MILNOR K-THEORY, F-ISOCRYSTALS AND SYNTOMIC REGULATORS

2.1 The category of Filtered F-isocrystals

Let $S={\operatorname {Spec}}(B)$ be an affine smooth variety over V. Let $\sigma \colon B^{\dagger }\to B^{\dagger }$ be a p-th Frobenius compatible with $F_V$ on V, which means that $\sigma $ is $F_V$ -linear and satisfies $\sigma (x)\equiv x^p$ mod $pB$ . The induced endomorphism $\sigma \otimes _{{\mathbb {Z}}}{\mathbb {Q}}\colon B_K^{\dagger }\to B_K^{\dagger }$ is also denoted by $\sigma $ . We define the category ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma )$ (which we call the category of filtered F-isocrystals on S) as follows.

Remark 2.2. (1) The category ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma )$ can also be described by using simpler categories as follows. Let ${{\mathrm {Fil}\text {-}\mathrm {MIC}}}(S_K)$ denote the category of filtered $S_K$ -modules with integrable connection – that is, the category of data $(M_{{\mathrm {d\hspace {-0.2pt}R}}},\nabla ,{\mathrm {Fil}}^{\bullet })$ with $M_{{{\mathrm {d\hspace {-0.2pt}R}}}}$ a coherent $B_K$ -module, $\nabla $ an integrable connection on $M_{{{\mathrm {d\hspace {-0.2pt}R}}}}$ and ${\mathrm {Fil}}^{\bullet }$ a finite descending filtration on $M_{{{\mathrm {d\hspace {-0.2pt}R}}}}$ of locally free $B_K$ -module that satisfies $\nabla ({\mathrm {Fil}}^i)\subset \Omega ^1_{B_K}\otimes {\mathrm {Fil}}^{i-1}$ . Let ${\mathop {\mathrm {MIC}}\nolimits }(B^{\dagger }_K)$ denote the category of coherent $B_K^{\dagger }$ -modules with integrable connections $(M_{\mathrm {rig}},\nabla )$ on $B^{\dagger }_K$ , and let ${{F\text {-}\mathrm {MIC}}}(B^{\dagger }_K,\sigma )$ denote the category of coherent $B^{\dagger }_K$ -modules with integrable connections equipped with ( $\sigma $ -linear) Frobenius isomorphisms $(M_{\mathrm {rig}},\nabla ,\Phi )$ . Then, ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma )$ is identified with the fiber product

$$\begin{align*}{{F\text{-}\mathrm{MIC}}}(B_K^{\dagger},\sigma)\times_{{\mathop{\mathrm{MIC}}\nolimits}(B_K^{\dagger})}{{\mathrm{Fil}\text{-}\mathrm{MIC}}}(S_K). \end{align*}$$

(2) Let ${{F\text {-}\mathrm {Isoc}^{\dagger }}}(B_k)$ denote the category of overconvergent F-isocrystals on $S_k$ . Then, there is the equivalence of categories ([Reference StumLS, Theorem 8.3.10])

$$\begin{align*}{{F\text{-}\mathrm{Isoc}^{\dagger}}}(B_k){\overset{\cong}{{\longrightarrow}}}{{F\text{-}\mathrm{MIC}}}(B_K^{\dagger},\sigma). \end{align*}$$

Therefore, by combining with the description in (1), we see that our category ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma )$ does not depend on $\sigma $ , which means that there is the natural equivalence ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma )\cong {{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma ')$ (see also Lemma 5.3 in the Appendix). By virtue of this fact, we often drop “ $\sigma $ ” in the notation.

For two objects H, $H'$ in the category ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma )$ , we have a direct sum $H\oplus H'$ and the tensor product $H\otimes H'$ in a customary manner. The unit object for the tensor product, denoted by B or $\mathscr {O}_S$ , is $(B_K, B_K^{\dagger }, c, \sigma _B, d, {\mathrm {Fil}}^{\bullet })$ , where c is the natural isomorphism, d is the usual differential and ${\mathrm {Fil}}^{\bullet }$ is defined by ${\mathrm {Fil}}^0B_K=B_K$ and ${\mathrm {Fil}}^1B_K=0$ . The category ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma )$ forms a tensor category with this tensor product and the unit object B or ${\mathscr {O}}_S$ .

The unit object can also be described as $B=B(0)$ or ${\mathscr {O}}_S={\mathscr {O}}_S(0)$ by using the following notion of Tate object.

Now, we discuss the Yoneda extension groups in the category ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S)$ . A sequence

$$\begin{align*}H_1{\longrightarrow} H_2{\longrightarrow} H_3 \end{align*}$$

in ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S)$ (or in ${{\mathrm {Fil}\text {-}\mathrm {MIC}}}(S_K)$ ) is called exact if

$$\begin{align*}{\mathrm{Fil}}^i H_{1,{{\mathrm{d\hspace{-0.2pt}R}}}}{\longrightarrow} {\mathrm{Fil}}^i H_{2,{{\mathrm{d\hspace{-0.2pt}R}}}}{\longrightarrow} {\mathrm{Fil}}^i H_{3,{{\mathrm{d\hspace{-0.2pt}R}}}} \end{align*}$$

are exact for all i. Then, the category ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S)$ forms an exact category which has kernel and cokernel objects for any morphisms. Thus, the Yoneda extension groups

$$\begin{align*}{\mathrm{Ext}}^{\bullet}(H,H')={\mathrm{Ext}}^{\bullet}_{{{\mathrm{Fil}\text{-}F\text{-}\mathrm{MIC}}}(S)}(H,H') \end{align*}$$

in ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S)$ are defined in the canonical way (or one can further define the derived category of ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S)$ [Reference Beilinson, Bernstein and DeligneBBDG, 1.1]). An element of ${\mathrm {Ext}}^j(H,H')$ is represented by an exact sequence

$$\begin{align*}0{\longrightarrow} H'{\longrightarrow} M_1{\longrightarrow}\cdots{\longrightarrow} M_j{\longrightarrow} H{\longrightarrow} 0 \end{align*}$$

and is subject to the equivalence relation generated by commutative diagrams

Note that ${\mathrm {Ext}}^{\bullet }(H,H')$ is uniquely divisible (i.e. a ${\mathbb {Q}}$ -module) as the multiplication by $m\in {\mathbb {Z}}_{>0}$ on H or $H'$ is bijective.

Next, we discuss the functoriality of the category ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S)$ with respect to S. Let $S'={\operatorname {Spec}}(B')$ be another affine smooth variety with a p-th Frobenius $\sigma '\colon B^{\prime \dagger }\to B^{\prime \dagger }$ compatible with $F_V$ and $i\colon S'\to S$ a morphism of V-schemes. Then, i induces a pullback functor

(2.1) $$ \begin{align} i^{\ast}\colon {\mathrm{Fil}}\text{-} F\text{-}{\mathop{\mathrm{MIC}}\nolimits}(S)\to {\mathrm{Fil}}\text{-} F \text{-} {\mathop{\mathrm{MIC}}\nolimits}(S') \end{align} $$

in a natural way. In fact, if $H=(H_{{{\mathrm {d\hspace {-0.2pt}R}}}}, H_{{\mathrm {rig}}}, c, \Phi , \nabla , {\mathrm {Fil}}^{\bullet })$ is an object of ${\mathrm {Fil}}$ -F- ${\mathop {\mathrm {MIC}}\nolimits }(S)$ , then we define $i^{\ast }H=(H_{{{\mathrm {d\hspace {-0.2pt}R}}}}\otimes _{B_K}B^{\prime }_K, H_{{\mathrm {rig}}}\otimes _{B^{\dagger }_K}B^{\prime \dagger }_K, c\otimes _{B^{\dagger }_K}B^{\prime \dagger }_K, \Phi ', \nabla ', {\mathrm {Fil}}^{\prime \bullet })$ , where $\nabla '$ and ${\mathrm {Fil}}^{\prime \bullet }$ are natural pullbacks of $\nabla $ and ${\mathrm {Fil}}^{\bullet }$ , respectively, and where $\Phi '$ is the natural Frobenius structure obtained as follows. We may regard $(H_{{\mathrm {rig}}}, \nabla _{{\mathrm {rig}}}, \Phi )$ as an overconvergent F-isocrystal on S via the equivalence $F\text {-}{\mathop {\mathrm {MIC}}\nolimits }(B_K^{\dagger })\simeq F\text {-}\mathrm {Isoc}^{\dagger }(B_k)$ [Reference StumLS, Theorem 8.3.10]. Then, its pullback along $i_k\colon S^{\prime }_k\to S_k$ is an overconvergent F-isocrystal on $S^{\prime }_k$ , which is again identified with an object of $F\text {-}{\mathop {\mathrm {MIC}}\nolimits }(B^{\prime \dagger }_K)$ . Thus, it is of the form $(H^{\prime }_{{\mathrm {rig}}}, \nabla ^{\prime }_{{\mathrm {rig}}}, \Phi ')$ , and $H^{\prime }_{\mathrm {rig}}$ is naturally isomorphic to $H_{{\mathrm {rig}}}\otimes _{B^{\dagger }_K}B^{\prime \dagger }_K$ ([Reference StumLS, Prop 8.1.15]). Now, $\Phi '$ gives the desired Frobenius structure.

2.2 The complex ${\mathscr {S}}(M)$

In this subsection, we introduce a complex ${\mathscr {S}}(M)$ for each object M in ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma )$ which is, in the case where $M={\mathscr {O}}_S(r)$ , close to the syntomic complex ${\mathscr {S}}_n(R)_{S,\sigma }$ of Fontaine–Messing.

