2.1.1 Let $\widetilde {\mathscr {C}}$ be a Tannakian category over $E$, neutralized by a fiber functor $\omega \colon \widetilde {\mathscr {C}}\to \operatorname {Vec}_F$, where $F$ is a finite extension of $E$. Suppose there exist $E$-linear tensor equivalences $\tau _i\colon \widetilde {\mathscr {C}}\to \widetilde {\mathscr {C}}$ and isomorphisms of tensor functors $\eta _i\colon \omega \circ \tau _i\xrightarrow {\sim } \omega$ for $i=1,2,\dots,n$. Moreover, we assume that there exist natural isomorphisms $\sigma _{ij}\colon \tau _i\circ \tau _j \simeq \tau _{j}\circ \tau _i$ for $i,j$ such that the following diagram commutes.

(2.1.1.1)

Since $\omega$ is faithful, such an isomorphism $\sigma _{ij}$ is unique. In the following, for every $i=1,\dots,n$, we fix a quasi-inverse $\tau _i^{-1}$ of $\tau _i$. For $m\in \mathbb {Z}$, we set $\tau _i^m$ to be the $|m|$th composition of $\tau _i$ (or $\tau _i^{-1}$ if $m<0$) and define $\eta _i^m\colon \omega \circ \tau _i^m \xrightarrow {\sim } \omega$ by the composition of $\eta _i$ (or the inverse of $\eta _i$ if $m<0$).

We define a category $\mathscr {C}_0$ as follows: an object $(\mathscr {E},\varphi _1,\dots,\varphi _{n})$ consists of an object $\mathscr {E}$ of $\widetilde {\mathscr {C}}$ and isomorphisms $\varphi _i\colon \tau _i(\mathscr {E})\xrightarrow {\sim } \mathscr {E}$ such that $\varphi _j\circ \tau _j(\varphi _i)=\varphi _i\circ \tau _i(\varphi _j)$ via $\sigma _{ij}$. Morphisms of $\mathscr {C}_0$ are morphisms of $\widetilde {\mathscr {C}}$ compatible with the $\varphi _i$. We have a canonical functor

\[ \mathscr{C}_0\to \widetilde{\mathscr{C}}, \quad (\mathscr{E},\varphi_i)\mapsto \mathscr{E}. \]

By [Reference DeligneDel07, § 2.5] and a similar argument to [Reference Xu and ZhuXZ22, Proposition 3.4.5], we can show that $\mathscr {C}_0$ is a Tannakian category over $E$ neutralized by $\omega$ over $\operatorname {Vec}_F$.

We say an object of $\mathscr {C}_0$ is constant if its image in $\widetilde {\mathscr {C}}$ is isomorphic to a finite direct sum of the unit object.

Recall [Reference BertolinBer08, § 2] that a Tannakian subcategory of $\widetilde {\mathscr {C}}$ is a strictly full abelian subcategory closed under $\otimes$, duals, and subobjects (and, thus, quotients). The constant objects of $\mathscr {C}_0$ form a Tannakian subcategory $\mathscr {C}_{0,\operatorname {cst}}$ of $\mathscr {C}_0$ (see [Reference BertolinBer08, § 2]).

Let $\mathscr {C}$ be the smallest Tannakian subcategory of $\widetilde {\mathscr {C}}$ containing the essential image of $\mathscr {C}_0$ (i.e. generated by the subquotients of the essential image of $\mathscr {C}_0$).

2.1.2 In the following, we assume that $\widetilde {\mathscr {C}}$ is neutral over $E$ by $\omega \colon \widetilde {\mathscr {C}}\to \operatorname {Vec}_E$. Then so are $\mathscr {C},\mathscr {C}_0$ and $\mathscr {C}_{0,\operatorname {cst}}$. We prove the following results by a similar approach of [Reference D'AddezioD'Ad20, Appendix], where D'Addezio treated the case $n=1$. A similar discussion appeared in [Reference Heinloth, Ngô and YunHNY13, Appendix 2] and [Reference Xu and ZhuXZ22, § 3.4].

Proof. (i) We construct a functor

\[ \mathscr{C}_0\to \mathbf{Rep}_E(W(\mathscr{C}_0)). \]

Given an object $(\mathscr {E},\varphi _i)$ of $\mathscr {C}_0$, we construct a representation $\rho$ of $W(\mathscr {C}_0)(E)$ on $\omega (\mathscr {E})$. For any element $(g,m_1,\dots,m_n)\in \pi _1(\mathscr {C})(E)\rtimes \mathbb {Z}^n$, we define $\rho (g,m_i)$ as the composition:

\[ \omega (\mathscr{E})\xrightarrow{\prod \eta_{i}^{-m_i}} \omega\Big(\prod\tau_{i}^{m_i}(\mathscr{E})\Big) \xrightarrow{g} \omega\Big(\prod\tau_{i}^{m_i}(\mathscr{E})\Big)\xrightarrow{\prod \eta_{i}^{m_i}} \omega(\mathscr{E}). \]

In view of the definition of $u_i$ (2.1.4.3), one checks that the above formula defines a representation. Then we obtain the above functor and we can check that it is an equivalence.

By the Tannakian reconstruction theorem [Reference Deligne and MilneDM82, Theorem 2.11], $\pi _1(\mathscr {C}_0)$ is the pro-algebraic completion of $W(\mathscr {C}_0)$ and the image of $\iota$ is therefore Zariski-dense.

(ii) The second assertion follows from the first one by Theorem 2.1.4(iii)(c). Let $H$ denote the normalizer of $\pi _1(\mathscr {C})$ in $\pi _1(\mathscr {C}_0)$. As $\pi _1(\mathscr {C})$ is normal in $W(\mathscr {C}_0)$, the image of $\iota \colon W(\mathscr {C}_0)\to \pi _1(\mathscr {C}_0)$ is contained in $H$. As $\iota$ has Zariski-dense image by assertion (i), this implies that $H=\pi _1(\mathscr {C}_0)$ and the assertion follows.

2.2.1 Let $X$ be a $k$-variety. We denote by $\operatorname {\mathbf {Isoc}^{{{\dagger}} }}(X/K)$ (respectively, $\operatorname {\mathbf {Isoc}}(X/K)$) the category of overconvergent (respectively, convergent) isocrystals over $X$ relative to $K$. If we apply the construction of § 2.1.1 to the Tannakian category $\widetilde {\mathscr {C}}=\operatorname {\mathbf {Isoc}^{{{\dagger}} }}(X/K)$ (respectively, $\operatorname {\mathbf {Isoc}}(X/K)$) over $k$ and the ($s$th) Frobenius pull-back functor $\tau =F_X^*$, then $\mathscr {C}_0$ corresponds to the category $\operatorname {\mathbf {F-Isoc}^{{{\dagger}} }}(X/K)$ (respectively, $\operatorname {\mathbf {F-Isoc}}(X/K)$) of overconvergent (respectively, convergent) $F$-isocrystals over $X/(K,\sigma )$. We set $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X/K):=\mathscr {C}$, the Tannakian full subcategory of $\operatorname {\mathbf {Isoc}^{{{\dagger}} }}(X/K)$ generated by subquotients of overconvergent isocrystals which admit a ($s$th) Frobenius structure, considered in [Reference AbeAbe18b].

2.2.2 In the following, we assume $(k,\sigma )=(\mathbb {F}_q,\operatorname {id}_{\mathscr {O}_K})$.

The above construction can be generalized as follows. For $i=1,\ldots,n$, let $X_i$ be a $k$-variety and set $X:=X_1\times _k \cdots \times _k X_n$. We define a morphism $F_i\colon X\to X$ by the ($s$th) Frobenius morphism $F_{X_i}$ of $X_i$ on the component $X_i$ and the identity map on other components. The morphisms $F_i$ commute with each other and their composition is equal to $F_X$. The morphisms $F_i$ induce tensor equivalences

\[ F_i^*\colon \operatorname{\mathbf{Isoc}^{{{\dagger}}}}(X/K)\to \operatorname{\mathbf{Isoc}^{{{\dagger}}}}(X/K)\quad \text{(respectively, } \operatorname{\mathbf{Isoc}}(X/K)\to \operatorname{\mathbf{Isoc}}(X/K)),\quad i=1,\ldots,n, \]

commuting with each other and their composition is equal to $F^*_X$.

Given an overconvergent (respectively, convergent) isocrystal $\mathscr {E}$ over $X/K$, a partial ( $s$th) Frobenius structure on $\mathscr {E}$ consists of isomorphisms $\varphi _i\colon F_i^{*}(\mathscr {E})\xrightarrow {\sim } \mathscr {E}$ for $i=1,\dots, n$ such that $\varphi _i\circ F_i^{*}(\varphi _j)=\varphi _j\circ F_j^{*}(\varphi _i)$. For $i=1,\dots,n$, $\varphi _i$ is called the $i$th partial Frobenius structure (on $\mathscr {E}$). Note that the composition of the $\varphi _i$ (in any order) forms a Frobenius structure on $\mathscr {E}$.

If we apply the construction of 2.1.1 to $(\widetilde {\mathscr {C}}=\operatorname {\mathbf {Isoc}^{{{\dagger}} }}(X/K)$ (respectively, $\operatorname {\mathbf {Isoc}}(X/K)$); $F_1^{*},\ldots,F_n^{*})$, then $\mathscr {C}_0$ corresponds to the category of overconvergent (convergent) isocrystals over $X$ with a partial Frobenius structure, which we denote by $\operatorname {\mathbf {\Phi -Isoc}^{{{\dagger}} }}(X/K)$ (respectively, $\operatorname {\mathbf {\Phi -Isoc}}(X/K)$). We set $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X,\Phi /K):=\mathscr {C}$, the full subcategory of $\operatorname {\mathbf {Isoc}^{{{\dagger}} }}(X/K)$, generated by subquotients of overconvergent isocrystals which admits a $s$th partial Frobenius structure.

