2.1.1 Rings of periods

We start with reviewing basic facts concerning the rings of periods. Consider the ring $R={\buildrel {\lim } \over \longleftarrow} {\mathcal O} _{\overline {K}}/p{\mathcal O} _{\overline {K}}$, where the maps in the projective system are the $p$th power maps. With addition and multiplication defined coordinate-wise, $R$ is a ring of characteristic $p$. Take its ring of Witt vectors $W(R)$. Then $\mathbf {A}_{\operatorname {cr} }$ is the $p$-adic completion of the divided power envelope $D_\xi (W(R))$ of the ideal $\xi W(R)$ in $W(R)$. Here $\xi =[p^{\flat }]-p$ and, for $x \in R$, $[x]=[x,0,0,\ldots ] \in W(R)$ is its Teichmüller representative.

(1) The rings $\mathbf {B}_{\operatorname {cr} }$ and $\mathbf {B}_{\operatorname {dR} }$. The ring $\mathbf {A}_{\operatorname {cr} }$ is a topological $W(k)$-module having the following properties:

1. (i) $W(\overline {k})$ is embedded as a subring of $\mathbf {A}_{\operatorname {cr} }$ and $\sigma$ extends naturally to a Frobenius $\phi$ on $\mathbf {A}_{\operatorname {cr} }$;

2. (ii) $\mathbf {A}_{\operatorname {cr} }$ is equipped with a decreasing separated filtration $F^{n}\mathbf {A}_{\operatorname {cr} }$, where $F^{n}\mathbf {A}_{\operatorname {cr} }$ is the closure of the $n$th divided power of the PD ideal of $D_\xi (W(S))$; for $n < p$, $\phi (F^{n}\mathbf {A}_{\operatorname {cr} })\subset p^{n}\mathbf {A}_{\operatorname {cr} }$;

3. (iii) $G_K$ acts on $\mathbf {A}_{\operatorname {cr} }$; the action is $W(\overline {k})$-semilinear, continuous, commutes with $\phi$, and preserves the filtration;

4. (iv) there exists an element $t\in F^1\mathbf {A}_{\operatorname {cr} }$ such that $\phi (t)=pt$ and $G_K$ acts on $t$ via the cyclotomic character: if we fix $\varepsilon \in R$, a sequence of nontrivial $p$-roots of unity, then $t=\log ([\varepsilon ])$.

The rings $\mathbf {B}_{\operatorname {cr} }^+$ and $\mathbf {B}_{\operatorname {cr} }$ are defined as the rings $\mathbf {A}_{\operatorname {cr} }[p^{-1}]$ and $\mathbf {A}_{\operatorname {cr} }[p^{-1},t^{-1}]$, respectively, with the induced topology, filtration, Frobenius, and the Galois action. For us, in this paper, it will be essential that the ring $\mathbf {A}_{\operatorname {cr} }$ can be thought of as a cohomology of an ‘arithmetic point’, namely that

\[ \mathbf{A}_{\operatorname{cr} ,n}\simeq H_{\operatorname{cr} }^*(\operatorname{Spec} ({\mathcal O}_{\overline{K},n})), \]

where, for a scheme $Y$ over $W(k)$, we set

\[ \mathrm{R}\Gamma_{\operatorname{cr} }(Y_n):=\mathrm{R}\Gamma_{\operatorname{cr} }(Y_n/W_n(k)),\quad H^*_{\operatorname{cr} }(Y):=H^*_{\operatorname{cr} }(Y/W(k)):=H^*\operatorname{holim} _n\mathrm{R}\Gamma_{\operatorname{cr} }(Y_n). \]

The canonical morphism $\mathbf {A}_{\operatorname {cr} ,n}\to {\mathcal O} _{\overline {K}}/p^n$ is surjective. Let $J_{\operatorname {cr} ,n}$ denote its kernel. Let

\[ \mathbf{B}^+_{\operatorname{dR} }={\mathrel{\mathop{\kern0pt\longleftarrow}\limits_r^{\lim }}}\Big(\mathbf{Q}\otimes {\mathrel{\mathop{\kern0pt\longleftarrow}\limits_n^{\lim }}} \mathbf{A}_{\operatorname{cr} ,n}/ J_{\operatorname{cr} ,n}^{[r]}\Big),\quad \mathbf{B}_{\operatorname{dR} }=\mathbf{B}^+_{\operatorname{dR} }[t^{-1}]. \]

The ring $\mathbf {B}_{\operatorname {dR} }^+$ has a discrete valuation given by powers of $t$. Its quotient field is $\mathbf {B}_{\operatorname {dR} }$. We will denote by $F^n\mathbf {B}^+_{\operatorname {dR} }$ the filtration induced on $\mathbf {B}^+_{\operatorname {dR} }$ by powers of $t$.

(2) The rings $\mathbf {B}_{\operatorname {st}}$ and $\widehat {\mathbf {B}}_{\operatorname {st}}$. Let us now recall the definition of the ring $\mathbf {B}_{\operatorname {st}}$ [Reference FontaineFon94]. Set $\mathbf {B}^+_{\operatorname {st}}:=\mathbf {B}^+_{\operatorname {cr} }[u]$, $\phi (u)=pu$, and $Nu=-1$. Let $\pi$ be a uniformizer of ${{\mathcal O} _K}$ (which we will fix in the rest of the paper). Let $\iota =\iota _{\pi }:\mathbf {B}^+_{\operatorname {st}}\hookrightarrow \mathbf {B}^+_{\operatorname {dR} }$ denote the embedding $u\mapsto u_{\pi }=\log ([\pi ^{\flat }]/\pi )$. We use it to induce the Galois action on $\mathbf {B}^+_{\operatorname {st}}$ from the one on $\mathbf {B}^+_{\operatorname {dR} }$. Let $\mathbf {B}_{\operatorname {st} }=\mathbf {B}_{\operatorname {cr} }[u_{\pi }]$.