Before the definition, we prepare a morphism attached to each object of ${{\mathrm {Fil}\text {-}\mathrm {MIC}}}(S_K)$ . Let $H=(H_{{\mathrm {d\hspace {-0.2pt}R}}},\nabla ,{\mathrm {Fil}}^{\bullet })$ be an object of ${{\mathrm {Fil}\text {-}\mathrm {MIC}}}(S_K)$ . Let $\Omega ^{\bullet }_{B_K}{\otimes }{\mathrm {Fil}}^{i-\bullet }H_{{\mathrm {d\hspace {-0.2pt}R}}}$ denote the de Rham complex

$$\begin{align*}{\mathrm{Fil}}^iH_{{\mathrm{d\hspace{-0.2pt}R}}}\to\Omega^1_{B_K}{\otimes} {\mathrm{Fil}}^{i-1}H_{{\mathrm{d\hspace{-0.2pt}R}}} \to\cdots \to\Omega^n_{B_K}{\otimes} {\mathrm{Fil}}^{i-n}H_{{\mathrm{d\hspace{-0.2pt}R}}}\to \cdots, \end{align*}$$

where ${\otimes }$ denotes ${\otimes }_{B_K}$ and the differentials are given by

(2.2) $$ \begin{align} \omega{\otimes} x\longmapsto d\omega^j{\otimes} x+(-1)^j\omega\wedge\nabla(x), \quad (\omega{\otimes} x\in \Omega^j_{B_K}{\otimes} H_{{\mathrm{d\hspace{-0.2pt}R}}}). \end{align} $$

Now, we define a natural map

(2.3) $$ \begin{align} {\mathrm{Ext}}^i_{{{\mathrm{Fil}\text{-}\mathrm{MIC}}}(S_K)}({\mathscr{O}}_{S_K},H){\longrightarrow} H^i(\Omega^{\bullet}_{B_K}{\otimes}{\mathrm{Fil}}^{-\bullet}H_{{\mathrm{d\hspace{-0.2pt}R}}}) \end{align} $$

in the following way. Let

(2.4) $$ \begin{align} 0{\longrightarrow} H{\longrightarrow} M_i{\longrightarrow} M_{i-1}{\longrightarrow}\cdots{\longrightarrow} M_0{\longrightarrow} {\mathscr{O}}_{S_K}{\longrightarrow}0 \end{align} $$

be an exact sequence in ${{\mathrm {Fil}\text {-}\mathrm {MIC}}}(S_K)$ . This induces an exact sequence

$$\begin{align*}0{\longrightarrow} \Omega^{\bullet}_{B_K}{\otimes}{\mathrm{Fil}}^{-\bullet}H{\longrightarrow} \Omega^{\bullet}_{B_K}{\otimes}{\mathrm{Fil}}^{-\bullet}M_i{\longrightarrow} \cdots{\longrightarrow} \Omega^{\bullet}_{B_K}{\otimes}{\mathrm{Fil}}^{-\bullet}M_0{\longrightarrow} \Omega^{\bullet}_{B_K}{\longrightarrow}0 \end{align*}$$

of complexes, and hence a connecting homomorphism $\delta \colon H^0(\Omega ^{\bullet }_{B_K})\to H^i(\Omega ^{\bullet }_{B_K}{\otimes }{\mathrm {Fil}}^{-\bullet }H)$ . Then, the map (2.3) is defined by sending the sequence (2.4) to $\delta (1)$ .

By the forgetful functor ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma )\to {{\mathrm {Fil}\text {-}\mathrm {MIC}}}(S)$ , the morphism (2.3) clearly induces a canonical morphism

(2.5) $$ \begin{align} {\mathrm{Ext}}^i_{{{\mathrm{Fil}\text{-}F\text{-}\mathrm{MIC}}}(S,\sigma)}({\mathscr{O}}_{S},H){\longrightarrow} H^i(\Omega^{\bullet}_{B_K}{\otimes}{\mathrm{Fil}}^{-\bullet}H_{{\mathrm{d\hspace{-0.2pt}R}}}). \end{align} $$

Let $F\text {-mod}={{F\text {-}\mathrm {MIC}}}({\operatorname {Spec}} K)$ , namely the category of finite-dimensional K-modules endowed with $F_V$ -linear bijective homomorphisms. Then, we have a functor

$$\begin{align*}{{F\text{-}\mathrm{MIC}}}(B_K^{\dagger},\sigma){\longrightarrow} D^b(F\text{-mod}),\quad (M_{\mathrm{rig}},\Phi)\longmapsto (\Omega^{\bullet}_{B^{\dagger}_K}{\otimes} M_{\mathrm{rig}},\Phi) \end{align*}$$

to the derived category of complexes in $F\text {-mod}$ , where $\Phi $ in the target is defined to be $\sigma {\otimes }\Phi $ (we use the same notation because we always extend the Frobenius action on the de Rham complex $\Omega ^{\bullet }_{B^{\dagger }_K}{\otimes } M_{\mathrm {rig}}$ by this rule). Here, we note that, by Beilinson’s lemma [Reference BeilinsonBei, Lemma 1.4] (as in [Reference BannaiBan3, Theorem 3.2], [Reference Déglise and NiziolDN, Theorem 2.17]), $D^b(F\text {-mod})$ is equivalent to the full subcategory of $D^b(F\text {-mod}')$ , where $F\text {-mod}'$ denotes the category of (possibly infinite-dimensional) K-modules with $F_V$ -linear (not necessarily bijective) endomorphism, consisting of complexes whose cohomology groups belong to F-mod. Since $\Omega ^{\bullet }_{B_K^{\dagger }}\otimes M_{{\mathrm {rig}}}$ belongs to the latter category, we are regarding it as an object of the former.

We also note that the above functor does not depend on $\sigma $ . Indeed, the composition

$$\begin{align*}F\text{-}\mathrm{Isoc}^{\dagger}(B_k){\overset{\cong}{{\longrightarrow}}}{{F\text{-}\mathrm{MIC}}}(B_K^{\dagger},\sigma) {\longrightarrow} D^b(F\text{-mod}) \end{align*}$$

is the functor $(E,\Phi )\mapsto (R{\varGamma }_{\mathrm {rig}}(S_k,E),\Phi _{\mathrm {rig}})$ , where $\Phi _{\mathrm {rig}}$ denotes the Frobenius action on the rigid cohomology ([Reference StumLS, Proposition 8.3.12]), and this does not depend on $\sigma $ .

Definition 2.4. For an object $M\in {{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma )$ , we define ${\mathscr {S}}(M)$ to be the mapping fiber of the morphism

$$\begin{align*}1-\Phi:\Omega^{\bullet}_{B_K}{\otimes} {\mathrm{Fil}}^{-\bullet}M_{{\mathrm{d\hspace{-0.2pt}R}}}{\longrightarrow} \Omega^{\bullet}_{B^{\dagger}_K}{\otimes} M_{\mathrm{rig}}, \end{align*}$$

where $1$ denotes the inclusion $ \Omega ^{\bullet }_{B_K}{\otimes } {\mathrm {Fil}}^{-\bullet }M_{{\mathrm {d\hspace {-0.2pt}R}}}{\hookrightarrow } \Omega ^{\bullet }_{B^{\dagger }_K}{\otimes } M_{\mathrm {rig}}$ via the comparison and $\Phi $ is the composition of it with $\Phi $ on $\Omega ^{\bullet }_{B^{\dagger }_K}{\otimes } M_{\mathrm {rig}}$ .

Note that, in a more down-to-earth manner, each term of ${\mathscr {S}}(M)$ is given by

(2.6) $$ \begin{align} {\mathscr{S}}(M)^i= \Omega^i_{B_K}{\otimes} {\mathrm{Fil}}^{-i}M_{{\mathrm{d\hspace{-0.2pt}R}}}{\oplus} \Omega^{i-1}_{B^{\dagger}_K}{\otimes} M_{\mathrm{rig}} \end{align} $$

and the differential ${\mathscr {S}}(M)^i\to {\mathscr {S}}(M)^{i+1}$ is given by

$$\begin{align*}(\omega,\xi)\longmapsto(d_M\omega, (1-\Phi)\omega-d_M\xi),\quad (\omega\in \Omega^i_{B_K}{\otimes} {\mathrm{Fil}}^{-i}M_{{\mathrm{d\hspace{-0.2pt}R}}},\,\xi\in \Omega^{i-1}_{B^{\dagger}_K}{\otimes} M_{\mathrm{rig}}), \end{align*}$$

where $d_M$ is the differential (2.2).

An exact sequence

$$\begin{align*}0{\longrightarrow} H{\longrightarrow} M_i{\longrightarrow} M_{i-1}{\longrightarrow}\cdots{\longrightarrow} M_0{\longrightarrow} {\mathscr{O}}_{S}{\longrightarrow}0 \end{align*}$$

in ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma )$ gives rise to an exact sequence

$$\begin{align*}0{\longrightarrow} {\mathscr{S}}(H){\longrightarrow} {\mathscr{S}}(M_i){\longrightarrow} {\mathscr{S}}(M_{i-1}){\longrightarrow}\cdots{\longrightarrow} {\mathscr{S}}(M_0){\longrightarrow} {\mathscr{S}}({\mathscr{O}}_{S}){\longrightarrow}0 \end{align*}$$

of complexes of ${\mathbb {Q}}_p$ -modules. Let $\delta \colon {\mathbb {Q}}_p\cong H^0({\mathscr {S}}({\mathscr {O}}_S))\to H^i({\mathscr {S}}(H))$ be the connecting homomorphism, where the first isomorphism is given as follows:

$$\begin{align*}H^0({\mathscr{S}}({\mathscr{O}}_S))=\{(f,0)\in B_K{\oplus}\{0\}={\mathscr{S}}({\mathscr{O}}_S)^0\mid df=0,\, \sigma(f)=f\}\overset{\sim}{\leftarrow}{\mathbb{Q}}_p. \end{align*}$$

We define a homomorphism

(2.7) $$ \begin{align} u\colon {\mathrm{Ext}}^i_{{{\mathrm{Fil}\text{-}F\text{-}\mathrm{MIC}}}(S,\sigma)}({\mathscr{O}}_{S},H){\longrightarrow} H^i({\mathscr{S}}(H)) \end{align} $$

by associating $\delta (1)$ to the above extension. The composition of u with the natural map $H^i({\mathscr {S}}(H))\to H^i(\Omega ^{\bullet }_{B_K}{\otimes }{\mathrm {Fil}}^{-\bullet }H_{{\mathrm {d\hspace {-0.2pt}R}}})$ agrees with (2.5).

The complex ${\mathscr {S}}({\mathscr {O}}_S(r))$ is close to the syntomic complex of Fontaine-Messing. More precisely, let $S_n:=S\times _W{\operatorname {Spec}} W/p^nW$ and $B_n:=B/p^nB$ . The syntomic complex ${\mathscr {S}}_n(r)_{S,\sigma }$ is the mapping fiber of the morphism

$$\begin{align*}1-p^{-r}\sigma\colon \Omega_{B_n}^{\bullet\geq r}{\longrightarrow} \Omega^{\bullet}_{B_n} \end{align*}$$

of complexes where we note that $p^{-r}\sigma $ is well-defined (see [Reference KatoKa1, p. 410–411]). The i-th term of ${\mathscr {S}}_n(r)_{S,\sigma }$ is

(2.8) $$ \begin{align} {\mathscr{S}}_n(r)^i_{S,\sigma}=\begin{cases} \{0\}{\oplus}\Omega^{i-1}_{B_n}&\text{if } i< r,\\ \Omega_{B_n}^i{\oplus}\Omega^{i-1}_{B_n}&\text{if } i\geq r, \end{cases} \end{align} $$

and the differential is given by

$$\begin{align*}(\omega,\xi)\longmapsto(d\omega, (1-p^{-r}\sigma)\omega-d\xi). \end{align*}$$

Hence, there is a natural map

(2.9) $$ \begin{align} H^i({\mathscr{S}}({\mathscr{O}}_S(r))){\longrightarrow} {\mathbb{Q}}{\otimes}\varprojlim_n H^i_{\mathrm{zar}}(S_1,{\mathscr{S}}_n(r)_{S,\sigma})=:H^i_{\mathrm{syn}}(S,{\mathbb{Q}}_p(r)). \end{align} $$

Let

(2.10) $$ \begin{align} {\mathrm{Ext}}^i_{{{\mathrm{Fil}\text{-}F\text{-}\mathrm{MIC}}}(S,\sigma)}({\mathscr{O}}_{S},{\mathscr{O}}_S(r)){\longrightarrow} H^i_{\mathrm{syn}}(S,{\mathbb{Q}}_p(r)) \end{align} $$

be the composition morphism. Apparently, both sides of (2.10) depend on $\sigma $ . However, if we replace $\sigma $ with $\sigma '$ , there is the natural transformation between ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma )$ and ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma ')$ , thanks to the theory of F-isocrystals, and there is also a natural transformation on the syntomic cohomology. The map (2.10) is compatible under these transformations. In this sense, (2.10) does not depend on the Frobenius $\sigma $ .

