-1}\mathcal {I}_1\cdot \mathcal {O}_{X'}= \mathcal {I}_E\cdot \tilde {\mathcal {I}}_1$, where $\mathcal {I}_E$ is the ideal sheaf of the exceptional divisor of $\rho $ and $\tilde {\mathcal {I}}_1$ is the ideal sheaf of $\tilde {Z}_1$; since $\mathcal {I}_E$ is invertible, the blow-ups of $X'$ in $\tilde {\mathcal {I}}_1$ and in $\rho ^{-1}\mathcal {I}_1\cdot \mathcal {O}_{X'}$ are isomorphic. Thus, by the remark above, the X-scheme $\rho \rho '$ is isomorphic to $\pi $ as well.

Proof. Note that the projection morphism $Y\to E$ makes Y into a $\mathbf {P}^1$-bundle over E and induces a morphism $(Y, \rho ^*L)\otimes \mathcal {X}\to (E, L')\otimes \mathcal {X}$. The latter morphism locally over E has the form of the projection ${\overline {\square }}\otimes \mathcal {W}\to \mathcal {W}$, for some $\mathcal {W}\in \operatorname {\mathbf {\underline {M}Cor}}$. Indeed, over an affine neighborhood $U\subset E$ intersecting (respectively, not intersecting) $L'$, the modulus pair $\mathcal {W}$ can be taken to be $(U,L'\cap U)\otimes \mathcal {X}$ (respectively, $(U,\emptyset )\otimes \mathcal {X}$). In both cases, the divisor $\{\infty \}\times U\times X$ on $\mathbf {P}^1\times U\times X$ is the restriction of the exceptional divisor to $q^{-1}(U)= \mathbf {P}^1\times U\times X$. Thus, the statement follows from the cube invariance of cohomology (see [Reference SaitoSai20a, Theorem 9.3].

Proof. The case $n=1$ is [Reference SaitoSai20a, Theorem 9.3]. Assume $n\ge 2$. Let $x\in \mathbf {P}^n$ be a k-rational point, $L\subset \mathbf {P}^n$ a hyperplane with $x\in L$ and $\rho : Y\to \mathbf {P}^n$ the blow-up in x. Then $R\rho _* F_{(Y,\rho ^*L)\otimes \mathcal {X}}= F_{(\mathbf {P}^n,L)\otimes \mathcal {X}}$ by Theorem 2.12. Thus, the statement follows from Lemma 2.17 and induction.

Proof. The question is local on X, hence, we can assume that V is trivial of rank $n+1$. Let $L\subset \mathbf {P}^n$ be a hyperplane and consider:

$$\begin{align*}F_{\mathcal{X}}\xrightarrow{\pi^*} \pi_*F_{\mathbf{P}^n\otimes \mathcal{X}}\hookrightarrow \pi_*F_{(\mathbf{P}^n,L)\otimes \mathcal{X}}.\end{align*}$$

The second map is injective by semipurity and [Reference SaitoSai20a, Theorem 3.1(2)]; the composition is an isomorphism by Theorem 2.18, hence, so is the first map.

Proof. The proof is a slight modification of [Reference Bloch and GiesekerBG71, Lemma 2.1]. First note that (4) follows from (2) and (1). Also, it suffices to prove the statement for D as a very ample divisor. Let $i: X\hookrightarrow \mathbf {P}^N:=\mathbf {P}$ be an immersion, such that $\mathcal {O}(D)=i^*\mathcal {O}_{\mathbf {P}}(1)$. By Bertini’s theorem (see, e.g. [Reference JouanolouJou83, Chapter I, Corollary 6.11]), we find hyperplanes $H_0,\ldots , H_N\subset \mathbf {P}$, such that all the intersections $H_{i_0}\cap \ldots \cap H_{i_r}$ and $H_{i_0}\cap \ldots \cap H_{i_r}\cap X_{\mathrm {sm}}$ are transversal (or empty), for all $\{i_0,\ldots , i_r\}\subset \{0,\ldots , N\}$ and all $0\le r\le N$. Let $Y_i$ be a linear polynomial defining $H_i$, so that $\mathbf {P}=\operatorname {Proj} k[Y_0,\ldots , Y_N]$. Let $\Pi : \mathbf {P}\to \mathbf {P}$ be the k-morphism defined by $Y_i\mapsto Y_i^n$, $i=0,\ldots , N$. Note that $\Pi $ is finite of degree $n^N$ and it is étale over $\mathbf {P}\setminus \cup _i H_i$. Form the cartesian diagram:

Then $X'\times _X X_{\mathrm {sm}}$ is smooth: this can be checked after base change to the algebraic closure of k, and then the argument is the same as in the second and third paragraph in the proof of [Reference Bloch and GiesekerBG71, Lemma 2.1] (the choice of the $H_i$ is crucial here). Let $X"\subset X'$ be an irreducible component (with reduced scheme structure), and denote by Y the normalisation of $X"$ and by $\pi : Y\to X$ the composition:

$$\begin{align*}Y\to X"\hookrightarrow X'\xrightarrow{\pi'} X\end{align*}$$

and by $E=\mathcal {O}_{\mathbf {P}}(1)_{|Y}$ the pullback of $\mathcal {O}_{\mathbf {P}}(1)$ along:

$$\begin{align*}Y\to X"\hookrightarrow X'\stackrel{i'}{\hookrightarrow} \mathbf{P}.\end{align*}$$

Then $\pi : Y\to X$ and E satisfy the conditions of the statement.

Proof. Denote by $H_l$ (respectively, $H_r$) the source (respectively, target) of (4.2.1), and take $\mathcal {X}=(X,D)\in \operatorname {\mathbf {\underline {M}Cor}}$. By definition, $\otimes _{\operatorname {\mathbf {\underline {M}PST}}}$ (respectively, $\otimes _{{\operatorname {\mathbf {PST}}}}$) is the Day convolution of the tensor product on $\operatorname {\mathbf {\underline {M}Cor}}$ (respectively, on $\operatorname {\mathbf {Cor}}$), so that we have the following presentations (which also hold for general $(X,D)$, cf. [Reference Suslin and VoevodskySV00, §2]):

$$\begin{align*}H_l(X,D)= \left(\bigoplus_{\mathcal{Y}, \mathcal{Z}\in \operatorname{\mathbf{\underline{M}Cor}}} F(\mathcal{Y}^o)\otimes_{\mathbb{Z}} G(\mathcal{Z}^o)\otimes_{\mathbb{Z}}\operatorname{\mathbf{\underline{M}Cor}}(\mathcal{X}, \mathcal{Y}\otimes \mathcal{Z})\right)/R_l,\end{align*}$$

where for $\mathcal {Y}=(\overline {Y}, Y_{\infty })$, we set $\mathcal {Y}^o=\overline {Y}\setminus Y_{\infty }$, and where $R_l$ is the subgroup generated by the elements:

$$\begin{align*}f^*a\otimes g^*b\otimes h - a\otimes b\otimes (f\otimes g)\circ h,\end{align*}$$

where $\mathcal {Y}, \mathcal {Y}', \mathcal {Z}, \mathcal {Z}'\in \operatorname {\mathbf {\underline {M}Cor}}$, $a\in F(\mathcal {Y}^o)$, $b\in G(\mathcal {Z}^o)$, $f\in \operatorname {\mathbf {\underline {M}Cor}}(\mathcal {Y}', \mathcal {Y})$, $g\in \operatorname {\mathbf {\underline {M}Cor}}(\mathcal {Z}',\mathcal {Z})$ and $h\in \operatorname {\mathbf {\underline {M}Cor}}(\mathcal {X}, \mathcal {Y}'\otimes \mathcal {Z}')$. Similarly,

$$\begin{align*}H_r(X,D)=\left(\bigoplus_{Y,Z\in \operatorname{\mathbf{Sm}}} F(Y)\otimes_{\mathbb{Z}} G(Z)\otimes_{\mathbb{Z}} \operatorname{\mathbf{Cor}}(X\setminus D, Y\times Z)\right)/R_r,\end{align*}$$

where $R_r$ is the subgroup generated by:

$$\begin{align*}f^*a\otimes g^*b\otimes h - a\otimes b\otimes (f\times g)\circ h,\end{align*}$$

where $Y, Y', Z, Z'\in \operatorname {\mathbf {Sm}}$, $a\in F(Y)$, $b\in G(Z)$, $f\in \operatorname {\mathbf {Cor}}(Y',Y)$, $g\in \operatorname {\mathbf {Cor}}(Z',Z)$ and $h\in \operatorname {\mathbf {Cor}}(X\setminus D, Y'\times Z')$.