We will need the following crystalline interpretation of the ring $\mathbf {B}^+_{\operatorname {st}}$ (see [Reference KatoKat94a, Reference TsujiTsu99a]). Let $R_{\pi ,n}$ denote the PD envelope of the ring $W_n(k)[x]$ with respect to the closed immersion $W_n(k)[x]\to {{\mathcal O} _{K,n}}$, $x\mapsto \pi$, equipped with the log-structure associated to $\mathbf {N}\to R_{\pi ,n},$ $1\mapsto x$. Set $R_{\pi }:={\mathrel{\mathop{\kern0pt\longleftarrow}\limits_n^{\lim }}}R_{\pi ,n}$. Let

\[ \widehat{\mathbf{A}}^+_{\operatorname{st}}={\mathrel{\mathop{\kern0pt\longleftarrow}\limits_n^{\lim }}}H^0_{\operatorname{cr} }(\operatorname{Spec} ({\mathcal O}_{\overline{K},n})/R_{\pi,n}),\quad \widehat{\mathbf{B}}^+_{\operatorname{st}}:=\widehat{\mathbf{A}}^+_{\operatorname{st}} [1/p]. \]

The ring $\widehat {\mathbf {B}}^+_{\operatorname {st}}$ has a natural action of $G_K$, Frobenius $\phi$, and a monodromy operator $N$. Kato showed [Reference KatoKat94a, 3.7] that the ring $\mathbf {B}^+_{\operatorname {st}}$ is canonically (and compatibly with all the structures) isomorphic to the subring of elements of $\widehat {\mathbf {B}}^+_{\operatorname {st}}$ annihilated by a power of the monodromy operator $N$. The map $\iota : \mathbf {B}^+_{\operatorname {st}}\to \mathbf {B}^+_{\operatorname {dR} }$ extends naturally to a map $\iota : \widehat {\mathbf {B}}^+_{\operatorname {st}}\to \mathbf {B}^+_{\operatorname {dR} }$.

2.1.2 Syntomic cohomology

We will recall briefly the definition of syntomic cohomology. For a log-scheme $X$, we denote by $X_{\operatorname {syn}}$ the small log-syntomic site of $X$. For a log-scheme $X$ log-syntomic over $\operatorname {Spec} (W(k))$, define

\[ {\mathcal O}^{\operatorname{cr} }_n(X) =H^0_{\operatorname{cr} }(X_n,{\mathcal O}_{X_n}),\quad {\mathcal{J}}_n^{[r]}(X) =H^0_{\operatorname{cr} }(X_n,{\mathcal{J}}^{[r]}_{X_n}), \]

where ${\mathcal O} _{X_n}$ is the structure sheaf of the absolute log-crystalline site (i.e., over $W_n(k)$), ${\mathcal {J}}_{X_n}=\operatorname {Ker} ({\mathcal O} _{X_n/W_n(k)}\to {\mathcal O} _{X_n})$, and ${\mathcal {J}}^{[r]}_{X_n}$ is its $r$th divided power of ${\mathcal {J}}_{X_n}$. Set ${\mathcal {J}}^{[r]}_{X_n}={\mathcal O} _{X_n}$ if $r\leq 0$. There is a canonical, compatible with Frobenius, and functorial isomorphism

\[ H^*(X_{\operatorname{syn}},{\mathcal{J}}_n^{[r]})\simeq H^*_{\operatorname{cr} }(X_n,{\mathcal{J}}^{[r]}_{X_n}). \]

It is easy to see that $\phi ({\mathcal {J}}_n^{[r]} )\subset p^r{\mathcal O} ^{\operatorname {cr} }_n$ for $0\leq r\leq p-1$. This fails in general and we modify ${\mathcal {J}}_n^{[r]}$:

\[ {\mathcal{J}}_n^{\langle r\rangle }:= \{x\in {\mathcal{J}}_{n+s}^{[r]}\mid \phi(x)\in p^r{\mathcal O}^{\operatorname{cr} }_{n+s}\}/p^n \]

for some $s\geq r$. This definition is independent of $s$. We can define the divided Frobenius

\[ \phi_r=\text{``}\phi/p^r\text{''}: {\mathcal{J}}_n^{\langle r\rangle } \to {\mathcal O}^{\operatorname{cr} }_n. \]

Set

\[ {\mathcal{S}}_n(r):=\operatorname{Cone}({\mathcal{J}}_n^{\langle r\rangle } \stackrel{1-\phi_r}{\longrightarrow}{\mathcal O}^{\operatorname{cr} }_n)[-1]. \]

We will write ${\mathcal {S}}_n(r)$ for the syntomic sheaves on $X_{m,\operatorname {syn}}$, $m\geq n$, as well as on $X_{\operatorname {syn}}$. We will also need the ‘undivided’ version of syntomic complexes of sheaves:

\[ {\mathcal{S}}'_n(r):=\operatorname{Cone}({\mathcal{J}}_n^{[r]} \stackrel{p^r-\phi}{\longrightarrow}{\mathcal O}^{\operatorname{cr} }_n)[-1]. \]

The natural map ${\mathcal {S}}^{\prime }_n(r)\to {\mathcal {S}}_n(r)$ induced by the maps $p^{r}: {\mathcal {J}}_n^{[r]}\to {\mathcal {J}}_n^{\langle r\rangle }$ and $\operatorname {Id} : {\mathcal O} ^{\operatorname {cr} }_n \to {\mathcal O} ^{\operatorname {cr} }_n$ has kernel and cokernel killed by $p^{r}$. We will also write ${\mathcal {S}}_n(r)$, ${\mathcal {S}}^{\prime }_n(r)$ for $\mathrm {R} \varepsilon _*{\mathcal {S}}_n(r)$, $\mathrm {R} \varepsilon _*{\mathcal {S}}^{\prime }_n(r)$, respectively, where $\varepsilon : X_{n,\operatorname {syn}}\to X_{n,\operatorname {\acute {e}t} }$ is the canonical projection to the étale site.

The $p$-adic syntomic cohomology of $X$ is defined as

\[ \mathrm{R}\Gamma_{\operatorname{\acute{e}t} }(X,{\mathcal{S}}(r)):=\operatorname{holim} _n\mathrm{R}\Gamma_{\operatorname{\acute{e}t} }(X,{\mathcal{S}}_n(r)),\quad \mathrm{R}\Gamma_{\operatorname{\acute{e}t} }(X,{\mathcal{S}}^{\prime}(r)):=\operatorname{holim} _n\mathrm{R}\Gamma_{\operatorname{\acute{e}t} }(X,{\mathcal{S}}^{\prime}_n(r)). \]

2.1.3 Cohomology with compact support

Let $X$ be a finite and saturated log-smooth log-scheme over ${\mathcal O} _K^{\times }$ (respectively over ${{\mathcal O} _K}$). Since $X$ is log-regular it is normal and the maximal open subset $U=X_{\operatorname {tr}}\subset X$, where the log-structure $M_X$ is trivial, is dense in $X$. We have $M_X={\mathcal O} _X\cap j_*{\mathcal O} _U^*$, where $j:U\hookrightarrow X$ is the open immersion. By [Reference NiziołNiz06a, Theorem 5.10], there exists a log-blow-up of $X$ that has Zariski log-structure and is (classically) regular.