Proof. Let

$$\begin{align*}0{\longrightarrow} H{\longrightarrow} M{\longrightarrow}{\mathscr{O}}_S{\longrightarrow}0 \end{align*}$$

be an exact sequence in ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma )$ . Since ${\mathrm {Fil}}^0H_{{\mathrm {d\hspace {-0.2pt}R}}}=0$ , there is the unique lifting $e\in {\mathrm {Fil}}^0M_{\mathrm {rig}}$ of $1\in \mathscr {O}_S$ . Then,

(2.11) $$ \begin{align} u(M)=(\nabla(e),(1-\Phi)e)\in H^1({\mathscr{S}}(H))\subset \Omega^1_{B^{\dagger}_K}{\otimes} {\mathrm{Fil}}^{-1}H_{\mathrm{rig}}{\oplus} H_{\mathrm{rig}} \end{align} $$

by definition of u. If $u(M)=0$ , then the datum $(B_Ke,B^{\dagger }_Ke, c,\Phi ,\nabla ,{\mathrm {Fil}}^{\bullet })$ forms a subobject of M which is isomorphic to the unit object ${\mathscr {O}}_S$ . This gives a splitting of the above exact sequence. The latter assertion is immediate as

$$\begin{align*}H^1_{\mathrm{syn}}(S,{\mathbb{Q}}_p(1))\subset {\mathbb{Q}}{\otimes}\varprojlim_n(\Omega^1_{B_n}{\oplus} B_n),\quad H^1_{\mathrm{syn}}(S,{\mathbb{Q}}_p(r))\subset {\mathbb{Q}}{\otimes}\varprojlim_n(\{0\}{\oplus} B_n),\,(r\geq2) \end{align*}$$

by (2.8) and

$$\begin{align*}H^1({\mathscr{S}}({\mathscr{O}}_S(1)))\subset \Omega^1_{B^{\dagger}_K}{\oplus} B^{\dagger}_K,\quad H^1({\mathscr{S}}({\mathscr{O}}_S(r)))\subset \{0\}{\oplus} B^{\dagger}_K,\,(r\geq2) \end{align*}$$

by (2.6).

2.3 Log objects

In this subsection, we introduce the “log object” in ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S)$ concerning a p-adic logarithmic function. In the next subsection, it will be generalized to a notion of “polylog object”.

For $f\in B^{\times }$ , let

$$\begin{align*}\log^{(\sigma)}(f):=p^{-1}\log\left(\frac{f^p}{f^{\sigma}}\right) =-p^{-1}\sum_{n=1}^{\infty}\frac{(1-f^p/f^{\sigma})^n}{n}\in B^{\dagger}. \end{align*}$$

An elementary computation yields $\log ^{(\sigma )}(f)+\log ^{(\sigma )}(g)=\log ^{(\sigma )}(fg)$ for $f,g\in B^{\times }$ .

This is fit into the exact sequence

(2.12) $$ \begin{align} 0{\longrightarrow} {\mathscr{O}}_S(1){\overset{\epsilon}{{\longrightarrow}}} {\mathop{\mathscr{L}\!\!\mathit{o}\mathit{g}}\nolimits}(f){\overset{\pi}{{\longrightarrow}}} {\mathscr{O}}_S{\longrightarrow} 0 \end{align} $$

in ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S)$ , where the two arrows are defined by $\epsilon (1)=e_{-2}$ and $\pi (e_0)=1$ . This defines a class in ${\mathrm {Ext}}^1_{{{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma )}({\mathscr {O}}_S,{\mathscr {O}}_S(1))$ , which we write by $[f]_S$ . It is easy to see that $f\mapsto [f]_S$ is additive. We call the group homomorphism

(2.13) $$ \begin{align} [-]_S:{\mathscr{O}}(S)^{\times}{\longrightarrow} {\mathrm{Ext}}^1_{{{\mathrm{Fil}\text{-}F\text{-}\mathrm{MIC}}}(S,\sigma)}({\mathscr{O}}_S,{\mathscr{O}}_S(1)) \end{align} $$

the symbol map.

Lemma 2.7. The composition

$$\begin{align*}{\mathscr{O}}(S)^{\times}{\overset{[-]_S}{{\longrightarrow}}} {\mathrm{Ext}}^1_{{{\mathrm{Fil}\text{-}F\text{-}\mathrm{MIC}}}(S,\sigma)}({\mathscr{O}}_{S},{\mathscr{O}}_{S}(1)) {\longrightarrow} H^1_{\mathrm{syn}}(S,{\mathbb{Q}}_p(1)) \end{align*}$$

agrees with the symbol map by Kato [Reference KatoKa1] where the second arrow is the map (2.10). Namely, it is explicitly described as follows:

$$\begin{align*}f\longmapsto \left(\frac{df}{f},\log^{(\sigma)}(f)\right)\in H^1_{\mathrm{syn}}(S,{\mathbb{Q}}_p(1))\subset \Omega^1_{\widehat{B}_K}{\oplus} \widehat{B}_K, \end{align*}$$

where $\widehat {B}:=\varprojlim _n B/p^nB$ and $\widehat {B}_K:={\mathbb {Q}}{\otimes }\widehat {B}$ .

Proof. By definition of u in (2.7), one has

$$\begin{align*}u({\mathop{\mathscr{L}\!\!\mathit{o}\mathit{g}}\nolimits}(f))=(\nabla(e_0),(1-\Phi)e_0)= \left(\frac{df}{f},\log^{(\sigma)}(f)\right) \end{align*}$$

as desired.

Lemma 2.8. The symbol map (2.13) is functorial with respect to the pullback. Namely, for a morphism $i\colon S'\to S$ , the diagram

is commutative, where $i^*$ denotes the map induced from the pullback functor (2.1).

2.4 Polylog objects

In this subsection, we generalize the log object to polylog objects. To define the polylog objects, we need the p-adic polylog function.

For an integer r, we denote the p-adic polylog function by

(2.14) $$ \begin{align} \ln^{(p)}_r(z):=\sum_{n\geq 1,\,p\not{\hspace{2pt}|}\,n}\frac{z^n}{n^r} = \lim_{s\to\infty}\frac{1}{1-z^{p^s}} \sum_{1\leq n<p^s,\,p\not{\hspace{2pt}|}\,n}\frac{z^n}{n^r} \in {\mathbb{Z}}_p\left[ z,\frac{1}{1-z}\right]^{\wedge}, \end{align} $$

where $A^{\wedge }$ denotes the p-adic completion of a ring A. As is easily seen, one has

$$\begin{align*}\ln^{(p)}_r(z)=(-1)^{r+1}\ln^{(p)}_r(z^{-1}),\quad z\frac{d}{dz}\ln_{r+1}^{(p)}(z)=\mathrm{ln}_r^{(p)}(z). \end{align*}$$

If $r\leq 0$ , this is a rational function. Indeed,

$$\begin{align*}\ln^{(p)}_0(z)=\frac{1}{1-z}-\frac{1}{1-z^p},\quad \ln^{(p)}_{-r}(z)=\left(z\frac{d}{dz}\right)^r\ln^{(p)}_0(z). \end{align*}$$

If $r\geq 1$ , it is no longer a rational function but an overconvergent function.

Proof. Since $\ln ^{(p)}_r(z)$ has ${\mathbb {Z}}_p$ -coefficients, it is enough to show that $\ln ^{(p)}_r(z)\in (x-x^2){\mathbb {Q}}_p[x]^{\dagger }$ . We first note that

(2.15) $$ \begin{align} (x^2-x)\frac{d}{dx}\mathrm{ln}_{k+1}^{(p)}(z)=\mathrm{ln}_k^{(p)}(z). \end{align} $$

The limit in (2.14) can be rewritten as

$$\begin{align*}\lim_{s\to\infty}\frac{1}{x^{p^s}-(x-1)^{p^s}} \sum_{1\leq n<p^s,\,p\not{\hspace{2pt}|}\,n}\frac{x^{p^s}(1-x^{-1})^n}{n^r}. \end{align*}$$

This shows that $\ln ^{(p)}_r(z)$ vanishes at $x=0, 1$ . Therefore, it suffices to show that $\ln _r^{(p)}(z)\in {\mathbb {Q}}_p[x]^{\dagger }$ because then it is divisible by $x-x^2$ in ${\mathbb {Q}}_p[x]^{\dagger }$ by [Reference MacarroMa, Theorem 3.5].

Let $w(x)\in {\mathbb {Z}}_p[x]$ be defined by

$$\begin{align*}\frac{1-z^p}{(1-z)^p}=x^p-(x-1)^p=1-pw(x). \end{align*}$$

Then,

$$\begin{align*}\ln_1^{(p)}(z)=p^{-1}\log\left(\frac{1-z^p}{(1-z)^p}\right) =-p^{-1}\sum_{n=1}^{\infty}\frac{p^nw(x)^n}{n}\in {\mathbb{Z}}_p[x]^{\dagger}. \end{align*}$$

This shows that $\ln ^{(p)}_1(z)\in (x-x^2){\mathbb {Q}}_p[x]^{\dagger }$ , as required in case $r=1$ .