We denote by $F_i\text {-}\operatorname {\mathbf {Isoc}^{{{\dagger}} }}(X/K)$ (respectively, $F_i\text {-}\operatorname {\mathbf {Isoc}}(X/K)$) the category of pairs $(\mathscr {E},\varphi _i)$ consisting of an overconvergent (respectively, convergent) isocrystal with an $i$th partial Frobenius structure.

Let $k'/k$ be a finite extension of degree $a$ and set $K':=\operatorname {W}(k')\otimes _{\operatorname {W}(k)}K$. (We denote the Witt vector functor by $\operatorname {W}$ to reduce confusion with the notation $W(\mathscr {C}_0)$ from Lemma 2.1.5.) Set $X_{k'}:=X\otimes _k k'$. Then we have a canonical functor of extension of scalars:

(2.2.2.1)\begin{equation} \operatorname{\mathbf{\Phi-Isoc}}(X/K)\to \operatorname{\mathbf{\Phi-Isoc}}(X_{k'}/K'),\quad (\mathscr{E},\varphi_i)\mapsto (\mathscr{E}\otimes_K K',\varphi_i^{a}\otimes_K K'). \end{equation}

2.2.3 The category $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X,\Phi /K)$ defined above is a Tannakian category over $K$ and may not be neutral. For any algebraic extension $L$ of $K$ in $\overline {\mathbb {Q}}_p$, we set $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X,\Phi /L):=\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X,\Phi /K)\otimes _K L$. When $L=\overline {\mathbb {Q}}_p$, this category is a neutral Tannakian category over $\overline {\mathbb {Q}}_p$ (with respect to a fiber functor) and we omit $/\overline {\mathbb {Q}}_p$ from the notation. We denote by $\pi _1^{\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }}(X,\Phi )$ (respectively, $\pi _1^{\operatorname {\mathbf {\Phi -Isoc}^{{{\dagger}} }}}(X)$, respectively $\pi _1^{\operatorname {\mathbf {\Phi -Isoc}}}(X)$) the Tannakian group of $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X,\Phi )$ (respectively, $\operatorname {\mathbf {\Phi -Isoc}^{{{\dagger}} }}(X)$, $\operatorname {\mathbf {\Phi -Isoc}}(X)$) over $\overline {\mathbb {Q}}_p$. When $n=1$ (i.e. partial Frobenius structures reduce to a Frobenius structure), we omit $\Phi$ from the notation or replace it with $F$.

The pullback functor $p_{i}^*$ induces a canonical tensor functor

(2.2.3.1) \begin{equation} \begin{gathered} p_i^{*}\colon \operatorname{\mathbf{F-Isoc}^{{{\dagger}}}}(X_i) \to \operatorname{\mathbf{\Phi-Isoc}^{{{\dagger}}}}(X)\\ \text{(respectively, } \mathbf{Isoc}^{{{\dagger}}{{\dagger}}}(X_i)\to \mathbf{Isoc}^{{{\dagger}}{{\dagger}}}(X,\Phi),\ \text{respectively, } \operatorname{\mathbf{F-Isoc}}(X_i)\to \operatorname{\mathbf{\Phi-Isoc}}(X))\\ (\mathscr{E}_i,\varphi_i) \mapsto (p_i^{*}\mathscr{E}_i, \operatorname{id},\ldots,p_i^{*}(\varphi_i),\ldots,\operatorname{id}). \end{gathered} \end{equation}

By the Künneth formula, the functor $p_i^*\colon \mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X_i)\to \mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X,\Phi )$ is fully faithful. It induces a canonical $\overline {\mathbb {Q}}_p$-homomorphism

\begin{gather*} p_i^{\circ}\colon \pi_1^{\operatorname{\mathbf{\Phi-Isoc}^{{{\dagger}}}}}(X) \to \pi_1^{\operatorname{\mathbf{F-Isoc}^{{{\dagger}}}}}(X_i)\\ \text{(respectively, } \pi_1^{\mathbf{Isoc}^{{{\dagger}}{{\dagger}}}}(X,\Phi)\to \pi_{1}^{\mathbf{Isoc}^{{{\dagger}}{{\dagger}}}}(X_i),\ \text{respectively, } \pi_1^{\operatorname{\mathbf{\Phi-Isoc}}}(X) \to \pi_1^{\operatorname{\mathbf{F-Isoc}}}(X_i)). \end{gather*}

Proof. (i) Given an object $\mathscr {E}$ of $\operatorname {\mathbf {\Phi -Isoc}^{{{\dagger}} }}(X)$, the partial Frobenius pullbacks permute the isomorphism classes of the irreducible constituents of $\mathscr {E}$ in $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X)$. We thus conclude that an irreducible object $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X,\Phi )$ admits $s'$th partial Frobenius structure for some $s|s'$.

On the other hand, if an object $\mathscr {E}$ of $\operatorname {\mathbf {Isoc}^{{{\dagger}} }}(X)$ is irreducible equipped with an $s'(=st)$th partial Frobenius structure for some $t\in \mathbb {N}$, then $\mathscr {E}$ is a subobject of

\[ \mathscr{F}:=\bigoplus_{j_1=0}^{t-1} \cdots \bigoplus_{j_n=0}^{t-1} F_1^{j_1*}\circ\cdots\circ F_{n}^{j_n*}(\mathscr{E}), \]

which admits an $s$th Frobenius structure. Therefore, $\mathscr {E}$ belongs to $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X,\Phi )$.

(ii) When $n=1$, assertion (ii) is proved in [Reference D'Addezio and EsnaultDE22, Theorem 5.4]. The general case can be showed in a similar way as in [Reference D'Addezio and EsnaultDE22, Theorem 5.4].

(iii) Assertion (iii) follows from assertions (i) and (ii).

2.2.6 We will use the theory of holonomic (arithmetic) $\mathscr {D}$-modules and their six functors formalism developed in [Reference Abe and CaroAC18, Reference AbeAbe18b]. Let $L$ be an extension of $K$ in $\overline {\mathbb {Q}}_p$, $\mathfrak {T}:=\{k,\mathscr {O}_K,K,L\}$ the associated geometric base tuple and $\mathfrak {T}_F:=\{k,\mathscr {O}_K,K,L,s,\operatorname {id}_L\}$ the associated arithmetic base tuple [Reference AbeAbe18b, 1.4.10, 2.4.14].

Let $X$ be a $k$-variety. There exists an $L$-linear triangulated category $\operatorname {D}(X/L)$ (respectively, $\operatorname {D}(X/L_F)$) relative to the geometric base tuple $\mathfrak {T}$ (respectively, arithmetic base tuple $\mathfrak {T}_F$). This category is denoted by $\operatorname {D}^{\operatorname {b}}_{\operatorname {hol}}(X/\mathfrak {T})$ or $\operatorname {D}^{\operatorname {b}}_{\operatorname {hol}}(X/L)$ (respectively, $\operatorname {D}^{\operatorname {b}}_{\operatorname {hol}}(X/\mathfrak {T}_F)$ or $\operatorname {D}^{\operatorname {b}}_{\operatorname {hol}}(X/L_F)$) in [Reference AbeAbe18b, 1.1.1, 2.1.16]. For $\blacktriangle \in \{\emptyset,F\}$, there exists a holonomic t-structure on $\operatorname {D}(X/L_{\blacktriangle })$, whose heart is denoted by $\operatorname {Hol}(X/L_{\blacktriangle })$, called the category of holonomic modules. We denote by $\mathcal {H}^*$ the cohomological functor for holonomic t-structure.

The six functors formalism for $\operatorname {D}(X/L)$ (respectively, $\operatorname {D}(X/L_F)$) has been established recently. We refer to [Reference AbeAbe18b, § 2.3] for details and to [Reference AbeAbe18b, 1.1.3] for a summary.

2.2.7 Let $X$ be a smooth and quasiprojective geometrically connected $k$-variety of dimension $d_X$. The category $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X)$ (respectively, $\operatorname {\mathbf {F-Isoc}^{{{\dagger}} }}(X)$) is equivalent to the full subcategory $\operatorname {Sm}(X/\overline {\mathbb {Q}}_p)$ (respectively, $\operatorname {Sm}(X/\overline {\mathbb {Q}}_{p,F})$) consisting of smooth objects of the category $\operatorname {Hol}(X/\overline {\mathbb {Q}}_p)[-d_X]$ (respectively, $\operatorname {Hol}(X/\overline {\mathbb {Q}}_{p,F})[-d_X]$) of holonomic arithmetic $\mathscr {D}$-modules shifted by the dimension $-d_X$ (see [Reference AbeAbe18b, 1.1.3 (12)]).

We briefly review the pushforward and pullback functors for a smooth morphism $f\colon X\to Y$ of relative dimension $d$ between quasiprojective geometrically connected $k$-varieties following [Reference AbeAbe18b, 1.2.8]. We have an adjoint pair $(f_+,f^+[2d])$

\[ f_+:\operatorname{D}(X/L_{\blacktriangle}) \rightleftarrows \operatorname{D}(Y/L_{\blacktriangle}): f^+[2d]. \]

The functor $f^+[d]$ is exact and induces a fully faithful functor [Reference Xu and ZhuXZ22, Proposition 2.1.6]:

(2.2.7.1)\begin{equation} f^+[d]\colon \operatorname{Hol}(Y/L_{\blacktriangle})\to \operatorname{Hol}(X/L_{\blacktriangle}). \end{equation}

If $d_X$ (respectively, $d_Y$) denotes the dimension of $X$ (respectively, $Y$), the functor

(2.2.7.2)\begin{equation} f_{*}:=\mathcal{H}^{d}(f_+(-))[-d]\colon \operatorname{Hol}(X/L_{\blacktriangle})[-d_X]\to \operatorname{Hol}(Y/L_{\blacktriangle})[-d_Y] \end{equation}

is a right adjoint of $f^*:=f^+$ and is left exact. Its right derived functor $\operatorname {R} f_*$ is compatible with $f_+$.