Assume that $X$ itself has these properties. Then $U$ is a complement of a divisor with simple normal crossings that is a union $D_0\cup D$ (respectively $D$) of the reduced special fiber and the horizontal part $D$. The scheme $X$ has generalized semistable reduction, i.e., Zariski locally on $X$, there exists an étale morphism over ${{\mathcal O} _K}$:

\[ X\to \operatorname{Spec} ({{\mathcal O}_K}[T_1,\ldots,T_u]/(T_1^{n_1}\cdots T_u^{n_u}-\pi)[U_1,\ldots,U_m,V_{1},\ldots,V_t]) \]

for some integers $u\geq 1$ (respectively $u=0$), $m,t\geq 0, n_i>0$. The divisor $D$ is the inverse image of $U_1\cdots U_m=0$. In particular, all the closed strata of $D$ are log-smooth over ${\mathcal O} _K^{\times }$ and regular (respectively smooth over ${{\mathcal O} _K}$). If all $n_i=1$, we say that $X$ has semistable reduction.

Take $X$ as above with semistable reduction. Recall the following definitions. The $p$-adic étale cohomology of $X_{\overline {K}}$ with compact support.Footnote ²

\[ \mathrm{R}\Gamma_{\operatorname{\acute{e}t} ,c}(X_{\overline{K}},\mathbf{Q}_p)=\mathrm{R}\Gamma_{\operatorname{\acute{e}t} }(X_{\overline{K}},\overline{j}_{K!}\mathbf{Q}_p). \]

The de Rham cohomology of $X_K$ with compact support [Reference TsujiTsu99b, Definition 3.2]

\[ \mathrm{R}\Gamma_{\operatorname{dR} ,c}(X_K)=\mathrm{R}\Gamma(X_K,{\mathcal{I}}_{D_K}\Omega^{\scriptscriptstyle\bullet}_{X_K}), \]

where ${\mathcal {I}}_{D_K}\subset j_{K*}{\mathcal O} ^*_{U_K}\cap {\mathcal O} _{X_K}$ is the ideal of ${\mathcal O} _{X_K}$ corresponding to $D_K$. We filter it by

\[ F^r\mathrm{R}\Gamma_{\operatorname{dR} ,c}(X_K)=\mathrm{R}\Gamma(X_K,{\mathcal{I}}_{D_K}\Omega^{\geq r}_{X_K}), \quad r\in\mathbf{Z}. \]

The crystalline cohomology of $X_0$ over $W(k)^0$ with compact support [Reference TsujiTsu99b, Definition 5.4]

\[ \mathrm{R}\Gamma_{\operatorname{cr} ,c}(X_0/W(k)^0)=\mathrm{R}\Gamma_{\operatorname{cr} }(X_0/W(k)^0,{\mathcal{K}}_{D_0}), \]

where ${\mathcal {K}}_{D_0}$ is an ideal sheaf induced by the sheaf ${\mathcal {I}}_{D_0}$ [Reference TsujiTsu99b, Lemma 5.3]. The crystalline cohomology $\mathrm {R}\Gamma _{\operatorname {cr} ,c}(X)$ is defined in a similar way. We filter it by setting $F^r\mathrm {R}\Gamma _{\operatorname {cr} ,c}(X)=\mathrm {R}\Gamma _{\operatorname {cr} }(X,{\mathcal {K}}_{D_0}{\mathcal {J}}^{[r]}_X)$, $r\in \mathbf {Z}$. This allows us to define the syntomic cohomology with compact support $\mathrm {R}\Gamma _{\operatorname {syn},c}(X,{\mathcal {S}}_n(r))$ and $\mathrm {R}\Gamma _{\operatorname {syn},c}(X,{\mathcal {S}}^{\prime }_n(r))$.

The above cohomologies with compact support are special cases of cohomologies of finite simplicial schemes. Define $C(X,D):=\operatorname {cofiber}(\widetilde {D}_{{\scriptscriptstyle \bullet }}\stackrel {i_*}{\to } X)$, where $\widetilde {D}_{{\scriptscriptstyle \bullet }}$ is the Čech nerve of the map $\coprod _i D_i\to D$, $D_i$ being an irreducible component of $D$. The log-structure on the schemes in $C(X,D)$ is trivial if $X$ is over ${\mathcal O} _K$ and induced from the special fiber if $X$ is over ${\mathcal O} _K^{\times }$.

Lemma 2.1 Let $\mathrm {R}\Gamma (X)$ denote one of the cohomologies mentioned above. We have a natural (filtered) quasi-isomorphism

\[ \mathrm{R}\Gamma_c(X)\simeq \mathrm{R}\Gamma(C(X,D)). \]

It is compatible with products.Footnote ³

Proof. The étale and de Rham cases follow immediately from the following exact sequences ($r\in \mathbf {Z}$):

(2.2)\begin{equation} \begin{gathered} 0\to \overline{j}_{K!}\mathbf{Q}_p\to \mathbf{Q}_{p,X_{\overline{K}}} \to\overline{ i}_{1*}\mathbf{Q}_{p,D^1_{\overline{K}}}\to \overline{i}_{2*}\mathbf{Q}_{p,D^2_{\overline{K}}}\to \cdots,\\ 0\to {\mathcal{I}}_{D_K}\Omega^{\geq r}_{X_K}\to \Omega^{\geq r}_{X_K}\to i_{1*}\Omega^{\geq r}_{D^{1}_K}\to i_{2*}\Omega^{\geq r}_{D^2_K}\to \cdots. \end{gathered} \end{equation}

Here $D^m:=\widetilde {D}_m$ is the direct sum of the intersections of $m$ irreducible components of $D$. We note that $C(X,D)_{\overline {K}}\simeq C(X_{\overline {K}},D_{\overline {K}})$ even if $(X,D)$ is not geometrically irreducible.

The crystalline case over $W(k)^0$ follows from a mixed characteristic analog of the second sequence. And the case over $W(k)$ reduces to this sequence as well. Indeed, if ${\mathcal O} _K=W(k)$, this is clear. In general, locally, we have an embedding into such a situation. Because, by assumption, this embedding is regular, the above-mentioned sequence remains exact after tensoring with the divided power envelope and computes cohomology with compact support.

For the syntomic case, it suffices to check that the above crystalline quasi-isomorphism preserves filtrations. But this follows easily from the fact that the associated grading of the filtration on the divided power envelope is free over ${\mathcal O} _X$.