Now, let

$$\begin{align*}-(x-x^2)^{-1}\mathrm{ln}_1^{(p)}(z) =a_0+a_1x+a_2x^2+\cdots+a_nx^n+\cdots\in{\mathbb{Z}}_p[x]^{\dagger}. \end{align*}$$

By (2.15), one has

$$\begin{align*}\ln_2^{(p)}(z)=c+a_0x+\frac{a_1}{2}x^2+\cdots +\frac{a_n}{n+1}x^{n+1}+\cdots\in {\mathbb{Q}}_p[x]^{\dagger}, \end{align*}$$

and hence, $\ln _2^{(p)}(z)\in (x-x^2){\mathbb {Q}}_p[x]^{\dagger }$ , as required in case $r=2$ . Continuing the same argument, we obtain $\ln _r^{(p)}(z)\in (x-x^2){\mathbb {Q}}_p[x]^{\dagger }$ for every r.

For a general $S={\operatorname {Spec}}(B)$ and $f\in B$ satisfying $f,1-f\in B^{\times }$ , we define the polylog object ${\mathop {\mathscr {P}\!\mathit {ol}}\nolimits }_n(f)$ to be the pullback $u^*{\mathop {\mathscr {P}\!\mathit {ol}}\nolimits }_n(T)$ , where $u\colon {\operatorname {Spec}}(B)\to {\operatorname {Spec}} V[T,T^{-1},(1-T)^{-1}]$ is given by $u(T)=f$ . When $n=1$ , ${\mathop {\mathscr {P}\!\mathit {ol}}\nolimits }_1(T)$ coincides with the log object ${\mathop {\mathscr {L}\!\!\mathit {o}\mathit {g}}\nolimits }(1-T)$ in ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(C)$ .

2.5 Relative cohomologies.

Let $S={\operatorname {Spec}}(B)$ be a smooth affine V-scheme and let $\sigma $ be a p-th Frobenius on $B^{\dagger }$ . In this subsection, we discuss objects in ${\mathrm {Fil}}$ -F- ${\mathop {\mathrm {MIC}}\nolimits }(S)$ arising as a relative cohomology of smooth morphisms.

Let $u\colon U\to S$ be a quasi-projective smooth morphism. We first describe a datum which we discuss in this subsection.

Definition 2.12. We define a datum

$$\begin{align*}H^i(U/S)=(H^i_{{{\mathrm{d\hspace{-0.2pt}R}}}}(U_K/S_K), H^i_{{\mathrm{rig}}}(U_k/S_k), c, \nabla, \Phi, {\mathrm{Fil}}^{\bullet}) \end{align*}$$

as follows.

• • $H^i_{{{\mathrm {d\hspace {-0.2pt}R}}}}(U_K/S_K)$ is the i-th relative algebraic de Rham cohomology of $u_K$ , namely the module of global sections of the i-th cohomology sheaf $R^i(u_K)_{\ast }\Omega ^{\bullet }_{U_K/S_K}$ .

• • $\nabla $ is the Gauss–Manin connection on $H^i_{{{\mathrm {d\hspace {-0.2pt}R}}}}(U_K/S_K)$ , and ${\mathrm {Fil}}^{\bullet }$ is the Hodge filtration defined from the theory of mixed Hodge modules by M. Saito [Reference SaitoSa1], [Reference SaitoSa2]Footnote ¹ (however, we shall only be concerned with the de Rham cohomology under the setting 2.14 or 2.16, and then the Hodge filtration can be defined from the Hodge to de Rham spectral sequences).

• • $(H^i_{{\mathrm {rig}}}(U_k/S_k), \Phi )$ is the $B_K^{\dagger }$ -module with $\sigma $ -linear Frobenius structure obtained as the i-th relative rigid cohomology of $u_k$ . In particular,

  $$\begin{align*}H^i_{{\mathrm{rig}}}(U_k/S_k) = \Gamma(S_K^{{\mathop{\mathrm{an}}\nolimits}}, R^iu_{{\mathrm{rig}}}j^{\dagger}_U{\mathscr{O}}_{U_K^{{\mathop{\mathrm{an}}\nolimits}}}), \end{align*}$$
  where $R^iu_{{\mathrm {rig}}}j^{\dagger }_U{\mathscr {O}}_{U_K^{{\mathop {\mathrm {an}}\nolimits }}}:=R^i(u_K^{{\mathop {\mathrm {an}}\nolimits }})_{\ast }j^{\dagger }_U\Omega ^{\bullet }_{U_K^{{\mathop {\mathrm {an}}\nolimits }}/S_K^{{\mathop {\mathrm {an}}\nolimits }}}$ is the i-th relative rigid cohomology sheaf (we justify this notation and definition in Remark 2.13 below).

• • $c\colon H^i_{{{\mathrm {d\hspace {-0.2pt}R}}}}(U_K/S_K)\otimes _{B_K}B_K^{\dagger }\to H^i_{{\mathrm {rig}}}(U_k/S_k)$ is the natural morphism between the algebraic de Rham cohomology and the rigid cohomology.

Let us give a construction of the comparison morphism c in this datum. We basically follow the construction in [Reference SolomonSo, 5.8.2]. Let $\iota \colon S_K^{{\mathop {\mathrm {an}}\nolimits }}\to S_K$ be the natural morphism of ringed topoi [Reference BerthelotBer3, 0.3]. Then, by adjunction, we have a natural morphism $R^iu_{K,\ast }\Omega ^{\bullet }_{U_K/S_K}\to \iota _{\ast }\big (R^iu_{K,\ast }\Omega _{U_K/S_K}^{\bullet }\big )^{{\mathop {\mathrm {an}}\nolimits }}$ . Now, together with the natural morphisms

$$\begin{align*}\big(R^iu_{K,\ast}\Omega^{\bullet}_{U_K/S_K}\big)^{{\mathop{\mathrm{an}}\nolimits}} \to R^i(u_K^{{\mathop{\mathrm{an}}\nolimits}})_{\ast}\Omega^{\bullet}_{U_K^{{\mathop{\mathrm{an}}\nolimits}}/S_K^{{\mathop{\mathrm{an}}\nolimits}}} \to R^i(u^{{\mathop{\mathrm{an}}\nolimits}}_K)_{\ast}j^{\dagger}_{U}\Omega^{\bullet}_{U_K^{{\mathop{\mathrm{an}}\nolimits}}/S_K^{{\mathop{\mathrm{an}}\nolimits}}}, \end{align*}$$

we get a morphism $R^iu_{K,\ast }\Omega ^{\bullet }_{U_K/S_K}\to \iota _{\ast }R^i(u^{{\mathop {\mathrm {an}}\nolimits }}_K)_{\ast }j^{\dagger }_{U}\Omega ^{\bullet }_{U_K^{{\mathop {\mathrm {an}}\nolimits }}/S_K^{{\mathop {\mathrm {an}}\nolimits }}}$ . By taking the module of global sections, we get a morphism $H^i_{{{\mathrm {d\hspace {-0.2pt}R}}}}(U_K/S_K)\to H^i_{{\mathrm {rig}}}(U_k/S_k)$ , and therefore, the desired morphism $c\colon H^i_{{{\mathrm {d\hspace {-0.2pt}R}}}}(U_K/S_K)\otimes _{B_K}B_K^{\dagger }\to H^i_{{\mathrm {rig}}}(U_k/S_k)$ because $H^i_{{\mathrm {rig}}}(U_k/S_k)$ is a $B_K^{\dagger }$ -module.

Remark 2.13. (1) Let us justify our description of the rigid cohomology $H^i_{{\mathrm {rig}}}(U_k/S_k)$ in Definition 2.12.

We begin by recalling a usual definition of the rigid cohomology of $u_k\colon U_k\to S_k$ over a frame $(S_k, \overline {S}_k, \widehat {\overline {S}})$ , where $\overline {S}$ is a closure of S in a projective space over V and $\widehat {\overline {S}}$ is its completion. Let $\overline {X}$ be the closure of U in a projective space over $\overline {S}$ and let $\overline {f}\colon \overline {X}\to \overline {S}$ be the extension of u (we choose this notation because, in this article, X usually denotes $\overline {f}^{-1}(S)$ and f denotes $\overline {f}|_X$ ). Then, by definition, the i-th relative rigid cohomology sheaf is $R^i(\overline {f}_K^{{\mathop {\mathrm {an}}\nolimits }})_{\ast }\overline {j}^{\dagger }_{U}\Omega ^{\bullet }_{\overline {X}_K^{{\mathop {\mathrm {an}}\nolimits }}/\overline {S}_K^{{\mathop {\mathrm {an}}\nolimits }}}$ , where $\overline {j}_U\colon \widehat {U}_K\hookrightarrow \overline {X}_K^{{\mathop {\mathrm {an}}\nolimits }}$ is the natural inclusion and $H^i_{{\mathrm {rig}}}(U_k/S_k)$ is the module of global sections of this sheaf on $\overline {S}_K$ .

However, by using the fact that $u\colon U\to S$ is a lift of $u_k\colon U_k\to S_k$ to a smooth morphism of smooth algebraic V-schemes, we may obtain the same module without referring to compactifications. In fact, first, since $S_K^{{\mathop {\mathrm {an}}\nolimits }}$ is a strict neighborhood of $\widehat {S}_K$ [Reference BerthelotBer3, (1.2.4)(ii)], $H^i_{{\mathrm {rig}}}(U_k/S_k)$ is also the module of global sections of $R^i(\overline {f}_K^{{\mathop {\mathrm {an}}\nolimits }})_{\ast }j^{\dagger }_{U}\Omega ^{\bullet }_{\overline {X}_K^{{\mathop {\mathrm {an}}\nolimits }}/\overline {S}_K^{{\mathop {\mathrm {an}}\nolimits }}}$ on $S_K^{{\mathop {\mathrm {an}}\nolimits }}$ by overconvergence. Moreover, the restriction of this sheaf on $S_K^{{\mathop {\mathrm {an}}\nolimits }}$ is isomorphic to $R^i(u_K^{{\mathop {\mathrm {an}}\nolimits }})_{\ast }j^{\dagger }_{U}\Omega ^{\bullet }_{U_K^{{\mathop {\mathrm {an}}\nolimits }}/S_K^{{\mathop {\mathrm {an}}\nolimits }}}$ by [Reference StumLS, 6.2.2] because, again, $U_K^{{\mathop {\mathrm {an}}\nolimits }}$ is a strict neighborhood of $\widehat {U}_K$ . Our definition of $R^iu_{{\mathrm {rig}}}j^{\dagger }_U{\mathscr {O}}_{U_K^{{\mathop {\mathrm {an}}\nolimits }}}$ and the description of $H^i_{{\mathrm {rig}}}(U_k/S_k)$ in Definition 2.12 are thus justified.