Let $\sum _i a_i\otimes b_i\otimes \gamma _i\in H_r(X,D)$, where $a_i\in F(Y_i)$, $b_i\in G(Y_i)$ and $\gamma _i\in \operatorname {\mathbf {Cor}}(X\setminus D, Y_i\times Z_i)$. By [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21a, Theorem 1.6.2], we find a proper morphism $\rho :X'\to X$ inducing an isomorphism $X'\setminus |\rho ^*D|\xrightarrow {\simeq } X\setminus |D|$, such that the closure of any irreducible component of $\gamma _i$ in $X'\times Y_i\times Z_i$ is finite over $X'$, for all i. By (1.2.4) and Lemma 1.3, we are reduced to show the following:

Let V be as above. Since the closure $\overline {V}\subset X\times Y\times Z$ of V is integral and finite over X, it is local. Denote by $v\in \overline {V}$ the closed point and by $y\in Y$, $z\in Z$ the images of v, respectively. We get induced maps $\mathcal {O}_{Y,y}\to \Gamma (\overline {V}, \mathcal {O}_{\overline {V}})$ and $\mathcal {O}_{Z,z}\to \Gamma (\overline {V}, \mathcal {O}_{\overline {V}})$. Hence:

$$\begin{align*}V\subset (X\setminus D)\times U_1\times U_2,\end{align*}$$

where $j_1:U_1\hookrightarrow Y$ and $j_2:U_2\hookrightarrow Z$ are open affines containing y and z, respectively. Denote by $V'\in \operatorname {\mathbf {Cor}}(X\setminus D, U_1\times U_2)$ the induced prime correspondence. Then $V= (j_1\times j_2)\circ V'$, and, thus:

$$\begin{align*}a\otimes b\otimes V= j_1^*a\otimes j_2^* b\otimes V'\quad \text{in }H_r(X,D).\end{align*}$$

Hence, Claim 4.2.1 follows from the following:

We prove the claim. First we reduce to k infinite by a standard trick: If k is finite, denote by $k(\ell )$ a $\mathbb {Z}_{\ell }$-Galois extension of k for a prime $\ell $; by a trace argument, the (diagonal) pullback $H_r(X,D)\to H_r(X_{k(\ell )}, D_{k(\ell )})\times H_r(X_{k(\ell ')}, D_{k(\ell ')})$ is injective for $\ell \neq \ell '$.

In the following, we assume k infinite. By assumption, we find proper modulus pairs $\mathcal {Y}=(\overline {Y}, Y_{\infty })$ and $\mathcal {Z}=(\overline {Z}, Z_{\infty })$, such that $\overline {Y}$ and $\overline {Z}$ are projective and $Y=\overline {Y}\setminus |Y_{\infty }|$ and $Z=\overline {Z}\setminus |Z_{\infty }|$. Since V is closed in $X\setminus |D|\times \overline {Y}\times \overline {Z}$, we find an integer $n_0$, such that $V\in \operatorname {\mathbf {\underline {M}Cor}}((X, n_0 D), \mathcal {Y}\otimes \mathcal {Z})$. Choose $n \ge n_0$ with $(n,p)=1$. By Lemma 4.1, we find a modulus pair $\mathcal {Y}'=(\overline {Y}', Y_{\infty }')$ together with a finite and surjective morphism $\bar {\pi }_{Y,n}: \overline {Y}'\to \overline {Y}$, such that $\deg \bar {\pi }_{Y,n}$ divides a power of n and $\bar {\pi }_{Y,n}^*(Y_{\infty })= n Y^{\prime }_{\infty }$, similarly for $\overline {Z}$. Denote by $\pi _{Y,n}: Y'\to Y$ the induced finite and surjective morphism in $\operatorname {\mathbf {Sm}}$ and by $\pi _{Y,n}^t\in \operatorname {\mathbf {Cor}}(Y, Y')$ the correspondence induced by the transpose of the graph. In $H_r(X,D)$, we obtain:

$$ \begin{align*} \deg(\pi_{Y,n})\deg(\pi_{Z,n})\cdot (a\otimes b\otimes V) &= \pi_{Y,n*}\pi_{Y,n}^*a\otimes \pi_{Z,n*}\pi_{Z,n}^*b\otimes V\\ &=(\pi_{Y,n}^t)^*\pi_{Y,n}^*a\otimes (\pi_{Z,n}^t)^*\pi_{Z,n}^*b\otimes V\\ &=\pi_{Y,n}^*a\otimes \pi_{Z,n}^*b\otimes (\pi_{Y,n}^t\times \pi_{Z,n}^t)\circ V. \end{align*} $$