Concerning compatibility with products, the étale, the de Rham, and the crystalline cases are immediate from the expressions (2.2). In the syntomic case, compatibility follows from the fact that syntomic cohomology is defined as a mapping fiber of (filtered) crystalline cohomology and the syntomic product is the mapping fiber product induced from the crystalline product.

2.1.4 Fontaine–Lafaille theory

The main reference for this section is [Reference Fontaine and LafailleFL82]. Assume first that ${{\mathcal O} _K}=W(k)$. For the integral crystalline theory (Fontaine–Laffaille theory), we will need the following abelian categories:

1. (i) $\mathcal {M }{\mathcal {F}}_{\rm big}({{\mathcal O} _K})$: an object is given by a $p$-torsion ${{\mathcal O} _K}$-module $M$ and a family of $p$-torsion ${{\mathcal O} _K}$-modules $F^{i}M$ together with ${{\mathcal O} _K}$-linear maps $F^{i}M\rightarrow F^{i-1}M,$ $F^{i}M\rightarrow M$, and $\sigma$-semilinear maps $\phi _{i} : F^{i}M\rightarrow M$ satisfying certain compatibility conditions;

2. (ii) ${\mathcal {M}}{\mathcal {F}} ({{\mathcal O} _K})$: the full subcategory of ${\mathcal {M}}{\mathcal {F}} _{\rm big}({{\mathcal O} _K})$ with objects: finite ${{\mathcal O} _K}$-modules $M$ such that $F^{i}M=0\text { for } i\gg 0$, the maps $F^{i}(M)\rightarrow M$ are injective and $\sum \text {Im}\, \phi _{i} = M$;

3. (iii) ${\mathcal {M}}{\mathcal {F}} _{[a,b]}({{\mathcal O} _K})$: the full subcategory of objects $M$ of ${\mathcal {M}}{\mathcal {F}} ({{\mathcal O} _K})$ such that $F^{a}M=M$ and $F^{b+1}M=0$.

Consider the category ${\mathcal {M}}{\mathcal {F}} _{[a,b]}({{\mathcal O} _K})\text { with } b-a\leq p-2$. There exists an exact and fully faithful functor

\[ \mathbf{L}(M)=\ker(F^0(M\otimes \mathbf{A}_{\operatorname{cr} }\{-b\}(-b)) {\buildrel {1-\phi_0} \over \longrightarrow} M\otimes \mathbf{A}_{\operatorname{cr} }(-b)), \]

where $\{-b\}$ and $(-b)$ are the ${\mathcal {M}}{\mathcal {F}}$ and Tate twistsFootnote ⁴, respectively, from ${\mathcal {M}}{\mathcal {F}} _{[a,b]}({{\mathcal O} _K})$ to finite ${\mathbf {Z}}_{p}$-Galois representations. Its essential image is called the category of crystalline representations of weight between $a$ and $b$. This category is closed under taking tensor products and duals (assuming that we stay in the admissible range of the filtration).

The following proposition generalizes [Reference Fontaine, Messing and RibetFM87, 2.7], Faltings [Reference Faltings and IgusaFal89, 4.1], and [Reference NiziołNiz09, Lemma 2.3] from schemes to finite simplicial schemes.

Proposition 2.3 Let $X$ be a smooth and proper $m$-truncated simplicial scheme over ${{\mathcal O} _K}=W(k)$ whose components have dimension smaller than $d$. Then, for $d\leq p-2$ or for $i\leq p-2$, the filtered Frobenius module $H^i_{\operatorname {cr} }(X_n)$ lies in ${\mathcal {M}}{\mathcal {F}}_{[0,d]} ({{\mathcal O} _K})$ or ${\mathcal {M}}{\mathcal {F}}_{[0,i]} ({{\mathcal O} _K})$, respectively. Moreover, then the natural morphism

\[ \psi_n: H^i_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}_n(r))\stackrel{\sim}{\to} {\mathbf{L}}(H^i_{\operatorname{cr} }(X_n)\{-r\}) \simeq F^rH^i_{\operatorname{cr} }(X_{{{\mathcal O}_{\overline{K}}},n})^{\phi_r=1} \]

is an isomorphism for $p-2\geq r\geq d$ or for $0\leq i\leq r\leq p-2$, respectively.

Here

\[ H^i_{\operatorname{cr} }(X_n)\simeq H^i_{\operatorname{dR} }(X_n/{{\mathcal O}_{K,n}}):=H^i(X_n,\Omega^{\scriptscriptstyle\bullet}_{X_n/{{\mathcal O}_{K,n}}}) \]

and the maps

\[ \phi_k=\text{``}\phi/p^k\text{''}: F^kH^i_{\operatorname{cr} }(X_n)\to H^i_{\operatorname{cr} }(X_n), \]

where $\phi$ denotes the crystalline Frobenius. The Hodge filtration is

\[ F^kH^i_{\operatorname{cr} }(X_n)\simeq \operatorname{Im}(H^i(X_n,\Omega^{\geq k}_{X_n/{{\mathcal O}_{K,n}}})\to H^i(X_n,\Omega^{\scriptscriptstyle\bullet}_{X_n/{{\mathcal O}_{K,n}}})) \]

since the Hodge–de Rham spectral sequence of $X_n$ degenerates: by devissage, we can reduce to $n=1$ and then it follows from the results of Deligne and Illusie [Reference Deligne and IllusieDI87, Corollary 3.7].

Proof. The proof of [Reference Fontaine, Messing and RibetFM87, 2.7] for schemes goes through for truncated simplicial schemes, proving the first claim of the proposition. For the second claim, we argue by induction on $m\geq 0$ such that $X\simeq \operatorname {sk} _mX$. The case of $m=0$ is treated in [Reference Fontaine, Messing and RibetFM87, 2.7]. Assume that our proposition is true for $m-1$. To show it for $m$, consider the homotopy cofiber sequence

\[ \operatorname{sk} _{m-1}X_{{\mathcal O}_{\overline{K}}}\to \operatorname{sk} _mX_{{\mathcal O}_{\overline{K}}}\to \operatorname{sk} _{m}X_{{\mathcal O}_{\overline{K}}}/\operatorname{sk} _{m-1}X_{{\mathcal O}_{\overline{K}}} \]

and apply the maps $\psi _n$ to it. We get the following map of sequences.