Moreover, there exists a canonical isomorphism (Poincaré duality [Reference AbeAbe18b, 1.5.13]):

\[ f^+(d)[2d]\xrightarrow{\sim} f^!. \]

2.2.9 Let $Y:=Y_1\times _k Y_2$ be a product of two geometrically connected $k$-varieties, $k'$ a perfect field over $k$, $\mathscr {O}_{K'}:=\operatorname {W}(k')\otimes _{\operatorname {W}(k)}\mathscr {O}_K$, equipped with a lift $\sigma '$ of the $s$th Frobenius automorphism of $k'$ defined by that of $\operatorname {W}(k')$ and $\operatorname {id}_{\mathscr {O}_K}$. We set $K':=\mathscr {O}_{K'}[{1}/{p}]$.

We denote by $F_1\text {-}\operatorname {\mathbf {Isoc}^{{{\dagger}} }}(Y_{1,k'}/K')$ (respectively, $F_1\text {-}\operatorname {\mathbf {Isoc}}(Y_{1,k'}/K')$) the category of pairs $(\mathscr {E},\phi )$ consisting of an object $\mathscr {E}$ of $\operatorname {\mathbf {Isoc}^{{{\dagger}} }}(Y_{1,k'}/K')$ (respectively, $\operatorname {\mathbf {Isoc}}(Y_{1,k'}/K')$) and an isomorphism $\phi \colon (F_{Y_1}\otimes \operatorname {id}_{k'})^*(\mathscr {E})\xrightarrow {\sim } \mathscr {E}$.

A point $i\colon \operatorname {Spec}(k')\to Y_2$ induces a natural functor:

(2.2.9.1)\begin{align} \begin{aligned} \iota\colon F_1\text{-}\operatorname{\mathbf{Isoc}^{{{\dagger}}}}(Y/K) & \to F_1\text{-}\operatorname{\mathbf{Isoc}^{{{\dagger}}}}(Y_{1,k'}/K'), \enspace \text{(respectively, } F_1\text{-}\operatorname{\mathbf{Isoc}}(Y/K)\to F_1\text{-}\operatorname{\mathbf{Isoc}}(Y_{1,k'}/K') ),\\ (\mathscr{E},\varphi) & \mapsto ((\operatorname{id}_{Y_1}\times i)^*(\mathscr{E}),(\operatorname{id}_{Y_1}\times i)^*\varphi). \end{aligned} \end{align}

When $k'$ is a finite extension of $k$ of degree $a$, we associate an $F$-isocrystal over $Y_{1,k'}/K'$ to an object of $F_1\text {-}\operatorname {\mathbf {Isoc}^{{{\dagger}} }}(Y_{1,k'}/K')$ by composing its Frobenius structure $a$ times.

When $Y_2=\operatorname {Spec}(k)$, we regard the above functor as the functor of extension of scalars:

\[ \iota_{k'/k}\colon \operatorname{\mathbf{F-Isoc}^{{{\dagger}}}}(Y/K) \to F_1\text{-}\operatorname{\mathbf{Isoc}^{{{\dagger}}}}(Y_{1,k'}/K'). \]

2.2.10. Proof of Proposition 2.2.8

(i) It suffices to prove the assertion for objects of $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X,\Phi /K)$. We may assume $i=1$. We set $X':=\prod _{i=2}^n X_i$.

(a) We first prove the assertion for an object $\mathscr {E}$ of $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X,\Phi /K)$ equipped with a partial Frobenius structure $(\varphi _1,\ldots,\varphi _n)$. There exists an open subscheme $U_1$ of $X_1$ such that $p_{1,*}(\mathscr {E})|_{U_1}$ is smooth. In view of their fibers, the adjoint morphism of overconvergent isocrystals:

\[ p_1^*(p_{1,*}(\mathscr{E})|_{U_1})\to \mathscr{E}|_{U_1\times_k X'} \]

is injective. The first partial Frobenius $\varphi _1$ on $\mathscr {E}$ induces a Frobenius structure $\phi _1$ on $p_{1,*}(\mathscr {E})|_{U_1}$ and the above morphism is compatible with the first partial Frobenius structures $p_1^*(\phi _1)$ and $\varphi _1$. Let $k'$ be a perfect closure of $k(X')$ and $K':=\operatorname {W}(k')[{1}/{p}]$. By the exactness of the functor $\iota$, defined in (2.2.9.1), we obtain an injection in $F_1\text {-}\operatorname {\mathbf {Isoc}^{{{\dagger}} }}(U_{1,k'}/K')$:

\[ \iota_{k'/k}(p_{1,*}(\mathscr{E})|_{U_1}) \hookrightarrow \iota(\mathscr{E}|_{U_{1}\times_k X'}). \]

By [Reference KedlayaKed07, Proposition 5.3.1], the left-hand side extends to a subobject $\mathscr {F}'$ of $\iota (\mathscr {E})$ of $\operatorname {\mathbf {Isoc}^{{{\dagger}} }}(X_{1,k'}/K')$.

We claim that there exists an overconvergent isocrystal $\mathscr {F}$ of $\operatorname {\mathbf {Isoc}^{{{\dagger}} }}(X_1/K)$ extending $p_{1,*}(\mathscr {E})|_{U_1}$ such that $\iota _{k'/k}(\mathscr {F})\simeq \mathscr {F}'$. By Kedlaya and Shiho's purity theorem [Reference KedlayaKed22a, Theorem 5.1], we may assume that the boundary $D:=X_1-U_1$ is a smooth divisor. Since the local monodromy of $\mathscr {F}'$ around $D_{k'}$ is constant [Reference KedlayaKed07, Theorem 5.2.1], then so is the local monodromy of $p_{1,*}(\mathscr {E})|_{U_1}$ around $D$. Then the claim follows.

Moreover, the Frobenius structure on $p_{1,*}(\mathscr {E})|_{U_1}$ extends to $\mathscr {F}$. Finally, by full faithfullness of pullback for holonomic $\mathscr {D}$-modules along $U_1\to X_1$ (see [Reference Xu and ZhuXZ22, Proposition 2.1.6]), we deduce that $\mathscr {F}$ is isomorphic to $p_{1,*}(\mathscr {E})$. Then the assertion in this case follows.

(b) An object $\mathscr {E}$ of $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X,\Phi )$ can be embedded into an object $\mathscr {F}$, which can be equipped with a partial Frobenius structure (Proposition 2.2.5(ii)). As $p_{1,*}$ is left exact, we have an injection to a smooth object $\mathscr {N}$ of $\operatorname {Hol}(X_1/\overline {\mathbb {Q}}_{p,F})[-d_{X_1}]$:

\[ \mathscr{M}:=p_{1,*}(\mathscr{E})\hookrightarrow \mathscr{N}:=p_{1,*}(\mathscr{F}). \]

It remains to show that $\mathscr {M}$ is also smooth. We first consider the case where $X_1$ is a curve. Let $U_1$ be an open subset of $X_1$ on which $\mathscr {M}$ is smooth. By applying [Reference Xu and ZhuXZ22, Corollary 2.3.4] to $\mathscr {M}|_{U_1}$, the exactness of pullback on smooth modules and of the nearby cycle functor, we deduce the smoothness of $\mathscr {M}$ from that of $\mathscr {N}$.

In general, there is a dense open subscheme $j\colon U_1\to X_1$ and a smooth object $\mathscr {L}$ on $U_1$ such that $\mathscr {M}|_{U_1}\simeq \mathscr {L}$. Let $c\colon C\to X_1$ be a morphism from a smooth curve to $X$ such that $c(C)\cap U_1$ is non-empty and $p_c\colon C\times _k X'\to C$ the projection. By the above argument, $p_{c,*}( (c\times \operatorname {id})^*\mathscr {E})$ is smooth and the pullback $c^*\mathscr {L}$ extends to an overconvergent isocrystal on $C$. By Shiho's cut-by-curves theorem [Reference ShihoShi11a] and Kedlaya and Shiho's purity theorem, $\mathscr {L}$ can be extended to an overconvergent isocrystal over $X$. By full faithfulness of pullback along $U_1\to X_1$, we deduce that $\mathscr {M}$ is isomorphic to the extension of $\mathscr {L}$ to $X_1$. This finishes the proof.

(ii) The isomorphism $\operatorname {id}\to p_{i,*}p_i^*$ follows from the Künneth formula [Reference AbeAbe18b, Proposition 1.1.7]. In view of fibers at closed points, the injectivity of $p_i^*p_{i,*}\to \operatorname {id}$ follows.

2.2.11 We now turn to the proof of Theorem 2.2.4. We first prove Theorem 2.2.4 for $\pi _1^{\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }}$ under the additional hypothesis that each $X_i$ is smooth and quasiprojective over $k$.

Proof. As we work with arithmetic $\mathscr {D}$-modules with coefficients in $\overline {\mathbb {Q}}_p$, we may enlarge the base field $k$ as in [Reference AbeAbe18b, 1.4.11]. Therefore, we may assume there exists a $k$-point $x$ of $X_1$. We consider the following diagram.