(2) We will use the datum $H^i(U/S)$ only in the case where the rigid cohomology sheaf $R^iu_{{\mathrm {rig}}}j^{\dagger }_U{\mathscr {O}}_{U_K^{{\mathop {\mathrm {an}}\nolimits }}}$ is known to be a coherent $j^{\dagger }_{S}{\mathscr {O}}_{S_K^{{\mathop {\mathrm {an}}\nolimits }}}$ -module for all $i\geq 0$ . In this case, we also have

$$\begin{align*}H^i_{{\mathrm{rig}}}(U_k/S_k)=\Gamma(S_K^{{\mathop{\mathrm{an}}\nolimits}},R^iu_{{\mathrm{rig}}}j^{\dagger}_U{\mathscr{O}}_{U_K^{{\mathop{\mathrm{an}}\nolimits}}}) \end{align*}$$

by the vanishing of higher sheaf cohomologies for coherent $j^{\dagger }_S{\mathscr {O}}_{S_K^{{\mathop {\mathrm {an}}\nolimits }}}$ -modules (this vanishing is perhaps well-known, but we included it as Lemma 2.19 at the end of this subsection because we could not find an appropriate reference).

Now, we have defined the datum $H^i(U/S)$ . This, however, does not immediately mean that it is an object of ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S)$ . For this datum to be an object in ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S)$ , we need the i-th relative cohomology $H^i_{{\mathrm {rig}}}(U_k/S_k)$ to be a coherent $B_K^{\dagger }$ -module with Frobenius structure, and we need the morphism c to be an isomorphism. In the rest of this subsection, we discuss two settings under which these conditions hold. Briefly said, these two settings are: the case of proper smooth morphisms (Setting 2.14) and the case of smooth families of general dimension with “good compactification” of both the source and the target (Setting 2.16).

We start with the first setting.

Proposition 2.15. Under Setting 2.14, the relative rigid cohomology sheaf $R^iu_{{\mathrm {rig}}}j^{\dagger }_U{\mathscr {O}}_{U_K^{{\mathop {\mathrm {an}}\nolimits }}}$ is a coherent $j^{\dagger }_S{\mathscr {O}}_{S_K^{{\mathop {\mathrm {an}}\nolimits }}}$ -module with Frobenius structure for each $i\geq 0$ . Consequently, $H^i_{{\mathrm {rig}}}(U_k/S_k)$ is a coherent F- $B_K^{\dagger }$ -module for each $i\geq 0$ . Moreover, the comparison morphism

(2.16) $$ \begin{align} c\colon H^i_{{{\mathrm{d\hspace{-0.2pt}R}}}}(U_K/S_K)\otimes_{B_K}B_K^{\dagger}\to H^i_{{\mathrm{rig}}}(U_k/S_k) \end{align} $$

is bijective for each $i\geq 0$ .

In particular, the datum

$$\begin{align*}H^i(U/S)=(H^i_{{{\mathrm{d\hspace{-0.2pt}R}}}}(U_K/S_K),H^i_{{\mathrm{rig}}}(U_k/S_k),c,\nabla,\Phi,{\mathrm{Fil}}^{\bullet}) \end{align*}$$

is an object of ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S)$ .

The second sufficient condition for $H^i(U/S)$ to be an object of ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S)$ is, briefly said, that u has a “good compactification”.

The notation in Setting 2.16 can be summarized by the diagram

where the notation above $\hookrightarrow $ shows the complement of the subscheme.

Proposition 2.17. Under Setting 2.16, the relative rigid cohomology sheaf $R^iu_{{\mathrm {rig}}}j^{\dagger }_U{\mathscr {O}}_{U_K^{{\mathop {\mathrm {an}}\nolimits }}}$ is a coherent $j^{\dagger }_S{\mathscr {O}}_{S_K^{{\mathop {\mathrm {an}}\nolimits }}}$ -module with Frobenius structure for each $i\geq 0$ . Consequently, $H^i_{{\mathrm {rig}}}(U_k/S_k)$ is a coherent F- $B_K^{\dagger }$ -module for each $i\geq 0$ . Moreover, the comparison morphism

$$\begin{align*}c\colon H^i_{{{\mathrm{d\hspace{-0.2pt}R}}}}(U_K/S_K)\otimes_{B_K}B_K^{\dagger}\to H^i_{{\mathrm{rig}}}(U_k/S_k) \end{align*}$$

is bijective for each $i\geq 0$ .

In particular, the datum

$$\begin{align*}H^i(U/S)=(H^i_{{{\mathrm{d\hspace{-0.2pt}R}}}}(U_K/S_K),H^i_{{\mathrm{rig}}}(U_k/S_k),c,\nabla,\Phi,{\mathrm{Fil}}^{\bullet}) \end{align*}$$

is an object of ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S)$ .

We also have a Gysin exact sequence in ${\mathrm {Fil}}$ -F- ${\mathop {\mathrm {MIC}}\nolimits }(S)$ for curves under this setting.

Proposition 2.18 (Gysin exact sequence).

Let $U={\operatorname {Spec}}(A)$ and $S={\operatorname {Spec}}(B)$ be smooth affine V-schemes and let $u\colon U\to S$ be a smooth morphism of relative dimension one with connected fibers. Assume that there exists a projective smooth curve $f\colon X\to S$ with an open immersion $U\hookrightarrow X$ such that $f|_U=u$ and that the complementary divisor $D:=X\setminus U$ is finite étale over S. Moreover, assume that u satisfies the conclusions of Proposition 2.17 (namely, the coherence of the rigid cohomology and the bijectivity of the comparison morphism), e.g. that we are in Setting 2.16.

Then, we have an exact sequence

$$\begin{align*}0\to H^1(X/S)\to H^1(U/S)\to H^0(D/S)(-1)\to H^2(X/S)\to 0. \end{align*}$$

in ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S)$ .

Proof. First, $H^i(X/S)$ and $H^i(D/S)$ are objects of ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S)$ by Proposition 2.15, and so is $H^1(U/S)$ by assumption. Next, it is a standard fact about de Rham cohomology that we have an exact sequence

$$\begin{align*}0\to H^1_{{{\mathrm{d\hspace{-0.2pt}R}}}}(X_K/S_K)\to H^1_{{{\mathrm{d\hspace{-0.2pt}R}}}}(U_K/S_K)\to H^0_{{{\mathrm{d\hspace{-0.2pt}R}}}}(D_K/S_K)(-1)\to H^2_{{{\mathrm{d\hspace{-0.2pt}R}}}}(X_K/S_K)\to 0 \end{align*}$$

whose morphisms are horizontal and compatible with respect to ${\mathrm {Fil}}$ . Therefore, by the comparison isomorphism on each term and by the flatness of $B_K^{\dagger }$ over $B_K$ , we get a corresponding exact sequence

$$\begin{align*}0\to H^1_{{\mathrm{rig}}}(X_k/S_k)\to H^1_{{\mathrm{rig}}}(U_k/S_k)\to H^0_{{\mathrm{rig}}}(D_k/S_k)(-1)\to H^2_{{\mathrm{rig}}}(X_k/S_k)\to 0 \end{align*}$$

for rigid cohomologies. The compatibility of this sequence with Frobenius structures on each term can be checked on each closed point of $\widehat {S}_K$ and therefore reduced to the absolute case [Reference BannaiBan2, Theorem 2.19].

The following lemma is the promised statement in Remark 2.13 (2).

Proof. This lemma is essentially given in the proof of [Reference StumLS, 6.2.12]. We recall the argument for the convenience for the reader. Choose a closed immersion $X\hookrightarrow {\mathbb {A}}_V^N$ to an affine space and let Y be the closure of X in $X\hookrightarrow {\mathbb {A}}_V^N\hookrightarrow {\mathbb {P}}_V^N$ . Then, $V_{\rho }:=X_K^{{\mathop {\mathrm {an}}\nolimits }}\cap \mathbb {B}^N(0,\rho ^+)$ for $\rho>1$ form a cofinal family of strict neighborhoods of $\widehat {X}_K$ .

By the coherence of ${\mathscr {M}}$ , we can take a coherent ${\mathscr {O}}_{V_{\rho _0}}$ -module M for some $\rho _0>1$ such that ${\mathscr {M}}|_{V_{\rho _0}}=j^{\dagger }_{X}M$ [Reference StumLS, 5.4.4]. Then, if $j_{\rho }\colon V_{\rho }\hookrightarrow V_{\rho _0}$ denotes the inclusion for $1<\rho <\rho _0$ , we have isomorphisms

$$\begin{align*}R\Gamma(X_K^{{\mathop{\mathrm{an}}\nolimits}}, {\mathscr{M}})=R\Gamma(V_{\rho_0}, {\mathscr{M}}|_{V_{\rho_0}})=\varinjlim_{\rho}R\Gamma(V_{\rho_0},(j_{\rho})_{\ast}M)=\varinjlim_{\rho}R\Gamma(V_{\rho},M|_{V_{\rho}}). \end{align*}$$

In fact, the first identification holds because ${\mathscr {M}}$ is a $j^{\dagger }_X{\mathscr {O}}_{X_K^{{\mathop {\mathrm {an}}\nolimits }}}$ -module by a standard argument (as in [Reference Baldassarri and BerthelotBal-Ber, 1.1]), the second one follows from quasi-compactness and separatedness of $V_{\rho _0}$ and the third one holds because $j_{\rho }$ is affinoid. Now, the claim follows because each $V_{\rho }$ is affinoid and $M|_{V_{\rho }}$ is coherent.

2.6 Extensions associated to Milnor symbols

In this subsection, we discuss how we associate an extension in ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(U)$ to a Milnor symbol.

Let $u\colon U={\operatorname {Spec}}(A)\to S={\operatorname {Spec}}(B)$ be a smooth morphism of smooth affine V-scheme. We assume that u satisfies the consequences of Proposition 2.17.

Assumption 2.20. For each $i\geq 0$ , the i-th relative rigid cohomology sheaf $R^iu_{{\mathrm {rig}}}j^{\dagger }_U{\mathscr {O}}_{U_K^{{\mathop {\mathrm {an}}\nolimits }}}$ is a coherent $j^{\dagger }_{S}{\mathscr {O}}_{S_K^{{\mathop {\mathrm {an}}\nolimits }}}$ -module with Frobenius structure and the natural morphism

(2.17) $$ \begin{align} H^i_{{{\mathrm{d\hspace{-0.2pt}R}}}}(U_K/S_K)\otimes_{B_K} B_K^{\dagger}{\longrightarrow} H^i_{{\mathrm{rig}}}(U_k/S_k) \end{align} $$

is bijective.