Here we set $H^*_{\operatorname {syn}}(Y)=H^*_{\operatorname {\acute {e}t} }(Y_{{\mathcal O} _{\overline {K}}}, {\mathcal {S}}_n(r))$, ${\mathbf {L}}(H^*_{\operatorname {cr} }(Y))={\mathbf {L}}(H^*_{\operatorname {cr} }(Y_n)\{-r\})$. We also put

\[ H^{*}_{\alpha}(X^{\prime}_{m},*) =H^{*}_{\alpha}(X_{m},*)\cap\ker s_0^*\cap\cdots\cap\ker s_{m-1}^*,\quad \alpha=\operatorname{syn},\operatorname{cr} , \]

where each $s_i: X_{m-1}\to X_m$ is a degeneracy map. The top sequence is exact. So is the bottom: it is clearly exact before applying ${\mathbf {L}}$ and it stays exact because the relevant categories ${\mathcal {M}}{\mathcal {F}}$ are closed under taking subobjects and the functor ${\mathbf {L}}$ is exact.

By the inductive hypothesis, we have the isomorphisms shown. It follows that the map

\[ \psi_n: H^i_{\operatorname{\acute{e}t} }(\operatorname{sk} _{m}X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}_n(r)) \to {\mathbf{L}}(H^*_{\operatorname{cr} }(\operatorname{sk} _mX_n)\{-r\}) \]

is an isomorphism as well. Since $H^i_{\operatorname {\acute {e}t} }(\operatorname {sk} _{m}X_{{\mathcal O} _{\overline {K}}},{\mathcal {S}}_n(r))\stackrel {\sim }{\to }H^i_{\operatorname {\acute {e}t} }(X_{{\mathcal O} _{\overline {K}}},{\mathcal {S}}_n(r))$ and $H^*_{\operatorname {cr} }(\operatorname {sk} _mX_n)\stackrel {\sim }{\to }H^*_{\operatorname {cr} }(X_n)$, we are done.

The above proposition can be applied to cohomology with compact support.

Corollary 2.4 Let $X$ be a smooth and proper scheme over ${{\mathcal O} _K}=W(k)$ with a divisor $D$ that has relative simple normal crossings and all the closed strata smooth over ${{\mathcal O} _K}$. Equip $X$ with the log-structure coming from $D$. Then, if the relative dimension $d$ of $X$ is $\leq p-2$ or if $i\leq p-2$, the filtered Frobenius module $H^i_{\operatorname {cr} }(X_n)$ lies in ${\mathcal {M}}{\mathcal {F}}_{[0,d]} ({{\mathcal O} _K})$ or ${\mathcal {M}}{\mathcal {F}}_{[0,i]} ({{\mathcal O} _K})$, respectively. Moreover, then the natural morphism

\[ \psi_n: H^i_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}_n(r))\stackrel{\sim}{\to} {\mathbf{L}}(H^i_{\operatorname{cr} ,c}(X_n)\{-r\}) \simeq F^rH^i_{\operatorname{cr} ,c}(X_{n})^{\phi_r=1} \]

is an isomorphism for $p-2\geq r\geq d$ or for $0\leq i\leq r\leq p-2$, respectively.

Proof. By Lemma 2.1, we have a canonical isomorphism

\[ H^i_{\operatorname{cr} ,c}(X_n)\simeq H^i_{\operatorname{cr} }(C(X,D)_n). \]

Our corollary follows now from Proposition 2.3.

2.1.5 More cohomological identities

Let ${\mathcal O} _K$ be general and let $X$ be an ${{\mathcal O} _K}$-scheme. Recall that, if $X$ is smooth and proper, Kato and Messing [Reference Kato and MessingKM92] have constructed the following isomorphisms:

\begin{align*} h_{\operatorname{cr}}:\ &H ^i_{\operatorname{cr} } (X_{0})_\mathbf{Q} \otimes \mathbf{B}^+_{\operatorname{cr} } \stackrel{\sim}{\to} H^i_{\operatorname{cr} }(X_{{\mathcal O}_{\overline{K}}})_{\mathbf{Q}} \quad [\text{KM92}, 1.2],\\ &H^i_{\operatorname{dR} } (X_K) \otimes \mathbf{B}^+_{\operatorname{dR} } \simeq {\mathrel{\mathop{\kern0pt\longleftarrow}\limits_N^{\lim }}}\Bigl({\mathrel{\mathop{\kern0pt\longleftarrow}\limits_n^{\lim }}} H^i_{\operatorname{cr} }(X_{{\mathcal O}_{\overline{K}},n},{\mathcal O}_{n}/J_n^{[N]})\Bigr)_{\mathbf{Q}} \quad [\text{KM92}, 1.4],\\ h_{\operatorname{dR} }:\ &F^r (H^i_{\operatorname{dR} }(X_K) \otimes \mathbf{B}^+_{\operatorname{dR} }) \stackrel{\sim}{\to} {\mathrel{\mathop{\kern0pt\longleftarrow}\limits_N^{\lim }}}\Bigl({\mathrel{\mathop{\kern0pt\longleftarrow}\limits_n^{\lim }}} H^i_{\operatorname{cr} }(X_{{\mathcal O}_{\overline{K},n}},J_n^{[r]}/J_n^{[N]})\Bigr)_{\mathbf{Q}}. \end{align*}

We will need also to know the following lemma [Reference NiziołNiz98, Lemma 2.2].

Lemma 2.5 The following two compositions of maps are equal:

\begin{align*} \mathbf{Q} &\otimes{\mathrel{\mathop{\kern0pt\longleftarrow}\limits_n^{\lim }}} H^i_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}_n'(r)) \to {\mathrel{\mathop{\kern0pt\longleftarrow}\limits_N^{\lim }}}\Bigl( \mathbf{Q}\otimes{\mathrel{\mathop{\kern0pt\longleftarrow}\limits_n^{\lim }}} H^i_{\operatorname{cr} }(X_{{\mathcal O}_{\overline{K}},n},J_n^{[r]}/J_n^{[N]})\Bigr) {\buildrel { h^{-1}_{\operatorname{dR} }} \over \longrightarrow} F^r(H^i_{\operatorname{dR} }(X_K) \otimes \mathbf{B}^+_{\operatorname{dR} })\\ &\quad \to H^i_{\operatorname{dR} }(X_K) \otimes \mathbf{B}^+_{\operatorname{dR} },\\ \mathbf{Q} &\otimes{\mathrel{\mathop{\kern0pt\longleftarrow}\limits_n^{\lim }}} H^i_{\operatorname{\acute{e}t} }( X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}_n'(r)) \to \mathbf{Q}\otimes{\mathrel{\mathop{\kern0pt\longleftarrow}\limits_n^{\lim }}} H^i_{\operatorname{cr} }(X_{{\mathcal O}_{\overline{K},n}}) {\buildrel { h^{-1}_{\operatorname{cr} } } \over \longrightarrow} H ^i _{\operatorname{cr} }(X_{0})\otimes_{W(k)} \mathbf{B}^+_{\operatorname{cr} } {\buildrel { \delta} \over \longrightarrow} H^i_{\operatorname{dR} }(X_K) \otimes \mathbf{B}^+_{\operatorname{dR} }, \end{align*}

where $\delta$ is induced by the Berthelot–Ogus isomorphism [Reference Berthelot and OgusBO83, 2.2] $H^i_{\operatorname {cr} }(X_{0})\otimes _{W(k)} K \simeq H^i_{\operatorname {dR} }(X_K)$.