Observe that the components of $(\pi _{Y,n}^t\times \pi _{Z,n}^t)\circ V\in \operatorname {\mathbf {Cor}}(X\setminus D, Y'\times Z')$ are the irreducible components of:

$$\begin{align*}V\times_{Y\times Z} (Y'\times Z')= ({\operatorname{id}}_{X\setminus |D|}\times \pi_{n,Y}\times\pi_{n,Z})^{-1}(V).\end{align*}$$

Let W be such a component, it comes with a finite and surjective map $W\to V$. Denote by $\overline {V}\subset X\times \overline {Y}\times \overline {Z}$ and $\overline {W}\subset X\times \overline {Y}'\times \overline {Z}'$ the closure of V and W, respectively, and denote by $\tilde {V}\to \overline {V}$ and $\tilde {W}\to \overline {W}$ the normalisations. Since $\overline {W}$ is contained in $\overline {V}\times _{\overline {Y}\times \overline {Z}} (\overline {Y}'\times \overline {Z}')$, the natural maps from $\tilde {W}$ to $\overline {Y}$ and $\overline {Z}$ factor via a morphism $\tilde {W}\to \tilde {V}$. We obtain:

$$\begin{align*}n D_{|\tilde{W}}\ge n_0 D_{|\tilde{W}}\ge Y_{\infty|\tilde{W}} + Z_{\infty|\tilde{W}} = n Y^{\prime}_{\infty|\tilde{W}}+ n Z^{\prime}_{\infty|\tilde{W}},\end{align*}$$

where the second inequality follows from $V\in \operatorname {\mathbf {\underline {M}Cor}}((X, n_0 D), \mathcal {Y}\otimes \mathcal {Z})$. Hence:

$$\begin{align*}(\pi_{Y,n}^t\times \pi_{Z,n}^t)\circ V\in \operatorname{\mathbf{\underline{M}Cor}}((X,D), \mathcal{Y}'\otimes\mathcal{Z}').\end{align*}$$

It follows that $\delta _n\cdot (a\otimes b\otimes V)$ lies in the image of $H_l(X,D)\to H_r(X,D)$, where $\delta _n :=\deg (\pi _{Y,n})\deg (\pi _{Z,n})$. Choose $r\ge n_0$ with $(r,p)=1=(r,n)$. Since $\delta _n$ divides a power of n and $\delta _r$ divides a power of r, we find integers $s, t$ with:

$$\begin{align*}a\otimes b\otimes V= s \delta_n\cdot(a\otimes b\otimes V)+ t\delta_r\cdot(a\otimes b\otimes V).\end{align*}$$

This proves Claim 4.2.2, and, hence, also the lemma.

Proof. We begin by recalling from [Reference Merici and SaitoMS20, Proposition 3.2] that for $F, G\in \operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$, the formula $F\otimes _{\operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}} G = \tau _!h_{0, {\operatorname {Nis}}}^{{\overline {\square }}, \mathrm {sp}}(\tau ^* F\otimes _{\operatorname {\mathbf {MPST}}} \tau ^* G)$ defines a symmetric monoidal structure on $\operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$. Next, note that $\underline {\omega }^*H\in \operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$ for $H\in \operatorname {\mathbf {HI}}_{\operatorname {Nis}}$ by [Reference Kahn, Saito and YamazakiKSY22, Lemma 2.3.1] and [Reference Kahn, Miyazaki, Saito and YamazakiKMSY21b, Proposition 6.2.1b)]. Moreover:

$$ \begin{align*} h^{{\overline{\square}}, \mathrm{sp}}_{0, {\operatorname{Nis}}}(\underline{\omega}^*F_1\otimes_{\operatorname{\mathbf{\underline{M}PST}}} \underline{\omega}^*F_2) & = h^{{\overline{\square}}, \mathrm{sp}}_{0, {\operatorname{Nis}}}(\tau_! \omega^*F_1\otimes_{\operatorname{\mathbf{\underline{M}PST}}} \tau_! \omega^* F_2) \\ & = \tau_! h^{{\overline{\square}}, \mathrm{sp}}_{0, {\operatorname{Nis}}}( \omega^*F_1\otimes_{\operatorname{\mathbf{MPST}}} \omega^* F_2) \\ &= \tau_! h^{{\overline{\square}}, \mathrm{sp}}_{0, {\operatorname{Nis}}}( \tau^* (\underline{\omega}^*F_1)\otimes_{\operatorname{\mathbf{MPST}}} \tau^* (\underline{\omega}^* F_2))= \underline{\omega}^* F_1 \otimes_{\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}}\underline{\omega}^* F_2, \end{align*} $$

for every $F_1, F_2 \in \operatorname {\mathbf {HI}}_{{\operatorname {Nis}}}$. Here, the isomorphisms follow from (1.1.3) and the exactness of $\tau _!$.