Recall that, for a commutative ring R, the r-th Milnor K-group $K_r^M(R)$ is defined to be the quotient of $(R^{\times })^{{\otimes } r}$ by the subgroup generated by

$$\begin{align*}a_1{\otimes}\cdots{\otimes} b{\otimes}\cdots{\otimes}(-b){\otimes}\cdots{\otimes} a_r,\quad a_1{\otimes}\cdots{\otimes} b{\otimes}\cdots{\otimes}(1-b){\otimes}\cdots{\otimes} a_r. \end{align*}$$

Recall from §2.3 the log object ${\mathop {\mathscr {L}\!\!\mathit {o}\mathit {g}}\nolimits }(f)$ in ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(U)$ for $f\in {\mathscr {O}}(U)^{\times }$ and the extension (2.12) which represents the class

$$\begin{align*}[f]_U\in {\mathrm{Ext}}^1_{{{\mathrm{Fil}\text{-}F\text{-}\mathrm{MIC}}}(U)}({\mathscr{O}}_U,{\mathscr{O}}_U(1)). \end{align*}$$

For $h_0,h_1,\ldots ,h_n\in {\mathscr {O}}(U)^{\times }$ , we associate an $(n+1)$ -extension

$$\begin{align*}0{\longrightarrow} {\mathscr{O}}_{U}(n+1){\longrightarrow} {\mathop{\mathscr{L}\!\!\mathit{o}\mathit{g}}\nolimits}(h_n)(n){\longrightarrow}\cdots{\longrightarrow}{\mathop{\mathscr{L}\!\!\mathit{o}\mathit{g}}\nolimits}(h_1)(1) {\longrightarrow}{\mathop{\mathscr{L}\!\!\mathit{o}\mathit{g}}\nolimits}(h_0){\longrightarrow} {\mathscr{O}}_{U}{\longrightarrow} 0 \end{align*}$$

which represents the class

$$\begin{align*}[h_0]_U\cup\cdots\cup[h_n]_U\in {\mathrm{Ext}}^{n+1}_{{{\mathrm{Fil}\text{-}F\text{-}\mathrm{MIC}}}(U)}({\mathscr{O}}_U,{\mathscr{O}}_U(n+1)). \end{align*}$$

It is a standard argument to show that the above cup-product is additive with respect to each $h_i$ , so that we have an additive map

$$\begin{align*}({\mathscr{O}}(U)^{\times})^{{\otimes} n+1}{\longrightarrow} {\mathrm{Ext}}^{n+1}_{{{\mathrm{Fil}\text{-}F\text{-}\mathrm{MIC}}}(U)}({\mathscr{O}}_U,{\mathscr{O}}_U(n+1)). \end{align*}$$

Proposition 2.22. $[f]_U\cup [f]_U=0$ for $f\in {\mathscr {O}}(U)^{\times }$ and $[f]_U\cup [1-f]_U=0$ for $f\in {\mathscr {O}}(U)^{\times }$ such that $1-f\in {\mathscr {O}}(U)^{\times }$ . Hence, the homomorphism

$$\begin{align*}K^M_{n+1}({\mathscr{O}}(U)){\longrightarrow}{\mathrm{Ext}}^{n+1}_{{{\mathrm{Fil}\text{-}F\text{-}\mathrm{MIC}}}(U)}({\mathscr{O}}_{U},{\mathscr{O}}_U(n+1)),\quad \{h_0,\ldots,h_n\}\mapsto [h_0]_U\cup\cdots\cup[h_n]_U \end{align*}$$

is well-defined (note ${\mathop {\mathscr {L}\!\!\mathit {o}\mathit {g}}\nolimits }(-f)={\mathop {\mathscr {L}\!\!\mathit {o}\mathit {g}}\nolimits }(f)$ by definition).

Proof. To prove this, it follows from Lemma 2.8 that we may assume $U={\operatorname {Spec}} V[T,T^{-1}]$ and $f=T$ for the vanishing $[f]_U\cup [f]_U=0$ and $U={\operatorname {Spec}} V[T,(T-T^2)^{-1}]$ and $f=T$ for the vanishing $[f]_U\cup [1-f]_U=0$ .

Here, we show the latter vanishing. Let $C=V[T,(T-T^2)^{-1}]$ and $U={\operatorname {Spec}}(C)$ . Recall the dilog object $D:={\mathop {\mathscr {D}\!\mathit {ilog}}\nolimits }(T)$ which has a unique increasing filtration $W_{\bullet }$ (as an object of ${\mathrm {Fil}}$ -F- ${\mathop {\mathrm {MIC}}\nolimits }(U)$ ) that satisfies

$$\begin{align*}W_jD_{{\mathrm{d\hspace{-0.2pt}R}}}=\begin{cases} 0 & \text{if } j\leq -5,\\ C_Ke_{-4} & \text{if } j=-4, -3,\\ C_Ke_{-4}\oplus C_Ke_{-2} & \text{if } j=-2, -1,\\ {\mathop{\mathscr{D}\!\mathit{ilog}}\nolimits}(T)_{{{\mathrm{d\hspace{-0.2pt}R}}}} & \text{if } j\geq 0 \end{cases} \end{align*}$$

and the filtration ${\mathrm {Fil}}^{\bullet }$ on $W_jD_{{\mathrm {d\hspace {-0.2pt}R}}}$ is given to be ${\mathrm {Fil}}^iW_jD_{{\mathrm {d\hspace {-0.2pt}R}}}=W_jD_{{\mathrm {d\hspace {-0.2pt}R}}}\cap {\mathrm {Fil}}^iD_{{{\mathrm {d\hspace {-0.2pt}R}}}}$ . Then, it is straightforward to check that $W_{-4}\cong {\mathscr {O}}_U(2), W_{-2}\cong {\mathop {\mathscr {L}\!\!\mathit {o}\mathit {g}}\nolimits }(T)(1), W_0/W_{-4}\cong {\mathop {\mathscr {L}\!\!\mathit {o}\mathit {g}}\nolimits }(1-T)$ and $W_0/W_{-2}\cong {\mathscr {O}}_U$ . Thus, $[T]_U\cup [1-T]_U$ is realized by the extension

$$\begin{align*}0{\longrightarrow} W_{-4}{\longrightarrow} W_{-2}{\longrightarrow} W_0/W_{-4}{\longrightarrow} W_0/W_{-2}{\longrightarrow} 0.\end{align*}$$

Consider a commutative diagram

(2.18)

where $\iota $ is the first inclusion, $\pi _1$ is the first projection, $\pi _2$ is the composition with the second projection and the inclusion $W_{-2}\hookrightarrow W_0$ , $\pi _3$ is the quotient $W_0\to W_0/W_{-2}\cong {\mathscr {O}}_U$ , and $\mathrm {add}\colon (x,y)\mapsto x+y$ . The above diagram shows the vanishing $[T]_U\cup [1-T]_U=0$ .

The proof of the vanishing $[T]_U\cup [T]_U=0$ goes in a similar way by replacing ${\mathop {\mathscr {D}\!\mathit {ilog}}\nolimits }(T)$ with $\mathrm {Sym}^2{\mathop {\mathscr {L}\!\!\mathit {o}\mathit {g}}\nolimits }(T)$ .

Let n be a nonnegative integer, let $h_0,\ldots ,h_{n}\in {\mathscr {O}}(U)^{\times }$ and suppose that ${{\mathrm {Fil}}^{n+1}H^{n+1}(U/S)=0}$ . Under this setting, we define an object

$$\begin{align*}M(U/S)_{h_0,\ldots,h_{n}} \end{align*}$$

in ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(U)$ in the following way.

Let ${\mathscr {M}}_{h_0,\ldots ,h_{n}}$ be the complex

$$\begin{align*}{\mathop{\mathscr{L}\!\!\mathit{o}\mathit{g}}\nolimits}(h_n)(n){\longrightarrow} \cdots{\longrightarrow}{\mathop{\mathscr{L}\!\!\mathit{o}\mathit{g}}\nolimits}(h_1)(1){\longrightarrow}{\mathop{\mathscr{L}\!\!\mathit{o}\mathit{g}}\nolimits}(h_0) \end{align*}$$

in ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(U)$ , where the first term is placed in degree $0$ , and the differentials are the compositions of the projection ${{\mathscr {L}{og}}}(h_i)(i)\to {\mathscr {O}}_{U_K}e_{0}$ and the injection ${\mathscr {O}}_{U_K}e_{-2}\to {{\mathscr {L}{og}}}(h_{i-1})(i-1)$ defined by $e_{0}\mapsto e_{-2}$ . This fits into a distinguished triangle

(2.19) $$ \begin{align} 0{\longrightarrow} {\mathscr{O}}_U(n+1){\longrightarrow} {\mathscr{M}}_{h_0,\ldots,h_{n}}{\longrightarrow} {\mathscr{O}}_U[-n]{\longrightarrow}0 \end{align} $$

in the derived category of ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(U)$ .

First, we define the de Rham part of $M(U/S)_{h_0,\ldots ,h_n}$ . Let ${\mathscr {M}}_{h_0,\ldots ,h_n,{{\mathrm {d\hspace {-0.2pt}R}}}}$ denote the de Rham realization of ${\mathscr {M}}_{h_0,\ldots ,h_n}$ . This can be seen as the de Rham realization of a complex of mixed Hodge modules by M. Saito [Reference SaitoSa1], [Reference SaitoSa2]. Set

which fits into an exact sequence

$$\begin{align*}0{\longrightarrow} H^n_{{\mathrm{d\hspace{-0.2pt}R}}}(U_K/S_K){\longrightarrow} M(U/S)_{h_0,\ldots,h_n,{{\mathrm{d\hspace{-0.2pt}R}}}}{\longrightarrow} {\mathscr{O}}_{S_K}{\overset{\delta}{{\longrightarrow}}} H^{n+1}_{{\mathrm{d\hspace{-0.2pt}R}}}(U_K/S_K). \end{align*}$$

Note that this implies that $M(U/S)_{h_0,\ldots ,h_n,{{\mathrm {d\hspace {-0.2pt}R}}}}$ is locally free of finite rank. Again, by the theory of mixed Hodge modules, all the terms underly variations of mixed Hodge structures. Hence, they carry the Hodge filtration, which we write by ${\mathrm {Fil}}^{\bullet }$ , and the integrable connection $\nabla $ that satisfies the Griffiths transversality. All the arrows are compatible with respect to $\nabla $ and strictly compatible with respect to ${\mathrm {Fil}}^{\bullet }$ . Moreover, since the image of $\delta $ is contained in ${\mathrm {Fil}}^{n+1}H^{n+1}_{{\mathrm {d\hspace {-0.2pt}R}}}(U_K/S_K)$ and ${\mathrm {Fil}}^{n+1}H^{n+1}_{{\mathrm {d\hspace {-0.2pt}R}}}(U_K/S_K)=0$ by the assumption, we have an exact sequence

$$\begin{align*}0{\longrightarrow} H^n_{{\mathrm{d\hspace{-0.2pt}R}}}(U_K/S_K){\longrightarrow} M(U/S)_{h_0,\ldots,h_n,{{\mathrm{d\hspace{-0.2pt}R}}}}{\longrightarrow} {\mathscr{O}}_{S_K}{\longrightarrow}0 \end{align*}$$

in the category ${{\mathrm {Fil}\text {-}\mathrm {MIC}}}(S_K)$ .