Let $X$ be any fine log-scheme which is log-smooth and proper over ${\mathcal O} _K^{\times }$ with saturated log-structure on the generic fiber. We will need the crystalline interpretation of $\mathbf {B}^+_{\operatorname {dR} }\otimes _{K} H^i_{\operatorname {dR} }(X_K)$ from [Reference KatoKat94a] (see also [Reference TsujiTsu99a, 4.7]):

(2.6)\begin{equation} \begin{gathered} \mathbf{B}^+_{\operatorname{dR} }\otimes _{K} H^i_{\operatorname{dR} }(X_K)\stackrel{\sim}{\to} {\mathrel{\mathop{\kern0pt\longleftarrow}\limits_s^{\lim }}} H^{i}_{\operatorname{cr} }(X_{{\mathcal O}_{\overline{K}}}/ {\mathcal O}_K^{\times },{\mathcal O}/J^{[s]})_{\mathbf{Q}}\quad [\text{Tsu99a}, 4.7.6],\\ F^r(\mathbf{B}^+_{\operatorname{dR} }\otimes _{K} H^i_{\operatorname{dR} }(X_K))\stackrel{\sim}{\to} {\mathrel{\mathop{\kern0pt\longleftarrow}\limits_{s\geq r}^{\lim }}} H^{i}_{\operatorname{cr} }(X_{{\mathcal O}_{\overline{K}}}/ {\mathcal O}_K^{\times },J^{[r]}/J^{[s]})_{\mathbf{Q}}\quad [\text{Tsu99a}, 4.7.13]. \end{gathered} \end{equation}

Finally, let us recall briefly the Hyodo–Kato isomorphism. We define the Hyodo–Kato cohomology as

\[ H^i_{\operatorname{HK} }(X):= H^i_{\operatorname{cr} }(X_0/W(k)^0)_{\mathbf{Q}}. \]

If the special fiber of $X$ is of Cartier type, Kato defined [Reference KatoKat94a, 4.2 and 4.5] canonical morphisms (that however depend on the choice of $\pi$)

(2.7)\begin{equation} H^i_{\operatorname{cr} }(X_{{\mathcal O}_{\overline{K}}})_{\mathbf{Q}}\stackrel{h_{\pi}}{\rightarrow} (\widehat{\mathbf{B}}^+_{\operatorname{st}}\otimes_{F}H^i_{\operatorname{HK} }(X))^{N=0} \stackrel{\sim}{\leftarrow} (\mathbf{B}^+_{\operatorname{st}} \otimes _{F} H^i_{\operatorname{HK} }(X))^{N=0}. \end{equation}

It can be checked (see [Reference TsujiTsu99a, 4.5.6 and 4.5.7]) that these morphisms are compatible with Galois action and the Frobenius. Moreover, Hyodo and Kato [Reference Hyodo and KatoHK94, 5.1] have constructed a canonical $K$-isomorphism

(2.8)\begin{equation} \rho_{\pi }: K\otimes_{F}H^i_{\operatorname{HK} }(X) \stackrel{\sim}{\to}H^i_{\operatorname{dR} }(X_K). \end{equation}

Hence, the composition

\[ \rho_{\pi }h_{\pi }: H^i_{\operatorname{cr} }(X_{{\mathcal O}_{\overline{K}}})_{\mathbf{Q}}\to \mathbf{B}^+_{\operatorname{st}}\otimes _{F} H^i_{\operatorname{dR} }(X_K) \]

is functorial in $X$ and compatible with Galois action.

It is easy to check that all the above extends to finite simplicial (log-)schemes $X$.

1. (i) The map $h_{\operatorname {cr} }$ actually lifts in a functorial way to a statement in the $\infty$-derived categoryFootnote ⁵ and hence extends to simplicial schemes. Similarly for the morphisms in (2.7).

2. (ii) Similarly for the Hyodo–Kato isomorphism (2.8), though here finiteness of the simplicial scheme is an important assumption one needs to make to control the denominators (for details, see [Reference TsujiTsu98, 6.3] and [Reference KisinKis02, 2.8]).

3. (iii) Similarly for the map $h_{\operatorname {dR} }$, the maps in Lemma 2.5, and the maps in (2.6), where in addition one needs to use that the Hodge–de Rham spectral sequence for $X$ degenerates (which, by passing to the complex numbers, follows from the classical Hodge theory; see [Reference DeligneDel74, 7.2.8]).

2.1.6 A key isomorphism

Let $X$ be a proper semistable scheme over ${\mathcal O} _K$. The following lemma will be crucial in the comparison of period morphisms.

Lemma 2.9 Let $r\geq i$. There exists a natural isomorphism

\[ H^i_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}'(r))_{\mathbf{Q}}\stackrel{\sim}{\to} (H^i_{\operatorname{HK} }(X)\otimes_{F}\mathbf{B}_{\operatorname{st}})^{N=0,\phi=p^r}\cap F^r(H^i_{\operatorname{dR} }(X_K)\otimes_K\mathbf{B}_{\operatorname{dR} }). \]

Proof. This is well known; see [Reference Nekovář and NiziołNN16, Corollary 3.23] and [Reference Colmez and NiziołCN17, Proposition 5.22]. We will sketch here the construction of the map for future reference; see [Reference Nekovář and NiziołNN16, Corollary 3.23] for details. Consider the following sequence of maps of homotopy limits; they are all quasi-isomorphisms. Homotopy limits are taken in the $\infty$-derived category.