We denote by $\operatorname {\mathbf {\Phi }^{\prime }-\mathbf {Isoc}^{{{\dagger}} }}(X')$ the category of overconvergent isocrystals over $X'$ with a partial Frobenius structure and $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X',\Phi ')$ the Tannakian subcategory of $\operatorname {\mathbf {Isoc}^{{{\dagger}} }}(X')$ generated by those object which can be equipped with some partial Frobenius structure. The morphism $u$ induces a tensor functor

(2.2.11.1)\begin{align} u^{*}\colon \mathbf{Isoc}^{{{\dagger}}{{\dagger}}}(X,\Phi)\to \mathbf{Isoc}^{{{\dagger}}{{\dagger}}}(X',\Phi')\quad \text{(respectively, } \operatorname{\mathbf{\Phi-Isoc}^{{{\dagger}}}}(X) \to \operatorname{\mathbf{\Phi}^{\prime}-\mathbf{Isoc}^{{{\dagger}}}}(X')) \quad \mathscr{E}\mapsto u^{*}(\mathscr{E}) \end{align}

and a homomorphism $u^{\circ }\colon \pi _1^{\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }}(X',\Phi ')\to \pi _1^{\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }}(X,\Phi )$. Consider the following commutative diagram.

(2.2.11.2)

Here the upper sequence is defined as before and the lower sequence is exact. By induction, it suffices to show the exactness of the upper sequence. In the following, we will do it by checking the conditions of Theorem 2.1.4 for the functors:

\[ \mathbf{Isoc}^{{{\dagger}}{{\dagger}}}(X_1)\xrightarrow{p_1^{*}} \mathbf{Isoc}^{{{\dagger}}{{\dagger}}}(X,\Phi) \xrightarrow{u^{*}} \mathbf{Isoc}^{{{\dagger}}{{\dagger}}}(X',\Phi'), \]

where $p_1^{*}$ is defined as in (2.2.3.1). We use the functor $p_{i,*}$ defined in Proposition 2.2.8(i).

(i) Let $\mathscr {E}$ be an object of $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X_1)$ and $\mathscr {F}$ a subobject of $p_1^*(\mathscr {E})$. We need to show that $\mathscr {F}$ is the image under $p_1^*$ of a subobject of $\mathscr {E}$. If $\mathscr {G}$ denotes the quotient $p_1^*(\mathscr {E})/\mathscr {F}$, by applying $p_1^*p_{1,*}$ we obtain a commutative diagram.

Here $\beta$ is the identity, whereas $\alpha,\gamma$ are injective by Proposition 2.2.8(ii). Then $\delta$ is surjective. We deduce that $\alpha$ is an isomorphism. Then we conclude that $\pi _1^{\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }}(X,\Phi ) \to \pi _1^{\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }}(X_1)$ is faithfully flat by [Reference Deligne and MilneDM82, Proposition 2.21].

(ii) and (iii)(c) Since $u$ is a section of $p'\colon X=X_1\times _k X'\to X'$, we deduce condition (iii)(c) of Theorem 2.1.4 for $u^{\circ }\colon \pi _1^{\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }}(X',\Phi ')\to \pi _1^{\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }}(X,\Phi )$. Then $u^{\circ }$ is a closed immersion by [Reference Deligne and MilneDM82, Proposition 2.21].

(iii)(b) Given an object $\mathscr {E}$ of $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X,\Phi )$, the maximal trivial subobject of $u^{*}\mathscr {E}$ in the category $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X',\Phi ')$ is $g^{*}g_{*}u^{*}\mathscr {E}$. Since $u^+(\mathscr {E})[2d]\simeq u^!(\mathscr {E})$, we have $g^{*}g_{*}u^{*}\mathscr {E}\simeq u^* p_1^*p_{1,*}\mathscr {E}$ by smooth base change. Then condition (b) follows from the fact that $p_1^*p_{1,*}\to \operatorname {id}$ is injective by Proposition 2.2.8(ii).

(iii)(a) If an object $\mathscr {E}$ of $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X,\Phi )$ comes from the essential image of $p_1^{*}$, then $u^*(\mathscr {E})$ is trivial. Conversely, in view of condition (iii)(b), if $u^*(\mathscr {E})$ is trivial, then $\mathscr {E}\simeq p_1^*p_{1,*}(\mathscr {E})$. This verifies condition (a).

Proof. Descent for $\operatorname {\mathbf {Isoc}^{{{\dagger}} }}(X)$ is established in [Reference LazdaLaz22, Theorem 5.1]. This then implies descent for $\operatorname {\mathbf {\Phi -Isoc}^{{{\dagger}} }}(X)$; we deduce descent for $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X, \Phi )$ using Proposition 2.2.5.

2.2.13. Proof of Theorem 2.2.4 for $\pi _1^{\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }}$ with general $X_i$

By de Jong's alterations theorem [Reference de JongdeJ96], there exist smooth, connected, and quasiprojective $k$-varieties $Y_i$ and proper surjective, generic étale maps $\pi _i\colon Y_i\to X_i$ for $i=1,2,\ldots,n$. We use the hypercovering produced by the $\pi _i$ to show that the upper sequence in (2.2.11.2) is exact, by again checking the conditions of Theorem 2.1.4.

We set $Y:=\prod _{i=1}^n Y_i$ and $\pi \colon Y\to X$ to be the product $\prod _{i=1}^n \pi _i$. By induction, it suffices to treat the case where $Y_i=X_i$ for $i=2,\dots,n$. Consider the following diagram.

By induction hypotheses, we may assume the exactness of the second line. We will deduce the exactness of the first line from the second one. The above diagram corresponds to the top two lines in the following diagram.

Here $Y_1^{(2)}:=Y_1\times _{X_1}Y_1$ and $Y^{(2)}:=Y\times _X Y$. We check the conditions of 2.1.4 for the first line.

(i) Let $U$ be an object of $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X_1)$ and $V_0$ a subobject of $p_1^*(U)$. By exactness of the second line, we deduce that there exists a subobject $U_{0,Y}$ of $\pi _1^*(U)$ over $Y_1$ such that $p_1^*(U_{0,Y})\simeq \pi ^*(V_0)$ as subobjects of $\pi ^*(p_1^*(U))$. As the functor $p_1^*\colon \mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(Y_1^{(2)})\to \mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(Y^{(2)},\Phi )$ is fully faithful, the descent data on $\pi ^*(V_0)$ gives rise to a descent data on $U_0$. In this way, we obtain a subobject $U_{0}$ of $U$ over $X_1$, sent to $V_0\to p_1^*(U)$ via $p_1^*$. Hence, the map $p_1^\circ \colon \pi _1^{\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }}(X,\Phi )\to \pi _1^{\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }}(X_1)$ is faithfully flat.

Conditions (ii) and (iii)(c) follow from the same argument as in the smooth and quasiprojective case.

(iii)(a) Let $V$ be an object of $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X,\Phi )$ such that $u^*(V)$ is trivial. Then there exists an object $U_Y$ over $Y_1$ such that $p_1^*(U_Y)\simeq \pi ^*(V)$. The descent data on $\pi ^*(V)$ induces a descent data on $U_Y$, which gives rise to an object $U$ on $X_1$, sent to $V$ via $p_1^*$.

(iii)(b) Let $V$ be an object of $\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }(X,\Phi )$, $W_0$ be the maximal trivial subobject of $u^*(V)$. By exactness of the second line, there exists a subobject $V_0$ of $\pi ^*(V)$ on $Y$, sent to $W_0$ via $u^*$. Since $u^*$ is faithful, we deduce that the descent data on $\pi ^*(V)$ preserves $V_0$. This gives rise to a subobject of $V$, sent to the trivial object $W_0$.

2.2.14. Proof of Theorem 2.2.4 for $\pi _1^{\operatorname {\mathbf {\Phi -Isoc}^{{{\dagger}} }}}$

By Proposition 2.1.3(i), we have a commutative diagram:

where the first and second lines are exact. By Proposition 2.1.3(ii), the right vertical arrow identifies with the projection of the pro-algebraic completion of $\mathbb {Z}^n$ to the pro-algebraic completion of $\mathbb {Z}$ and is therefore an isomorphism. The assertion follows from that the left vertical arrow is an isomorphism.

3.1.1 We first define the partial Frobenius slopes at a closed point of $X$.

Let $(\mathscr {E},\varphi _i)$ be an object of $\operatorname {\mathbf {\Phi -Isoc}}(X/K)$, $x\in |X|$ a closed point, $K_x:=\operatorname {W}(k_x)\otimes _{\operatorname {W}(k)}K$, and $a:=[k_x:k]$. We take the extension of scalars to $k_x$ (2.2.2.1) and then take its fiber $(\mathscr {E}_x,\varphi _1^a|_x,\ldots,\varphi _n^a|_x)$ at $x$, which is an object of $\operatorname {\mathbf {\Phi -Isoc}}(\operatorname {Spec}(k_x)/K_x)$ with respect to the $sa$th power of Frobenius, that is a $K_x$-vector space together with commuting linear automorphisms $\varphi _i^a|_x$. This allows us to define the $i$th partial Frobenius slopes of $(\mathscr {E},\varphi _i)$ at $x$ for $i=1,\ldots,n$ by Dieudonné–Manin theory.