Let ${\mathscr {M}}_{h_0,\ldots ,h_n,{\mathrm {rig}}}$ be the corresponding complex in ${{F\text {-}\mathrm {MIC}}}(A^{\dagger }_K)$ to ${\mathscr {M}}_{h_0,\ldots ,h_n}$ which can be seen as a complex of overconvergent F-isocrystals. In particular, this can also be seen as a complex of coherent $j^{\dagger }_U{\mathscr {O}}_{U_K^{{\mathop {\mathrm {an}}\nolimits }}}$ -modules with Frobenius structure. We define

where $R^nu_{{\mathrm {rig}}}{\mathscr {M}}_{h_0,\dots ,h_n,{\mathrm {rig}}} =R^n(u_K^{{\mathop {\mathrm {an}}\nolimits }})_{\ast }\left (j^{\dagger }_U\Omega ^{\bullet }_{U_K^{{\mathop {\mathrm {an}}\nolimits }}/S_K^{{\mathop {\mathrm {an}}\nolimits }}}\otimes {\mathscr {M}}_{h_0,\dots ,h_n,{\mathrm {rig}}}\right )$ .

Now, we explain how we get a comparison isomorphism

$$\begin{align*}c\colon M(U/S)_{h_0,\ldots,h_n,{{\mathrm{d\hspace{-0.2pt}R}}}} \otimes_{B_K}{B_K^{\dagger}}\xrightarrow{\quad\cong\quad} M(U/S)_{h_0,\ldots,h_n,{\mathrm{rig}}}. \end{align*}$$

First, we have a canonical homomorphism

$$\begin{align*}H^i_{{\mathrm{d\hspace{-0.2pt}R}}}(U_K/S_K,{\mathscr{M}}_{h_0,\ldots,h_n,{{\mathrm{d\hspace{-0.2pt}R}}}}) \otimes_{B_K}{B_K^{\dagger}}{\longrightarrow} H^i_{\mathrm{rig}}(U_k/S_k,{\mathscr{M}}_{h_0,\ldots,h_n,{\mathrm{rig}}}) \end{align*}$$

for each i (the construction is the same as in the case of trivial coefficients in the beginning of the previous subsection). To prove that this is an isomorphism, as in the de Rham part, note that we have an exact sequence

$$\begin{align*}0{\longrightarrow} H^n_{{\mathrm{rig}}}(U_k/S_k){\longrightarrow} M(U/S)_{h_0,\dots,h_n,{\mathrm{rig}}}{\longrightarrow} B_K^{\dagger} {\longrightarrow} H^{n+1}_{{\mathrm{rig}}}(U_k/S_k) \end{align*}$$

because $H^1(S_K^{{\mathop {\mathrm {an}}\nolimits }}, R^nu_{{\mathrm {rig}}}j^{\dagger }_U{\mathscr {O}}_{U_K^{{\mathop {\mathrm {an}}\nolimits }}/S_K^{{\mathop {\mathrm {an}}\nolimits }}})=0$ by Assumption 2.20. Thus, the comparison homomorphism is an isomorphism by the flatness of $B_K^{\dagger }$ over $B_K$ , by Assumption 2.20 and by five lemma.

Now, we have constructed an object $M_{h_0,\ldots ,h_n}(U/S)$ in ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S)$ which fits into the exact sequence

(2.20) $$ \begin{align} 0{\longrightarrow} H^n(U/S)(n+1){\longrightarrow} M_{h_0,\ldots,h_n}(U/S){\longrightarrow} {\mathscr{O}}_S{\longrightarrow}0. \end{align} $$

The extension class (2.20) is additive with respect to each $h_i$ (one can show this in the same way as the proof of bi-additivity of $[h_0]_U\cup \cdots \cup [h_n]_U$ , but based on the theory of mixed Hodge modules concerning the strictness of the filtrations; for the rigid part, we use the functoriality of the relative rigid cohomology and the comparison to algebraic de Rham cohomology), so that we have a homomorphism

(2.21) $$ \begin{align} ({\mathscr{O}}(U)^{\times})^{{\otimes} n+1}{\longrightarrow}{\mathrm{Ext}}^1_{{{\mathrm{Fil}\text{-}F\text{-}\mathrm{MIC}}}(S)}({\mathscr{O}}_S,H^n(U/S)(n+1)). \end{align} $$

Theorem 2.23. Suppose ${\mathrm {Fil}}^{n+1}H^{n+1}_{{\mathrm {d\hspace {-0.2pt}R}}}(U_K/S_K)=0$ . Then, under the assumption 2.20, the homomorphism (2.21) factors through the Milnor K-group, so that we have a map

(2.22) $$ \begin{align} [-]_{U/S}:K^M_{n+1}({\mathscr{O}}(U)){\longrightarrow}{\mathrm{Ext}}^1_{{{\mathrm{Fil}\text{-}F\text{-}\mathrm{MIC}}}(S)}({\mathscr{O}}_S,H^n(U/S)(n+1)) \end{align} $$

which we call the symbol map for $U/S$ .

Remark 2.24. If it were possible to define a natural projection

$$\begin{align*}{\mathrm{Ext}}^{n+1}({\mathscr{O}}_U,{\mathscr{O}}_U(n+1)){\longrightarrow} {\mathrm{Ext}}^1({\mathscr{O}}_S,H^n(U/S)(n+1)) \end{align*}$$

under the assumption $H^{n+1}(U/S)=0$ , then the object $M_{h_0,\ldots ,h_n}(U/S)$ should correspond to the class $[h_0]_U\cup \cdots \cup [h_n]_U$ , and hence, Theorem 2.23 would be immediate from Proposition 2.22. However, this is impossible since we do not take into consideration the boundary conditions, such as admissibility, for constructing our category. We need to prove Theorem 2.23 independently while almost the same argument works as well.

Proof. Write the homomorphism (2.21) by ${\widetilde {\rho }}$ . It is enough to show the following.

$$\begin{align*}{\widetilde{\rho}}(h_0{\otimes}\cdots{\otimes} f{\otimes} f{\otimes}\cdots{\otimes} h_n)=0,\quad {\widetilde{\rho}}(h_0{\otimes}\cdots{\otimes} f{\otimes} (1-f){\otimes}\cdots{\otimes} h_n)=0. \end{align*}$$

We show the latter. Let $f\in {\mathscr {O}}(U)^{\times }$ such that $1-f\in {\mathscr {O}}(U)^{\times }$ . Let $u:U\to {\operatorname {Spec}} V[T,(T-T^2)^{-1}]$ be the morphism given by $u^*T=f$ . Recall the diagram (2.18), and take the pullback by u. It follows that

$$\begin{align*}0\to{\mathscr{O}}_U(n+1)\to \overbrace{{\mathop{\mathscr{L}\!\!\mathit{o}\mathit{g}}\nolimits}(h_n)\to\cdots\to{\mathop{\mathscr{L}\!\!\mathit{o}\mathit{g}}\nolimits}(f)(i+1)\to{\mathop{\mathscr{L}\!\!\mathit{o}\mathit{g}}\nolimits}(1-f)(i)\to\cdots \to{\mathop{\mathscr{L}\!\!\mathit{o}\mathit{g}}\nolimits}(h_0)}^{{\mathscr{M}}}\to{\mathscr{O}}_U\to0 \end{align*}$$

is equivalent to

$$\begin{align*}0\to{\mathscr{O}}_U(n+1)\to \overbrace{{\mathop{\mathscr{L}\!\!\mathit{o}\mathit{g}}\nolimits}(h_n)\to\cdots\to{\mathscr{O}}_U(i+2){\overset{0}{\to}} \underbrace{{\mathscr{O}}_U(i)\to\cdots\to{\mathop{\mathscr{L}\!\!\mathit{o}\mathit{g}}\nolimits}(h_0)}_{\mathscr N}}^{{\mathscr{M}}'}\to{\mathscr{O}}_U\to0 \end{align*}$$

as an $(n+1)$ -extension in ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(U)$ . This implies that ${\mathscr {M}}$ is quasi-isomorphic to ${\mathscr {M}}'$ as mixed Hodge modules or F-isocrystals. Hence, $\mathscr N$ gives rise to a splitting of

$$\begin{align*}0{\longrightarrow} H^n(U/S)(n+1){\longrightarrow} M_{h_0,\ldots,f,1-f,\ldots,h_n}(U/S){\longrightarrow} {\mathscr{O}}_S{\longrightarrow}0. \end{align*}$$

This completes the proof of ${\widetilde {\rho }}(h_0{\otimes }\cdots {\otimes } f{\otimes } (1-f){\otimes }\cdots {\otimes } h_n)=0$ .

To see ${\widetilde {\rho }}(h_0{\otimes }\cdots {\otimes } f{\otimes } f{\otimes }\cdots {\otimes } h_n)=0$ , we consider the similar diagram to (2.18) obtained from $\mathrm {Sym}^2{\mathop {\mathscr {L}\!\!\mathit {o}\mathit {g}}\nolimits }(T)$ . Then, the rest is the same.

2.7 Explicit formula

In this subsection, we continue to use the setting of the previous subsection. In particular, $U={\operatorname {Spec}}(A)\to S={\operatorname {Spec}}(B)$ is a smooth morphism of smooth V-schemes that satisfies Assumption 2.20. We fix a p-th Frobenius endomorphism $\varphi $ (resp. $\sigma $ ) on $A^{\dagger }$ (resp. $B^{\dagger }$ ).

Let $n\geq 0$ be an integer. Suppose that ${\mathrm {Fil}}^{n+1}H^{n+1}_{{\mathrm {d\hspace {-0.2pt}R}}}(U_K/S_K)=0$ . Recall the maps

$$\begin{align*}K_{n+1}^M(A){\overset{[-]_{U/S}}{{\longrightarrow}}} {\mathrm{Ext}}^1_{{{\mathrm{Fil}\text{-}F\text{-}\mathrm{MIC}}}(S)}({\mathscr{O}}_S,H^n(U/S)(n+1)) {\overset{(2.7)}{{\longrightarrow}}} H^1({\mathscr{S}}(H^n(U/S)(n+1))). \end{align*}$$

Note that the last cohomology group is

$$\begin{align*}\left\{\kern-1.2pt(\omega,\xi)\in \left(\Omega^1_{B_K}{\otimes} {\mathrm{Fil}}^nH^n_{{\mathrm{d\hspace{-0.2pt}R}}}(U_K/S_K)\right){\oplus}\left( B^{\dagger}_K{\otimes} H^n_{{\mathrm{d\hspace{-0.2pt}R}}}(U_K/S_K)\right) \bigg| (1-\varphi_{n+1})\omega=d\xi,\,d(\omega)=0 \kern-1.2pt\right\}\!, \end{align*}$$

where $\varphi _i:=p^{-i}\varphi $ and d is the differential map induced from the Gauss-Manin connection as in (2.2). Let

(2.23)

be the compositions of (2.7) and the projections $(\omega ,\xi )\mapsto \omega $ and $(\omega ,\xi )\mapsto \xi $ , respectively. The map $\delta $ agrees with the map (2.5), and hence, it does not depend on $\sigma $ , while $R_\sigma $ does. We put

(2.24) $$ \begin{align} D_{U/S}:=\delta\circ[-]_{U/S},\quad {\mathrm{reg}}^{(\sigma)}_{U/S}:=R_\sigma\circ[-]_{U/S}. \end{align} $$

The purpose of this section is to give an explicit description of these maps.