We now observe that the functor $\underline {\omega }^*$ is lax monoidal from ${\operatorname {\mathbf {PST}}}$ to $\operatorname {\mathbf {\underline {M}PST}}$ (this follows from the fact that $\underline {\omega }^*$ is right adjoint to $\underline {\omega }_!$, which is strict monoidal by construction). By applying $h^{{\overline {\square }},\mathrm {sp}}_{0,{\operatorname {Nis}}}$ to (4.3.1), we obtain the functorial map (4.3.2), which we can rewrite for $n=2$ as:

(4.3.3)$$ \begin{align} \underline{\omega}^* F_1 \otimes_{\operatorname{\mathbf{CI}}^{\tau,sp}_{{\operatorname{Nis}}}}\underline{\omega}^* F_2 \to \underline{\omega}^*( F_1 \otimes_{\operatorname{\mathbf{HI}}_{{\operatorname{Nis}}}} F_2). \end{align} $$

In particular, $\underline {\omega }^*$ restricts to a lax symmetric monoidal functor from $\operatorname {\mathbf {HI}}_{{\operatorname {Nis}}}$ to $\operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$, and the statement of the proposition is equivalent to the fact $\underline {\omega }^*$ is in fact (strictly) monoidal, that is, that the map (4.3.3) is an isomorphism (note that the identity for the tensor product is simply the constant sheaf $\mathbb {Z}$ and that $\underline {\omega }^* \mathbb {Z} = \mathbb {Z}$). Since the tensor products in $\operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$ and $\operatorname {\mathbf {HI}}_{{\operatorname {Nis}}}$ are, in particular, associative, it is enough to prove the claim when $n=2$.

By Lemma 4.2, the map (4.3.3) (or, equivalently, (4.3.2)) is surjective. On the other hand, we have:

$$ \begin{align*} \underline{\omega}_! h^{{\overline{\square}}, \mathrm{sp}}_{0, {\operatorname{Nis}}}( \underline{\omega}^*F_1\otimes_{\operatorname{\mathbf{\underline{M}PST}}} \underline{\omega}^*F_2) & = a^V_{\operatorname{Nis}}\underline{\omega}_! {\underline{h}}^{{\overline{\square}}}_0( \underline{\omega}^*F_1\otimes_{\operatorname{\mathbf{\underline{M}PST}}} \underline{\omega}^*F_2)^{\mathrm{sp}} \\ &=a^V_{\operatorname{Nis}}\underline{\omega}_! {\underline{h}}^{{\overline{\square}}}_{0}\tau_!(\omega^*F_1\otimes_{\operatorname{\mathbf{\underline{M}PST}}} \omega^*F_2)\\ &=a^V_{\operatorname{Nis}}\underline{\omega}_!\tau_!h^{{\overline{\square}}}_{0}( \omega^*F_1\otimes_{\operatorname{\mathbf{MPST}}} \omega^*F_2) \\ &= a^V_{\operatorname{Nis}}\omega_!h^{{\overline{\square}}}_{0}( \omega^*F_1\otimes_{\operatorname{\mathbf{MPST}}} \omega^*F_2),\end{align*} $$

where the first equality follows from the definition of $h^{{\overline {\square }},\mathrm {sp}}_{0,{\operatorname {Nis}}}$ (cf. (1.4.2)) and $\underline {\omega }_!{\underline {a}}_{\operatorname {Nis}}=a^V_{\operatorname {Nis}}\underline {\omega }_!$ (cf. (1.2.2)), the second holds by the fact $\underline {\omega }_! A^{\mathrm {sp}}=\underline {\omega }_! A$ for $A\in \operatorname {\mathbf {\underline {M}PST}}$ and $\underline {\omega }^*=\tau _!\omega ^*$ (cf. (1.1.3)) and the monoidality of $\tau _!$, the third follows from ${\underline {h}}^{{\overline {\square }}}_0(\tau _! B)=\tau _! h^{\square }_0(B)$ for $B\in \operatorname {\mathbf {MPST}}$, where $h^{\square }_0(B)\in \operatorname {\mathbf {MPST}}$ is the maximal cube invariant quotient of B defined by the same way as (1.4.3) and the last holds by $\underline {\omega }_!\tau _!=\omega _!$ (cf. (1.1.3)). Thus, $\underline {\omega }_!(4.3.2)$ is an isomorphism by [Reference Rülling, Sugiyama and YamazakiRSY22, Theorem 5.3], in view of $\omega ^*F=\tilde {F}$ (see [Reference Rülling, Sugiyama and YamazakiRSY22, (3.14.5)]) by [Reference Kahn, Saito and YamazakiKSY22, Lemma 2.3.1]. Since both sides of (4.3.2) are semipure, the map (4.3.2) is injective as well.