We have a surjection from points of $X$ to products of points of the $X_i$:

\[ \pi\colon |X|\twoheadrightarrow \prod_{i=1}^n |X_i|. \]

Let $(x_i)_{i=1}^n\in \prod |X_i|$ be a tuple of closed points. To simplify notation, assume that there exists a finite extension $k'$ of $k$ such that $k_{x_i}\simeq k'$ (this can always be enforced by enlarging $k$). We have a decomposition

(3.1.1.1)\begin{equation} \prod_k x_i\simeq \bigsqcup_G \operatorname{Spec}(k'), \end{equation}

indexed by $G:=\big ( \prod \operatorname {Gal}(k'/k) \big )/\operatorname {Gal}(k'/k)$, for the diagonal action. For an object $(\mathscr {E},\varphi _i)$ of $\operatorname {\mathbf {\Phi -Isoc}}(\prod _k x_i/K)$, $\mathscr {E}$ corresponds to a direct sum of vector spaces over $K'$, indexed by $G$. If $a=[k':k]$, then the $a$th power of each partial Frobenius $\varphi _{i}^a$ preserves each component.

Proof. We keep the above notation. Since $G$ acts transitively on each component of (3.1.1.1), the action induces isomorphisms between the pullback of $(\mathscr {E}_x,\varphi _{i,x}^a)$ to each component and the claim follows.

3.1.4 In the following, we assume each $X_i$ is a smooth geometrically connected $k$-variety. Then so is $X$.

Let $x$ be a closed point of $X$ and $\overline {x}$ the associated geometric point. Recall [Reference CrewCre87] that there exists a canonical equivalence between the category of continuous $K$-representations of $\pi _1^{{\unicode{x00E9}}{\text t}}(X,\overline {x})$ and the category of unit-root convergent $F$-isocrystals over $X/K$:

(3.1.4.1)\begin{equation} \mathbf{Rep}^{\operatorname{cont}}_{K}(\pi_1^{{\unicode{x00E9}}{\text t}}(X,\overline{x}))\xrightarrow{\sim} \operatorname{\mathbf{F-Isoc}}^{\operatorname{ur}}(X/K). \end{equation}

Via the equivalence between the left-hand side and the category $\operatorname {\mathbf {LocSys}}(X,K)$ of lisse $K$-sheaves over $X$, we have

(3.1.4.2)\begin{equation} \operatorname{\mathbf{LocSys}}(X,K)\xrightarrow{\sim} \operatorname{\mathbf{F-Isoc}}^{\operatorname{ur}}(X/K). \end{equation}

The above equivalence is compatible with the following operations.

1. (i) The pullback functoriality along a morphism between smooth connected $k$-varieties.

2. (ii) Extensions of scalars.

Recall that a lisse $K$-sheaf $\mathbb {L}$ over $X$ is equipped with the Frobenius correspondence [Reference GrothendieckSGA5, XIV=XV]:

(3.1.4.3)\begin{equation} \phi_{\mathbb{L}}\colon F_X^*(\mathbb{L})\xrightarrow{\sim} \mathbb{L}. \end{equation}

Proof. In view of the construction of (3.1.4.1), we may reduce to the case where $\mathbb {L}$ is a lisse $\mathscr {O}_K/p^n \mathscr {O}_K$-sheaf for some $n$, and we may assume $X$ is affine and admits a smooth lifting $\mathfrak {X}_n$ to $\mathscr {O}_K/p^n\mathscr {O}_K$, equipped with a Frobenius lift. After taking pullback along a finite étale morphism trivializing $\mathbb {L}$, we may moreover assume that $\mathbb {L}$ is a trivial $\mathscr {O}_K/p^n\mathscr {O}_K$-module. In this case, the assertion is clear.

3.1.7 Let $i$ be an integer $\in [1,n]$. The $i$th partial Frobenius $F_i$ of $X$ is a homeomorphism and induces an equivalence of étale topoi of $X$. Let $(\mathscr {E},\varphi _1,\ldots,\varphi _n)$ be a diagonally unit-root convergent $\Phi$-isocrystal over $X$ and $\mathbb {L}$ the lisse $K$-sheaf over $X$ associated to $(\mathscr {E},\varphi )$. The $i$th partial Frobenius structure $\varphi _i$ induces an isomorphism of sheaves:

\[ \phi_i\colon F_i^*(\mathbb{L})\xrightarrow{\sim} \mathbb{L}. \]

The isomorphisms $\phi _i$ commute with each other in the following sense: for any $1\le i,j \le n$, the identifications

\[ F_i^* (F_j^* (\mathbb{L})) \cong (F_i \circ F_j)^* (\mathbb{L}) = (F_j \circ F_i)^* (\mathbb{L}) \cong F_j^*( F_i^* (\mathbb{L})) \]

induce an equality $\phi _j\circ F_j^*(\phi _i)=\phi _i\circ F_i^*(\phi _j)$. The composition of $\phi _i$ coincides with the isomorphism $\phi _{\mathbb {L}}$ (3.1.4.3). The above construction is clearly functorial.

Proof. We construct a quasi-inverse of the functor in § 3.1.7. Let $(\mathbb {L},\{\phi _i\}_{i=1}^n)$ be a collection of data as above and $(\mathscr {E},\varphi )$ the unit-root $F$-isocrystal over $X$ associated to $\mathbb {L}$. By functoriality of (3.1.4.2), $\phi _i$ induces an $i$th partial Frobenius structure $\varphi _i$ on $\mathscr {E}$. The commutativity of the $\varphi _i$, and the fact that the composition of the $\varphi _i$ equals $\varphi$, follow from the corresponding properties of the $\phi _i$. This construction is clearly functorial and provide a quasi-inverse of § 3.1.7.

Proof. Let $(\mathbb {L},\{\phi _i\}_{i=1}^n)$ be the associated data in Proposition 3.1.8. Then the $i$th partial Frobenius slope of $\mathscr {E}$ at $x\in X(k)$ can be calculated by that of $\phi _{i,\overline {x}}$ on $\mathbb {L}_{\overline {x}}$ (with the convention of Remark 3.1.6). Since $\mathbb {L}$ admits a lisse $\mathscr {O}_K$-sheaf as an integral model, the slopes of $\phi _i$ at each fibers $\mathbb {L}_{\overline {x}}$ are constant as function on $x\in X(k)$.

The corollary follows from applying the previous argument to extensions of scalars of $(\mathscr {E},\varphi _i)$ (§ 3.4.1).

Proof. (i) There exists a slope filtration (3.1.10.1) as convergent $F$-isocrystals [Reference KedlayaKed22a, Corollary 4.2]. It suffices to show that each partial Frobenius $\varphi _i$ preserves this filtration, that is the composition $F_i^*(\mathscr {E}_j)\xrightarrow {\varphi _i} \mathscr {E} \to \mathscr {E}/\mathscr {E}_j$ vanishes. Then uniqueness follows from that of the slope filtration for $F$-isocrystals.

We reduce to checking the above claim at the fiber of each closed point $x\in |X|$. We may assume there exists a finite extension $k'/k$ of degree $a$ such that $x\simeq \operatorname {Spec}(k')$ as in § 3.1.1. Then the fiber of the Frobenius structure $\varphi _x^a$ is a linear automorphism of $\mathscr {E}_x$, and its generalized eigenspace decomposition is a refinement of the filtration $\mathscr {E}_{0,x}\subset \cdots \subset \mathscr {E}_{\ell,x}$. Since each partial Frobenius structure $\varphi _{i,x}$ commutes with $\varphi _{x}^a$, $\varphi _{i,x}$ preserves each generalized eigenspace of $\varphi _x^a$. Then the assertion follows.

(ii) By assertion (i), we may reduce to the case where $\mathscr {E}$ is diagonally unit-root after twisting. In this case, assertion (ii) follows from Corollary 3.1.9.

3.2.1 We denote by $\mathcal {C}(X,\Phi )$ the category of objects $(T,\{F_{ \{i\}}\}_{i=1}^n)$ consisting of a finite étale morphism $T\to X=X_1\times _k\cdots \times _k X_n$ and isomorphisms $F_{ \{i\}}\colon T\times _{X,F_i}X \xrightarrow {\sim } T$ commuting with each other, i.e. $F_{ \{i\}}\circ F_{ \{j\}}\times _{X,F_j}X= F_{ \{j\}}\circ F_{ \{i\}}\times _{X,F_i}X$, whose composition is the relative Frobenius morphism $F_{T/X}$ of $T$ over $X$. A morphism in this category is a morphism above $X$ compatible with the $F_{ \{i\}}$. Note that an object of $\mathcal {C}(X,\Phi )$ is equivalent to the data of a pair $(T, \{\varphi _{T,i}\}_{i=1}^n)$ consisting of a finite étale morphism $T\to X$ together with morphisms $\varphi _{T,i}\colon T\to T$ above $F_i$, commuting with each other, whose composition is the Frobenius morphism $F_T$.

This category is a Galois category and we denote by $\pi _1^{{\unicode{x00E9}}{\text t}}(X,\Phi,\overline {x})$ the Galois group defined by the fiber functor associated to $\overline {x}$. We have the following equivalence:

• • the category of continuous actions of $\pi _1(X,\Phi,\overline {x})$ on finite sets;

• • the category of locally constant constructible sheaves $\mathbb {L}$ of $X_{{\unicode{x00E9}}{\text t}}$, equipped with a partial Frobenius structure $\phi _i\colon F_i^*(\mathbb {L})\xrightarrow {\sim } \mathbb {L}$, commuting with each other, whose composition is the Frobenius correspondence $\phi _{\mathbb {L}}$.

3.2.3 Let $\pi$ be a uniformizer of $\mathscr {O}_K$. We set $R:=\mathscr {O}_K$ and $R_n:=\mathscr {O}_K/\pi ^n \mathscr {O}_K$ for $n\ge 1$. We may reduce to the case where each $X_i$ is affine and admits a smooth formal lifting $\mathfrak {X}_i$ to $R$ and a lifting $F_{\mathfrak {X}_i}\colon \mathfrak {X}_i\to \mathfrak {X}_i$ of the Frobenius $F_{X_i}$. We set $\mathfrak {X}:=\prod _{\operatorname {W}(k)} \mathfrak {X}_i$ and denote by $F_{i,\mathfrak {X}}\colon \mathfrak {X}\to \mathfrak {X}$ the product of $F_{\mathfrak {X}_i}$ and the identity maps on other components.