Theorem 2.25. Suppose ${\mathrm {Fil}}^{n+1}H^{n+1}_{{\mathrm {d\hspace {-0.2pt}R}}}(U_K/S_K)=0$ . Let $\xi =\{h_0,\ldots ,h_n\}\in K^M_{n+1}(A)$ . Then,

(2.25) $$ \begin{align} D_{U/S}(\xi)=(-1)^n\frac{dh_0}{h_0}\wedge\frac{dh_1}{h_1}\wedge\cdots \wedge\frac{dh_n}{h_n}, \end{align} $$

(2.26) $$ \begin{align} {\mathrm{reg}}^{(\sigma)}_{U/S}(\xi) =(-1)^n\sum_{i=0}^n(-1)^ip^{-1}\log\left(\frac{h_i^p}{h_i^{\varphi}}\right) \left(\frac{dh_0}{h_0}\right)^{\varphi_1}\wedge\cdots\wedge\left(\frac{dh_{i-1}}{h_{i-1}}\right)^{\varphi_1}\wedge \frac{dh_{i+1}}{h_{i+1}}\wedge\cdots\wedge\frac{dh_n}{h_n}. \end{align} $$

Here, one can think of (2.25) as an element of

$$\begin{align*}\Omega^1_{B_K} {\otimes} {\mathrm{Fil}}^nH^n_{{\mathrm{d\hspace{-0.2pt}R}}}(U_K/S_K) =\Omega_{B_K}^1{\otimes}{\varGamma}(X_K,\Omega^n_{X_K/S_K}(\log D_K)) \end{align*}$$

in the following way. Let ${\widetilde {\Omega }}^{\bullet }_{X_K}(\log D_K):=\Omega ^{\bullet }_{X_K}(\log D_K)/{{\mathrm {Im}}}(\Omega ^2_{S_K} {\otimes } \Omega ^{\bullet -2}_{X_K}(\log D_K))$ which fits into the exact sequence

$$\begin{align*}0{\longrightarrow} \Omega_{S_K}^1{\otimes}\Omega^{i-1}_{X_K/S_K}(\log D_K) {\longrightarrow} {\widetilde{\Omega}}^i_{X_K}(\log D_K){\longrightarrow} \Omega^i_{X_K/S_K}(\log D_K){\longrightarrow}0. \end{align*}$$

We think (2.25) of being an element of ${\varGamma }(X_K,{\widetilde {\Omega }}^{n+1}_{X_K}(\log D_K))$ . However, since ${\mathrm {Fil}}^{n+1}H^{n+1}_{{\mathrm {d\hspace {-0.2pt}R}}}(U_K/S_K)=0$ by the assumption, it turns out to be an element of

$$\begin{align*}{\varGamma}(X_K,\Omega_{S_K}^1{\otimes}\Omega^n_{X_K/S_K}(\log D_K)) = \Omega^1_{B_K}{\otimes}{\varGamma}(X_K,\Omega^n_{X_K/S_K}(\log D_K)). \end{align*}$$

Proof. In this proof, we omit to write the symbol “ $\wedge $ ”.

First, we describe the extension $[\xi ]_{U/S}$ . Let

$$\begin{align*}0{\longrightarrow} H^n_{{\mathrm{d\hspace{-0.2pt}R}}}(U/S)(n+1){\overset{\iota}{{\longrightarrow}}} M_\xi(U/S){\longrightarrow} {\mathscr{O}}_S{\longrightarrow}0 \end{align*}$$

be the extension $[\xi ]_{U/S}$ in ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma )$ . Let $e_\xi \in {\mathrm {Fil}}^0M_\xi (U/S)_{{\mathrm {d\hspace {-0.2pt}R}}}$ be the unique lifting of $1\in {\mathscr {O}}(S_K)$ . Then,

(2.27) $$ \begin{align} \iota (D_{U/S}(\xi))=\nabla(e_\xi),\quad \iota({\mathrm{reg}}_{U/S}^{(\sigma)}(\xi))=(1-\Phi)e_\xi \end{align} $$

by definition (see also (2.11)), where $\nabla $ and $\Phi $ are the data in $M_\xi (U/S)\in {{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma )$ .

We first write down the term $M_\xi (U/S)$ explicitly. Write $l_i:=p^{-1}\log (h_i^p/h_i^{\varphi })$ . Let $\{e_{i,0},e_{i,-2}\}$ be the basis of ${{\mathscr {L}{og}}}(h_i)(i)$ such that ${\mathrm {Fil}}^{-i}{\mathop {\mathscr {L}\!\!\mathit {o}\mathit {g}}\nolimits }(h_i)(i)_{{\mathrm {d\hspace {-0.2pt}R}}}=A_Ke_{i,0}$ and

$$\begin{align*}\nabla(e_{i,0})=\frac{dh_i}{h_i} e_{i,-2},\quad \Phi(e_{i,0})=p^{-i}e_{i,0}-p^{-i}l_ie_{i,-2},\quad \Phi(e_{i,-2})=p^{-i-1}e_{i,-2}.\end{align*}$$

Recall the $(n+1)$ -extension

$$\begin{align*}0\to {\mathscr{O}}_{U}(n+1)\to {{\mathscr{L}{og}}}(h_n)(n)\to\cdots\to{{\mathscr{L}{og}}}(h_1)(1)\to{{\mathscr{L}{og}}}(h_0)\to{\mathscr{O}}_{U}\to0. \end{align*}$$

Let $(T^{\bullet }_{A_K/B_K},D)$ be the total complex of the double complex $\Omega ^{\bullet }_{A_K/B_K}{\otimes } {{\mathscr {L}{og}}}(h_\star )_{{\mathrm {d\hspace {-0.2pt}R}}}$ . In a more down-to-earth manner, we have

$$\begin{align*}T^q_{A_K/B_K}:=\bigoplus_{i=0}^d\Omega^i_{A_K/B_K}{\otimes}{{\mathscr{L}{og}}}(h_{i-q})(i-q)_{{\mathrm{d\hspace{-0.2pt}R}}},\quad q\in{\mathbb{Z}}, \end{align*}$$

where we denote ${{\mathscr {L}{og}}}(h_j)(j):=0$ if $j<0$ or $j>n$ , and the differential $D:T^q\to T^{q+1}$ is defined by

$$\begin{align*}D(\omega^i{\otimes} x_j)=d\omega^i{\otimes} x_j+(-1)^i\omega^i\wedge \nabla(x_j)+(-1)^i\omega^i{\otimes} \pi(x_j) \end{align*}$$

for $\omega ^i{\otimes } x_j\in \Omega ^i_{A_K/B_K}{\otimes }{{\mathscr {L}{og}}}(h_j)(j)_{{\mathrm {d\hspace {-0.2pt}R}}}$ , where $\pi \colon {{\mathscr {L}{og}}}(h_i)(i)\to {{\mathscr {L}{og}}}(h_{i-1})(i-1)$ is the composite of the projection ${{\mathscr {L}{og}}}(h_i)(i)\to A_Ke_{i,0}$ and the injection $A_Ke_{i,0}\cong A_Ke_{i-1,-2}\hookrightarrow {{\mathscr {L}{og}}}(h_{i-1})(i-1)$ defined by $e_{i,0}\mapsto e_{i-1,-2}$ . We have the exact sequence

(2.28)

where the first arrow is induced from ${\mathscr {O}}_{U_K}\cong {\mathscr {O}}_{U_K}e_{d,-2}{\hookrightarrow } {{\mathscr {L}{og}}}(h_d)_{{\mathrm {d\hspace {-0.2pt}R}}}$ , the second arrow is induced from the projection ${{\mathscr {L}{og}}}(h_0)_{{\mathrm {d\hspace {-0.2pt}R}}}\to {\mathscr {O}}_{U_K}e_{0,0}\cong {\mathscr {O}}_{U_K}$ and the differential on $\Omega ^{\bullet +n}_{A_K/B_K}$ is the usual differential operator d (not $(-1)^nd$ ). For the rigid part, let $(T_{A^{\dagger }_K/B^{\dagger }_K}^{\bullet },D)$ be defined in the same way by replacing $\Omega ^{\bullet }_{A_K/B_K}$ with $\Omega ^{\bullet }_{A^{\dagger }_K/B^{\dagger }_K}$ (see Remark 2.21 (3)). Then, we also have an exact sequence corresponding to (2.28). Now, we have a description

$$\begin{align*}M_\xi(U/S)_{{\mathrm{d\hspace{-0.2pt}R}}}=H^0(T^{\bullet}_{A_K/B_K}),\quad M_\xi(U/S)_{\mathrm{rig}}=H^0(T^{\bullet}_{A^{\dagger}_K/B^{\dagger}_K}) \end{align*}$$

(the description of the rigid part follows from the exact sequence (2.28) on two sides with Remark 2.21 (3)), and we have an exact sequence

in ${{\mathrm {Fil}\text {-}F\text {-}\mathrm {MIC}}}(S,\sigma )$ .

Before going to the proof of Theorem 2.25, we give an explicit description of $e_\xi $ in (2.27). Put

$$\begin{align*}\omega^0:=1,\qquad \omega^i:=\frac{dh_0}{h_0}\frac{dh_1}{h_1}\cdots \frac{dh_{i-1}}{h_{i-1}}\in \Omega^i_{A_K/B_K}\quad (1\leq i\leq n+1). \end{align*}$$

However, we note that $\omega ^{n+1}=\omega ^n\wedge \frac {dh_n}{h_n}=0$ as $\omega ^n \in {\mathrm {Fil}}^{n+1}H^{n+1}_{{\mathrm {d\hspace {-0.2pt}R}}}(U_K/S_K)=0$ . Put

$$\begin{align*}e^{\prime}_\xi=\sum_{i=0}^{n}\omega^ie_{i,0}\in T^0_{A_K/B_K}. \end{align*}$$

It is a direct computation to show $D(e^{\prime }_\xi )=0$ , so that one has