Proof. For a proper modulus pair $\mathcal {X}$, we have $\tau _!\tau ^*{\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}(\mathcal {X})={\operatorname {\mathbb {Z}_{{\operatorname {tr}}}}}(\mathcal {X})$. It follows that $\overline {\square }^{(1)}_{{\operatorname {red}}}\in \operatorname {\mathbf {\underline {M}PST}}^{\tau }$. By Lemma 2.4, we have $h^{{\overline {\square }}, \mathrm { sp}}_{0,{\operatorname {Nis}}}(\overline {\square }^{(1)}_{{\operatorname {red}}})=\underline {\omega }^*\mathbf {G}_m$. Thus:

(4.5.4)$$ \begin{align}\gamma F= \operatorname{\underline{Hom}}_{\operatorname{\mathbf{\underline{M}PST}}}(\overline{\square}^{(1)}_{\operatorname{red}}, F). \end{align} $$

Indeed (we drop the index $\operatorname {\mathbf {\underline {M}PST}}$ from $\operatorname {Hom}$ and $\operatorname {\underline {Hom}}$):

$$ \begin{align*} \operatorname{\underline{Hom}}(\overline{\square}^{(1)}_{\operatorname{red}}, F)(\mathcal{X}) &= \operatorname{Hom}({\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathcal{X})\otimes \overline{\square}^{(1)}_{\operatorname{red}}, F)\\ &=\operatorname{Hom}(\overline{\square}^{(1)}_{\operatorname{red}}, \operatorname{\underline{Hom}}({\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathcal{X}), F))\\ &=\operatorname{Hom}(h^{{\overline{\square}},\mathrm{ sp}}_{0,{\operatorname{Nis}}}(\overline{\square}^{(1)}_{\operatorname{red}}), \operatorname{\underline{Hom}}({\operatorname{\mathbb{Z}_{{\operatorname{tr}}}}}(\mathcal{X}), F))\\ &= \operatorname{\underline{Hom}}(h^{{\overline{\square}},\mathrm{ sp}}_{0,{\operatorname{Nis}}}(\overline{\square}^{(1)}_{\operatorname{red}}), F)(\mathcal{X})\\ & = \gamma F(\mathcal{X}), \end{align*} $$

where the third equality holds by Lemma 1.5(1), (2). This implies the first equality in (4.5.1) and also that $\gamma ^n F\in \operatorname {\mathbf {CI}}^{\tau ,sp}_{{\operatorname {Nis}}}$, for all $n\ge 0$ (by Lemma 1.5(1)). For the second equality in (4.5.1), first note that it follows from [Reference KerzKer10] and results by Voevodsky (see [Reference Rülling, Sugiyama and YamazakiRSY22, 5.5]), that we have:

(4.5.5)$$ \begin{align}K^M_n\cong \mathbf{G}_m^{\otimes_{\operatorname{\mathbf{HI}}_{\operatorname{Nis}}} n}\in \operatorname{\mathbf{HI}}_{{\operatorname{Nis}}}. \end{align} $$

Hence, by Proposition 4.3 and [Reference Merici and SaitoMS20, Lemma 1.14(iii)], we obtain:

(4.5.6)$$ \begin{align}\underline{\omega}^*K^M_n= h_{0,{\operatorname{Nis}}}^{{\overline{\square}},\mathrm{ sp}}((\underline{\omega}^*\mathbf{G}^m)^{\otimes_{\operatorname{\mathbf{\underline{M}PST}}} n})= h_{0,{\operatorname{Nis}}}^{{\overline{\square}},\mathrm{ sp}}((\overline{\square}^{(1)}_{{\operatorname{red}}})^{\otimes_{\operatorname{\mathbf{\underline{M}PST}}} n}). \end{align} $$

Thus, the second equality in (4.5.1) follows from the adjunction (1.4.2). The equalities in (4.5.2) follow similarly.