A unit-root $\Phi$-lattice on $\mathfrak {X}/R$ is a locally free $\mathscr {O}_{\mathfrak {X}}$-module of finite rank together with isomorphisms $\varphi _i\colon F_{i,\mathfrak {X}}^*(M)\xrightarrow {\sim } M$ commuting with each other. We first establish an equivalence following [Reference CrewCre87]:

(3.2.3.1)\begin{equation} \iota\colon \mathbf{Rep}_{R}^{\operatorname{cont}}(\pi_1(X,\Phi,\overline{x})) \xrightarrow{\sim} \{\text{Unit-root $\Phi$-lattices on $\mathfrak{X}/R$}\}. \end{equation}

Let $V$ be a continuous $R$-linear representation of $\pi _1^{{\unicode{x00E9}}{\text t}}(X,\Phi,\overline {x})$. For $n\ge 1$, let $G_n$ be the image of $\pi _1^{{\unicode{x00E9}}{\text t}}(X,\Phi,\overline {x})$ in $\operatorname {GL}(V/\pi ^nV)$, which corresponds to a Galois cover $Y^{(n)}\to X$ and $\pi _n\colon \mathfrak {Y}_n^{(n)}\to \mathfrak {X}_n$ its lifting to $R_n$. Moreover, each isomorphism $F_{ \{i\}}$ uniquely lifts to an isomorphism $F_{ \{i\}}\colon F_{i,\mathfrak {X}}^*(\mathfrak {Y}_n^{(n)})\xrightarrow {\sim } \mathfrak {Y}_n^{(n)}$, which commute with each other. As in [Reference CrewCre87], we consider a locally free $\mathscr {O}_{\mathfrak {X}_n}$-module (respectively, $\mathscr {O}_{\mathfrak {X}}$-module)

\[ M_n:= \pi_{n,*}(\mathscr{O}_{\mathfrak{Y}_n^{(n)}})\otimes_{R_n[G_n]}V/\pi^n V, \quad M:=\varprojlim M_n. \]

Then the isomorphism $F_{ \{i\}}$ induces an isomorphism $F_{i,\mathfrak {X}}^*(M_n)\xrightarrow {\sim } M_n$ and gives rise to a unit-root partial Frobenius structure

\[ \varphi_i\colon F_{i,\mathfrak{X}}^*(M)\xrightarrow{\sim} M. \]

In this way, we obtain the functor $\iota$ as in (3.2.3.1).

Conversely, given a unit-root $\Phi$-lattice $(\mathscr {E},\varphi _i)$ on $\mathfrak {X}/R$, the associated unit-root $F$-lattice $(\mathscr {E},\varphi )$ defines a lisse $R$-sheaf $\mathbb {L}$ on $X$ by [Reference CrewCre87, Theorem 2.2]. The partial Frobenius structures $\varphi _i$ define isomorphisms $\phi _i\colon F_i^*(\mathbb {L})\xrightarrow {\sim } \mathbb {L}$ commuting with each other. Then we obtain a continuous $R$-representation of $\pi _1(X,\Phi,\overline {x})$ on $\mathbb {L}_{\overline {x}}$. The above construction is functorial and defines a quasi-inverse of $\iota$.

Proof. Let $\Delta$ be the formal completion of the diagonal map $\mathfrak {X}\to \mathfrak {X}\times _{R}\mathfrak {X}$ and $p_1,p_2\colon \Delta \to \mathfrak {X}$ the natural projections. For $n\ge 1$, we set $\Delta _n:=\Delta \otimes _{R}R_n$. Let $p_i^*\mathfrak {Y}_{n}^{(n)}$ be the fiber product $\mathfrak {Y}_{n}^{(n)}\times _{\mathfrak {X}_n,p_i}\Delta _n$. The map $\mathfrak {Y}_n^{(n)}\to \mathfrak {X}_n \xrightarrow {diag} \Delta _n$ gives rise to formal thickenings $\mathfrak {Y}_n^{(n)}\to p_i^*\mathfrak {Y}_{n}^{(n)}$. Recall (see [Reference CrewCre87, Proposition 2.3]) that, since $\mathfrak {Y}_n^{(n)}\to \mathfrak {X}_n$ is étale, there exists a unique isomorphism $p_1^*\mathfrak {Y}_n^{(n)}\xrightarrow {\sim } p_2^*\mathfrak {Y}_n^{(n)}$ which fits into the following diagram.

By uniqueness, this isomorphism is compatible with the $G_n$-action, the partial Frobenius maps on $\mathfrak {Y}_n^{(n)}$, and the canonical connection on $M$. Then the lemma follows.

3.2.5. Proof of Proposition 3.2.2

By the above lemma, we have a fully faithful functor

\[ \{\text{Unit-root $\Phi$-lattices on $\mathfrak{X}/R$}\}\otimes_{R}K \to \{\text{Unit-root $\Phi$-isocrystals on $X/K$}\}. \]

It suffices to show that the functor in Proposition 3.2.2 is essentially surjective.

Let $(\mathscr {M},\varphi _i)$ be a unit-root $\Phi$-isocrystal on $X/K$ and $(\mathbb {L},\phi _i)$ the associated data in Proposition 3.1.8. There exists a $R$-lattice $\mathbb {L}^{\circ }$ of $\mathbb {L}$ such that $\phi _{\mathbb {L}}$ induces an isomorphism $F_X^*(\mathbb {L}^{\circ })\xrightarrow {\sim } \mathbb {L}^{\circ }$. As $\varphi _i$ is unit-root, we claim there exist integers $r,s$ such that for every integer $m\ge 0$, we have

\[ p^r \mathbb{L}^{\circ} \subset \phi_i^m(F_i^{m,*}\mathbb{L}^{\circ}) \subset p^s \mathbb{L}^{\circ}. \]

Indeed, let $x\in |X|$ be a closed point with degree $[k_x:k]=a$. The linear action of $\phi _{i,x}^a$ on the fiber $\mathbb {L}^{\circ }_{\overline {x}}$ is unit-root and satisfies the above properties [Reference CrewCre87, Proposition 1.11]. Then the claim follows.

Following [Reference CrewCre87], we consider

\[ \mathbb {L}':=\sum _{m\ge 0} \operatorname {Im} \phi _i^m\colon \mathbb {L}^{\circ }\to p^s \mathbb {L}\quad \textrm{ and }\quad \mathbb {L}_i:= \cap _{m\ge 1} \operatorname {Im} \phi _i^m\colon \mathbb {L}'\to \mathbb {L}'. \]

Then $\phi _i$ induces an isomorphism $F_i^*\mathbb {L}_i\xrightarrow {\sim } \mathbb {L}_i$.

By repeating the above argument to each $\phi _i$, we obtain a $R$-lattice $\widetilde {\mathbb {L}}$ of $\mathbb {L}$ such that $\phi _i$ induces an isomorphism $F_i^*(\widetilde {\mathbb {L}})\xrightarrow {\sim } \widetilde {\mathbb {L}}$ for every $i$. Then the data $(\widetilde {\mathbb {L}},\phi _i)$ descend to a representation of $\pi _1^{{\unicode{x00E9}}{\text t}}(X,\Phi,\overline {x})$. In view of Proposition 3.1.8, this defines a quasi-inverse of the previous construction. The proposition follows.

3.3.1 We say a representation of $\mathbf {Rep}^{\operatorname {cont}}_K(\pi _1^{{\unicode{x00E9}}{\text t}}(X,\Phi,\overline {x}))$ is smooth if the action of $\pi _1^{{\unicode{x00E9}}{\text t}}(X,\Phi,\overline {x})$ factors through a finite quotient. In view of étale descent for overconvergent $F$-isocrystals, the canonical functor

(3.3.1.1)\begin{equation} \mathbf{Rep}^{\operatorname{smooth}}_{K}(\pi_1^{{\unicode{x00E9}}{\text t}}(X,\Phi,\overline{x})) \to \operatorname{\mathbf{\Phi-Isoc}}(X) \end{equation}

factors through $\operatorname {\mathbf {\Phi -Isoc}^{{{\dagger}} }}(X)$. It induces canonical homomorphisms:

(3.3.1.2)\begin{equation} \pi_1^{\operatorname{\mathbf{\Phi-Isoc}}}(X)\to \pi_1^{\operatorname{\mathbf{\Phi-Isoc}^{{{\dagger}}}}}(X)\to \pi_1^{{\unicode{x00E9}}{\text t}}(X,\Phi,\overline{x}). \end{equation}

Since the functor (3.3.1.1) is fully faithful, the composition is an epimorphism.

Writing $\mathbf {Fet}(X_i)$ for the category of finite étale schemes over $X_i$, the canonical functor

\[ \mathbf{Fet}(X_i)\to \mathcal{C}(X,\Phi), \quad T_i\mapsto \prod_{j\neq i}X_j \times_k T_i \]

induces a canonical homomorphism $\widetilde {p}_i\colon \pi _1^{{\unicode{x00E9}}{\text t}}(X,\Phi,\overline {x})\to \pi _1^{{\unicode{x00E9}}{\text t}}(X_i,\overline {x}_i)$, where $\overline {x}_i:=p_i(\overline {x})$. In view of the construction, the above morphism is compatible with the projection, that is, the following diagram commutes.