(2.10)

Here the eigenspaces are taken in the derived sense and we used the brackets $[-]$ to denote a mapping fiber. The first two maps and the last map are the canonical maps. We wrote $p_{\pi }$ for the projection $x\mapsto \pi$. The second map is induced by the distinguished triangle

\[ \mathrm{R}\Gamma_{\operatorname{cr} }(X_{{\mathcal O}_{\overline{K}}})\to \mathrm{R}\Gamma_{\operatorname{cr} }(X_{{\mathcal O}_{\overline{K}}}/R_{\pi})\stackrel{N}{\to}\mathrm{R}\Gamma_{\operatorname{cr} }(X_{{\mathcal O}_{\overline{K}}}/R_{\pi}). \]

The third map is induced by the Künneth map; we also used here the quasi-isomorphism (2.6). The fourth map is induced by the section $\iota _{\pi }: \mathrm {R}\Gamma _{\operatorname {HK} }(X)\to \mathrm {R}\Gamma _{\operatorname {cr} }(X/R_{\pi })_{\mathbf {Q}}$ of the projection $x\mapsto 0$ (recall that $\rho _{\pi }=p_{\pi }\iota _{\pi }$).

3.1.1 Integral crystalline conjecture

We treat first its integral version. Let $X$ be a smooth proper finite simplicial scheme over ${{\mathcal O} _K}$, ${{\mathcal O} _K}=W(k)$. Assume that $X\simeq \operatorname {sk} _mX$ and that the dimension $d\leq p-2$, $d=\max _{s\leq m}\dim X_s$. We would like to construct functorial Galois-equivariant morphisms

\[ \alpha_{ab}: H^a_{\operatorname{\acute{e}t} }(X_{\overline{K}},\mathbf{Z}/p^n(b)) \to \mathbf{L}( H^a_{\operatorname{cr} }(X_n)\{-b\}). \]

We will be able to do it under certain additional restrictions on the integers $a,b$, and $d$. Our construction is based on the following diagram.

(3.1)

Here $1\leq b < p-1$, $2b-a\geq 3$, $p^n\geq 5$, and $p\neq 2$. The Chern class map

\[ \overline{c}_{b,2b-a}^{\operatorname{syn}}: F^b_{\gamma}K_j(X_{{\mathcal O}_{\overline{K}}},\mathbf{Z}/p^n) \to H^{a}_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}_n(b)) \]

is defined as the limit over finite extensions ${\mathcal O} _K'/{{\mathcal O} _K}$ of the syntomic Chern class maps

\[ F^b_{\gamma}K_{2b-a}(X_{{\mathcal O}_K'},\mathbf{Z}/p^n) \to H^{a}_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_K'},{\mathcal{S}}_n(b)). \]

Due to [Reference NiziołNiz16, Lemma 5.3], the Chern class maps $\overline {c}_{b,2b-a}^{\operatorname {\acute {e}t} }$ and $\overline {c}_{b,2b-a}^{\operatorname {syn}}$ factor through $F^{b+1}_{\gamma }$ yielding the maps in the above diagram. The restriction map

\[ j^*: F^b_{\gamma}/F^{b+1}_{\gamma}K_{2b-a}(X_{{\mathcal O}_{\overline{K}}},\mathbf{Z}/p^n) \to F^b_{\gamma}/F^{b+1}_{\gamma} K_{2b-a}(X_{\overline{K}},\mathbf{Z}/p^n) \]

is an isomorphism by Lemma 2.11. By Proposition 2.14, the étale Chern class map

\[ \overline{c}_{b,2b-a}^{\operatorname{\acute{e}t} }: F^b_{\gamma}/F^{b+1}_{\gamma} K_{2b-a}(X_{\overline{K}},\mathbf{Z}/p^n)\to H^{a}_{\operatorname{\acute{e}t} }(X_{\overline{K}},\mathbf{Z}/p^n(b)) \]

is an isomorphism if $p>d+m+2b-a+1$.

Assume now that $b\geq d$, $2b-a\geq 3$, and $p-2\geq d+m+2b-a$. Define the morphisms

\[ \alpha_{ab}: H^a_{\operatorname{\acute{e}t} }(X_{\overline{K}},\mathbf{Z}/p^n(b))\to \mathbf{L}(H^a_{\operatorname{cr} }(X_n)\{-b\}) \]

as the composition $\alpha _{ab}:= \psi _n \overline {c}^{\operatorname {syn}}_{b,2b-a}(j^*)^{-1}(\overline {c}^{\operatorname {\acute {e}t} }_{b,2b-a})^{-1}$, where $\psi _n$ is the natural map

\[ H^a_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}_n(b))\to \mathbf{L}(H^a_{\operatorname{cr} }(X_n)\{-b\}). \]

Note that, by Proposition 2.3, this map is an isomorphism.

The following theorem generalizes our [Reference NiziołNiz98, Theorem 4.1] from schemes to finite simplicial schemes.

Theorem 3.2 For any proper smooth finite simplicial scheme $X$ over ${{\mathcal O} _K}=W(k)$, $X\simeq \operatorname {sk} _mX$, the functorial Galois-equivariant morphism

\[ \alpha_{ab}:H^a_{\operatorname{\acute{e}t} }(X_{\overline{K}},\mathbf{Z}/p^n(b)) \stackrel{\sim}{\to} \mathbf{L}(H^a_{\operatorname{cr} }(X_n)\{-b\}) \]

is an isomorphism if the numbers $p,b,d$ are such that $b\geq 2d+3$, $p-2\geq 2b+d+m$ for $d=\max _{s\leq m}\dim X_s$.

Proof. By Lemma 2.11, Proposition 2.14, and Proposition 2.3, it suffices to show that the syntomic Chern class map

\[ \overline{c}^{\operatorname{syn}}_{b,2b-a}: \operatorname{gr}^b_{\gamma}K_{2b-a}(X_{{\mathcal O}_{\overline{K}}},\mathbf{Z}/p^n) \to H^a_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}_n(b)) \]

is an isomorphism. Note that for $a<0$ this is an isomorphism by Lemma 2.19.

We argue by induction on $m\geq 0$ such that $X\simeq \operatorname {sk} _mX$. The case of $m=0$ is treated by [Reference NiziołNiz98, Theorem 4.1]. Assume that our theorem is true for $m-1$. To show it for $m$, consider the homotopy cofiber sequence

\[ \operatorname{sk} _{m-1}X_{{\mathcal O}_{\overline{K}}}\to \operatorname{sk} _mX_{{\mathcal O}_{\overline{K}}}\to \operatorname{sk} _{m}X_{{\mathcal O}_{\overline{K}}}/\operatorname{sk} _{m-1}X_{{\mathcal O}_{\overline{K}}} \]

and apply the syntomic Chern class maps to it. We get the following map of sequences.