(3.3.1.3)

Proof. (i) We first show that for any finite group $\Gamma$, any tensor functor

(3.3.2.1)\begin{equation} \mathbf{Rep}_{\overline{\mathbb{Q}}_p}(\Gamma) \to \operatorname{\mathbf{\Phi-Isoc}}(X) \end{equation}

factors through $\mathbf {Rep}_{\overline {\mathbb {Q}}_p}^{\operatorname {cont}}(\pi _1(X,\Phi,\overline {x}))$. By Proposition 3.2.2, it suffices to show that objects in the essential image of the above functor are unit-root at each point of $|X|$. This can be shown by a similar argument to [Reference Drinfeld and KedlayaDK17, Proposition B.4.1].

Moreover, we deduce that the above functor factors through $\mathbf {Rep}_{\overline {\mathbb {Q}}_p}^{\operatorname {smooth}}(\pi _1(X,\Phi,\overline {x}))$ as in [Reference Drinfeld and KedlayaDK17, Proposition B.4.1].

Hence, the kernel of the canonical epimorphism $\pi _1^{\operatorname {\mathbf {\Phi -Isoc}}}(X)\twoheadrightarrow \pi _1(X,\Phi,\overline {x})$ is the neutral component of $\pi _1^{\operatorname {\mathbf {\Phi -Isoc}}}(X)$. We obtain the isomorphism $\pi _0(\pi _1^{\operatorname {\mathbf {\Phi -Isoc}}}(X))\xrightarrow {\sim } \pi _1(X,\Phi,\overline {x})$.

By a full faithfulness result à la Kedlaya (Proposition 3.3.3), $\pi _1^{\operatorname {\mathbf {\Phi -Isoc}^{{{\dagger}} }}}(X)\to \pi _1(X,\Phi,\overline {x})$ is an epimorphism. Then by [Reference Drinfeld and KedlayaDK17, Lemma B.7.4], we deduce the isomorphism $\pi _0(\pi _1^{\operatorname {\mathbf {\Phi -Isoc}^{{{\dagger}} }}}(X))\xrightarrow {\sim } \pi _1(X,\Phi,\overline {x})$.

(ii) Assertion (ii) follows from assertion (i), Theorem 2.2.4, and the commutative diagram (3.3.1.3).

Proof. (i) Consider the following diagram.

The vertical arrows are faithful and $\iota _F$ is fully faithful [Reference KedlayaKed04b]. Hence, it remains to show the fullness of $\iota _{\Phi }$.

Given two objects $(\mathscr {E},\varphi _i),(\mathscr {E}',\varphi '_i)$ of $\operatorname {\mathbf {\Phi -Isoc}^{{{\dagger}} }}(X)$ and a morphism $f\colon \mathscr {E}\to \mathscr {E}'$ of $\operatorname {\mathbf {\Phi -Isoc}}(X)$, $f$ extends to a morphism $f^{{{\dagger}} }\colon \mathscr {E}\to \mathscr {E}'$ of overconvergent $F$-isocrystals by the full faithfulness of $\iota _F$. Since the canonical functor $\operatorname {\mathbf {Isoc}^{{{\dagger}} }}(X)\to \operatorname {\mathbf {Isoc}}(X)$ is faithful, we deduce that $f^{{{\dagger}} }$ is compatible with partial Frobenius structures.

(ii) By assertion (i), it suffices to prove the assertion for convergent $\Phi$-isocrystals. By [Reference KedlayaKed22a, Theorem 5.3] and a similar argument as in assertion (i), it suffices to show its fullness.

Given two objects $(\mathscr {E},\varphi _i),(\mathscr {E}',\varphi '_i)$ of $\operatorname {\mathbf {\Phi -Isoc}}(X/K)$ and a morphism $f\colon \mathscr {E}\to \mathscr {E}'$ of $\operatorname {\mathbf {\Phi -Isoc}}(U/K)$. Then $f$ extends to a morphism $g\colon \mathscr {E}\to \mathscr {E}'$ of convergent $F$-isocrystals on $X$. The compatibility between $f$ and partial Frobenius structures follows from [Reference KedlayaKed22a, Theorem 5.3].

Proof. (i) Let $(\mathbb {L},\phi _i)$ be the data associated to $\mathscr {E}$ in Proposition 3.1.8. Let $x\in |X|$ be a closed point of degree $a$ and $(V=\mathbb {L}_{\overline {x}},\rho )$ the associated $\pi _1(X,\overline {x})$-representation. Since $\phi _{i,\overline {x}}$ commute with each other, $V$ admits a decomposition

\[ V\simeq \bigoplus_{(d_1,\ldots,d_n)} V_{d_1,\ldots,d_n}, \]

according to partial Frobenius slopes of $(\phi _{1,\overline {x}},\ldots,\phi _{n,\overline {x}})$. It remains to show that each component is invariant under the $\pi _1(X,\overline {x})$-action. Note that each $\phi _{i,\overline {x}}$ preserves $V_{d_1,\ldots,d_n}$, as then does their composition $\phi _{\mathbb {L},\overline {x}}$.

Since $F_X^a$ induces an identity on étale fundamental groups $F_X^a\colon \pi _1(X,\overline {x})=\pi _1(X,\overline {x})$, $\phi _{\mathbb {L},\overline {x}}^a$ is an automorphism of the representation $(V,\rho )$. Since $\phi _{\mathbb {L},\overline {x}}^a$ commutes with the action of $\pi _1(X,\overline {x})$ on $V$, we deduce that the $\pi _1(X,\overline {x})$-action also preserves each component $V_{d_1,\ldots,d_n}$. Then assertion (i) follows.

(ii) After taking extension of scalars and twisting, we may assume $\mathscr {F}=\mathscr {E}_{d_1,\ldots,d_n}$ is unit-root. We prove the assertion by induction on $n$. Let $\overline {x}$ be a geometric point of $X$. By Proposition 3.2.2 and Theorem 3.3.2, $\mathscr {E}$ corresponds to a continuous $K$-linear representation $V$ of $\pi _1(X_1,\overline {x})\times \cdots \times \pi _1(X_n,\overline {x})$. Let $W$ be an irreducible sub-$\pi _1(X_n,\overline {x})$-representation of $V$. Then $W$ corresponds to a unit-root $F$-isocrystal $\mathscr {F}_n$ on $X_n$, and $\operatorname {Hom}_{\pi _1(X_n,\overline {x})}(W,V)$, viewed as a $\pi _1(X_1,\overline {x})\times \cdots \times \pi _1(X_{n-1},\overline {x})$-representation, corresponds to a unit-root $\Phi$-isocrystal $\mathscr {E}'$ on $X_1\times _k\cdots \times _k X_{n-1}$. We obtain a monomorphism $\mathscr {E}'\boxtimes \mathscr {F}_n\to \mathscr {F}$. Applying the induction hypothesis to $\mathscr {E}'$, then assertion (ii) follows.

Proof. (i) Let $i$ be an integer, $X':=\prod _{j\neq i} X_j$ and $k'$ a closed point of $X'$. We apply the exact and faithful functor (2.2.9.1) to the pullback along $\operatorname {Spec}(k')\to X'$ to obtain convergent $F$-isocrystals on $X_{i,k'}/K'$. Then the assertion follows from the corresponding assertion for convergent $F$-isocrystals.

(ii) We will deduce this vanishing result of $\operatorname {Ext}$ from the case of $F$-isocrystals, which is known (it follows from the existence of the usual slope filtration). We set $\mathscr {F}:=\mathscr {H}om(\mathscr {E}',\mathscr {E})$. By Lemma 3.4.2(ii), we may reduce to the case where $\mathscr {F}\simeq \mathscr {F}_1\boxtimes \cdots \boxtimes \mathscr {F}_n$, with $\mathscr {F}_i\in \operatorname {\mathbf {F-Isoc}}(X_i/K)$ after taking extension of scalars. Suppose $\mu _i'>\mu _i$ for some integer $i\in [1,n]$. We consider the extension group $\operatorname {Ext}^1_{F_i\text {-}\operatorname {\mathbf {Isoc}}(X/K)}(\mathscr {E}',\mathscr {E})$, which fits into the following diagram.

(3.4.3.1)\begin{equation} 0\to \operatorname{H}^0(X,\mathscr{F})/(\varphi_i-\operatorname{id})\to \operatorname{Ext}^1_{F_i\text{-}\operatorname{\mathbf{Isoc}}(X/K)}(\mathscr{E}',\mathscr{E}) \to \operatorname{H}^1(X,\mathscr{F})^{\varphi_i=\operatorname{id}}\to 0 \end{equation}

Since the slope of $\mathscr {F}_i$ is $>0$, we obtain the vanishing of $\operatorname {H}^0(X_i,\mathscr {F}_i)^{\varphi _i=\operatorname {id}},\operatorname {H}^0(X_i,\mathscr {F}_i)/(\varphi _i-\operatorname {id})$, and $\operatorname {H}^1(X_i,\mathscr {F}_i)^{\varphi _i=\operatorname {id}}$; note that the last case follows from the assertion (ii) for convergent $F$-isocrystals. We conclude that the middle term of (3.4.3.1) vanishes by the Künneth formula [Reference AbeAbe14, Lemma 4.5]. Then the assertion follows.

3.4.4. Proof of Theorem 3.4.1

By Theorem 3.1.10, each partial Newton polygon of $\mathscr {E}$ is also constant. Then we apply Lemma 3.4.2 to the successive quotients of the diagonal slope filtration; this yields a filtration in which each successive quotient has the property that every $i$th partial Frobenius slope has a fixed value, but these values may not occur in ascending order. However, using Lemma 3.4.3 we can reorder the successive quotients to enforce this condition.

Proof. We may assume $i=n$ and set $X':=\prod _{i=1}^{n-1}X_i$.

(i) We first prove the full faithfulness. By Proposition 3.3.3, we may assume each $X_i$ is affine and admits a smooth formal lifting $\mathfrak {X}_i$ over $\mathscr {O}_K$. Let $(\mathscr {E},\varphi )$ be an object of $\operatorname {\mathbf {F-Isoc}}(X_n)$. Since each $X_i$ is smooth and geometrically connected, the canonical morphism

\[ \operatorname{H}^0(X_n,\mathscr{E})\xrightarrow{\sim} \operatorname{H}^0(X,p_n^*(\mathscr{E})) \]

is an isomorphism by the Künneth formula for the sheaf of differential operators [Reference AbeAbe14, Lemma 4.5] and for quasicoherent modules. Then we deduce a canonical isomorphism:

\[ \operatorname{H}^0(X_n,\mathscr{E})^{\varphi=\operatorname{id}}\to \operatorname{H}^0(X,p_n^*(\mathscr{E}))^{\Phi=\operatorname{id}}. \]

The full faithfulness follows from the above isomorphism applied to internal homomorphisms.

(ii) Next we treat the second assertion. Let $\mathscr {E}$ be an object of $\operatorname {\mathbf {F-Isoc}}(X_n)$ and $p_n^*(\mathscr {E})\to \mathscr {F}$ a surjection in $\operatorname {\mathbf {\Phi -Isoc}}(X)$. By enlarging $k$ (2.2.2.1), we may assume there exists a $k$-point $x$ of $X'$. We will show that the surjection $\mathscr {E}\to \mathscr {G}:=(x\times _k \operatorname {id}_{X_n})^*(\mathscr {F})$ over $X_n$ is isomorphic to $p_n^*(\mathscr {E})\to \mathscr {F}$ after pullback via $p_n^*$.

Let $U$ be a dense open subset of $X$ such that the diagonal Newton polygon of $\mathscr {F}$ is constant, which is preserved by partial Frobenius morphisms. By Drinfeld's lemma for open immersions [Reference LauLau04, Lemma 9.2.1], [Reference KedlayaKed19, Theorem 4.3.6], and Proposition 3.3.3, we may assume that the diagonal Newton polygons of $p_n^*(\mathscr {E})$ and $\mathscr {F}$ are constant on $|X|$ after shrinking $X_i$. Then the set of each partial Newton polygon of $\mathscr {F}$ is also constant on $|X|$. To show the claim, we may reduce to the case where the Frobenius slopes of $\mathscr {E}$ are constant on $X_n$ by applying Theorem 3.1.10 to $p_n^*(\mathscr {E})\to \mathscr {F}$. Moreover, we may assume $\mathscr {E}$ is unit-root by twisting. Then $\mathscr {F}$ is also a unit-root $\Phi$-isocrystal. By Proposition 3.2.2 and Theorem 3.3.2, for a geometric point $\overline {y}$ of $X$, the action of $\prod _{i=1}^{n-1}\pi _1(X_i,\overline {y})$ on $p_n^*(\mathscr {E})_{\overline {y}}$ is trivial. Then the same holds for $\mathscr {F}$ and the assertion follows.

Proof. (i) Let $\mathscr {E}$ be an object of $\operatorname {\mathbf {\Phi -Isoc}}(X)$. By Proposition 3.5.2, the collection of subobjects of $\mathscr {E}$ which belong to the essential image of $p^*_i$ has a maximal element $\mathscr {F}$. Then we define $p_{i,*}(\mathscr {E})$ to be the object of $\operatorname {\mathbf {F-Isoc}}(X_i)$ of which $\mathscr {F}$ is a pullback.

Let $f:\mathscr {E}_1\to \mathscr {E}_2$ be a morphism of $\operatorname {\mathbf {\Phi -Isoc}}(X)$. Let $\mathscr {G}$ be the image of $p_i^*(p_{i,*}(\mathscr {E}_1))$ in $\mathscr {E}_2$. By Proposition 3.5.2, $\mathscr {G}$ belongs to the essential image of $p_i^*$ and the functoriality of $p_{i,*}$ follows. The second assertion is clear.

(ii) By Lemma 3.3.3, we may assume that the diagonal Newton polygon of $\mathscr {E}$ is constant on $|X|$. By Theorem 3.1.10, we may moreover assume that $\mathscr {E}$ is diagonally unit-root after twisting the $i$th partial Frobenius structure. Then $p_{i,*}(\mathscr {E})$ is unit-root and $p_i^*(p_{i,*}(\mathscr {E}))$ lies in the unit-root component $\mathscr {E}_{(0,\ldots,0)}$ of $\mathscr {E}$ (cf. Lemma 3.4.2). Moreover, the unit-root $F$-isocrystal $p_{i,*}(\mathscr {E})$ over $X_i$ can be identified with the $\prod _{j\neq i}\pi _1(X_j,\overline {x})$-invariant part of the $\pi _1(X,\Phi,\overline {x})$-representation associated to $\mathscr {E}_{(0,\ldots,0)}$ via Proposition 3.2.2. The base change property follows from this identification in view of Theorem 3.3.2.

3.5.4. Proof of Theorem 3.5.1

The proof is similar to that of § 2.2.11(a). We keep the notation of § 2.2.11. It suffices to show the exactness of the following sequence:

\[ \pi_1^{\operatorname{\mathbf{\Phi-Isoc}}}(X',\Phi') \xrightarrow{u^{\circ}} \pi_1^{\operatorname{\mathbf{\Phi-Isoc}}}(X,\Phi) \xrightarrow{p_1^{\circ}} \pi_1^{\operatorname{\mathbf{F-Isoc}}}(X_1). \]

We verify the exactness using Theorem 2.1.4.

Condition (i) follows from Proposition 3.5.2. Conditions (ii) and (iii)(c) can be verified in the same way as in § 2.2.11(a). Finally, as in § 2.2.11, we verify conditions (iii)(a) and (b) using functor $p_{i,*}$ and its base change property Proposition 3.5.3. This finishes the proof.

Proof. It suffices to show the assertion about the quotient. By induction, we can reduce to the case where $n=2$. We set $G:=\pi _1^{\mathbf {Isoc}^{{{\dagger}} {{\dagger}} }}(X,\Phi ), G_i:=\pi _1^{\operatorname {\mathbf {Isoc}^{{{\dagger}} }}}(X_i)$. Let $V$ be a representation of $G\simeq G_1\times G_2$ corresponding to $\mathscr {E}$. We have morphisms of $G_1\times G_2$-representations:

\[ (V\otimes V^{\vee})^{G_2}\otimes V \hookrightarrow V\otimes V^{\vee}\otimes V \xrightarrow{\operatorname{id}_V\otimes \operatorname{ev}} V, \]

where $G_1$ only acts at the first component of $V\otimes V^{\vee }\otimes V$ and $G_2$ acts diagonally. By composing with $V\xrightarrow {\operatorname {coev}\otimes \operatorname {id}} V\otimes V^{\vee }\otimes V$, we see that the above composition is surjective. This finishes the proof.

Proof. After twisting, we may assume $\mathscr {E}$ is unit-root. Let $\rho$ be the character of $\pi _1(X,\Phi,\overline {x})$ associated to $\mathscr {E}$ (Proposition 3.2.2) and $\rho _i$ the restriction of $\rho$ to $\pi _1(X_i,\overline {x})$ (Theorem 3.3.2). We may shrink each $X_i$ and, in particular, we may assume that $X_i$ admits a smooth compactification $X_i\to \overline {X}_i$ such that the complement is a simple normal crossing divisor. Using [Reference ShihoShi11b, Theorem 4.3], [Reference Katz and LangKL81, Theorem 2], and the same argument of [Reference CrewCre87, Corollary 4.13], one can show that some power $\mathscr {E}^{\otimes N}$ is geometrically trivial, i.e. its underlying overconvergent isocrystal is trivial. Then $\rho ^{\otimes N}$ factors through the quotient $\pi _1(X,\Phi,\overline {x})\to \widehat {\mathbb {Z}}^n$ and each $\rho _i^{\otimes N}$ is also geometrically trivial. Hence, the unit-root convergent $F$-isocrystal $\mathscr {E}_i$ over $X_i$ associated to $\rho _i$ is overconvergent. We may take suitable twists $\chi _i$ such that each $\mathscr {E}_i(\chi _i)$ has finite order [Reference AbeAbe18a, Lemma 6.1]. Then the assertion follows.

4.1.4 Let $\mathscr {E}$ be an object of $\operatorname {\mathbf {\Phi -Isoc}^{{{\dagger}} }}(X)$. We denote by $G_{\operatorname {geo}}(\mathscr {E})$ (respectively, $G_{\operatorname {arith}}(\mathscr {E})$) its geometric (respectively, arithmetic) monodromy group over $\overline {\mathbb {Q}}_p$, which is defined by the Tannakian full subcategory of $\operatorname {\mathbf {\Phi -Isoc}^{{{\dagger}} }}(X)$ (respectively, $\operatorname {\mathbf {Isoc}^{{{\dagger}} }}(X)$) whose objects are subquotients of $\mathscr {E}^{\otimes m}\otimes \mathscr {E}^{\vee, \otimes n}$ for some $m,n$. Recall that the radical of $G_{\operatorname {geo}}(\mathscr {E})$ is unipotent [Reference CrewCre92, Theorem 4.9].

Given a field $E$, by a semisimple group (respectively, reductive group) over $E$, we mean an algebraic group of finite type over $E$ whose neutral component is semisimple (respectively, reductive).