Here we set $K^*_{*}(Y)=\operatorname {gr}^*_{\gamma }K_{*}(Y_{{\mathcal O} _{\overline {K}}})$, $H^*(Y, *)=H^*_{\operatorname {\acute {e}t} }(Y_{{\mathcal O} _{\overline {K}}},{\mathcal {S}}_n(*))$, and skipped the coefficients $\mathbf {Z}/p^n$ in $K$-theory. We also put

\[ K^*_{*}(X^{\prime}_m) =K^*_{*}(X_m)\cap\ker s_0^*\cap\cdots\cap\ker s_{m-1}^*,\enspace H^{*}(X^{\prime}_{m},*) =H^{*}(X_{m},*)\cap\ker s_0^*\cap\cdots\cap\ker s_{m-1}^*, \]

where each $s_i: X_{m-1}\to X_m$ is a degeneracy map. The bottom sequence is exact. By Lemma 2.18, so is the top. By the inductive hypothesis and by the case $m=0$ of this theorem plus Lemmas 2.20 and 2.11, we have the isomorphisms shown. It follows that the syntomic Chern class map

\[ \overline{c}^{\operatorname{syn}}_{b,2b-a}: \operatorname{gr}^b_{\gamma}K_{2b-a}(\operatorname{sk} _{m}X_{{\mathcal O}_{\overline{K}}},\mathbf{Z}/p^n)\to H^a_{\operatorname{\acute{e}t} }(\operatorname{sk} _{m}X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}_n(b)) \]

is an isomorphism as well. Since

\[ K_{2b-a}(\operatorname{sk} _{m}X_{{\mathcal O}_{\overline{K}}},\mathbf{Z}/p^n)\stackrel{\sim}{\to}K_{2b-a}(X_{{\mathcal O}_{\overline{K}}},\mathbf{Z}/p^n) \quad {\rm and}\quad H^a_{\operatorname{\acute{e}t} }(\operatorname{sk} _{m}X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}_n(b))\stackrel{\sim}{\to}H^a_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}_n(b)), \]

we are done.

Example 3.4 Integral crystalline conjecture for cohomology with compact support. As a corollary of the above comparison theorem, we obtain a comparison theorem for cohomology with compact support. Consider a proper smooth scheme $X$ over ${{\mathcal O} _K}=W(k)$. Let $i:D\hookrightarrow X$, built from $m$ irreducible components that are smooth over ${{\mathcal O} _K}$, be the divisor at infinity of $X$. Let $U=X\setminus D$. Consider the simplicial scheme $C(X,D):=\operatorname {cofiber}(\widetilde {D}_{{\scriptscriptstyle \bullet }}\stackrel {i_*}{\to } X)$. We have $C(X,D)\simeq \operatorname {sk} _mC(X,D)$. Equip $X$ with the log-structure associated to $D$. Applying the above constructions to $C(X,D)$, we obtain the basic following diagram

and the induced period morphism

\[ \alpha^{\prime}_{ab}: H^{a}_{\operatorname{\acute{e}t} }(C(X_{\overline{K}},D_{\overline{K}}),\mathbf{Z}/p^n(b))\to H^{a}_{\operatorname{\acute{e}t} }(C(X_{{\mathcal O}_{\overline{K}}},D_{{\mathcal O}_{\overline{K}}}),{\mathcal{S}}_n(b)). \]

But, by Lemma 2.1,

\begin{align*} H^{a}_{\operatorname{\acute{e}t} }(C(X_{\overline{K}},D_{\overline{K}}),\mathbf{Z}/p^n(b))&\simeq H^{a}_{\operatorname{\acute{e}t} ,c}(U_{\overline{K}},\mathbf{Z}/p^n(b)), \\ H^{a}_{\operatorname{\acute{e}t} }(C(X_{{\mathcal O}_{\overline{K}}},D_{{\mathcal O}_{\overline{K}}}),{\mathcal{S}}_n(b))&\simeq H^{a}_{\operatorname{\acute{e}t} ,c}(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}_n(b)). \end{align*}

Hence, we obtain a period morphism

\[ \alpha^{\prime}_{ab}: H^{a}_{\operatorname{\acute{e}t} ,c}(U_{\overline{K}},\mathbf{Z}/p^n(b))\to H^{a}_{\operatorname{\acute{e}t} ,c}(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}_n(b)) \]

that composed with the map $H^{a}_{\operatorname {\acute {e}t} ,c}(X_{{\mathcal O} _{\overline {K}}},{\mathcal {S}}_n(b)) \to \mathbf {L}(H^a_{\operatorname {cr} ,c}(X_n)\{-b\})$ yields a Galois-equivariant map

\[ \alpha_{ab}: H^a_{\operatorname{\acute{e}t} ,c}(U_{\overline{K}},\mathbf{Z}/p^n(b)) \to \mathbf{L}(H^a_{\operatorname{cr} ,c}(X_n)\{-b\}). \]

We get the following corollary of Theorem 3.2.

Corollary 3.5 The Galois-equivariant morphism

\[ \alpha_{ab}: H^a_{\operatorname{\acute{e}t} ,c}(U_{\overline{K}},\mathbf{Z}/p^n(b)) \to \mathbf{L}(H^a_{\operatorname{cr} ,c}(X_n)\{-b\}) \]

is an isomorphism if the numbers $p,b,d$ are such that $b\geq 2d+3$ and $p-2\geq 2b+d+m$.

3.1.2 Rational crystalline conjecture

We will treat now the rational crystalline conjecture. Let $X$ be a smooth proper finite simplicial scheme over ${{\mathcal O} _K}$, where the ring ${{\mathcal O} _K}$ is possibly ramified over $W(k)$. Assume that $X\simeq \operatorname {sk} _mX$ and set $d=\max _{s\leq m}\dim X_s$. For large $b$, we will construct Galois-equivariant functorial period morphisms

\[ \alpha_{ab}:H^a_{\operatorname{\acute{e}t} }(X_{\overline{K}},\mathbf{Q}_p(b))\to H^a_{\operatorname{cr} }(X_{0})\otimes \mathbf{B}^+_{\operatorname{cr} }. \]

Assume that $p^n\geq 5$, $2b-a\geq \max \{2d,2\}$, $2b-a\geq 3$ for $d=0$ and $p=2$, and $a\geq 0$. Nizioł [Reference NiziołNiz16, Lemma 5.3] and Lemma 2.11 give us the following diagram.

Define the morphisms

\[ \alpha^n_{ab}: H^a_{\operatorname{\acute{e}t} }(X_{\overline{K}},\mathbf{Z}/p^n(b))\to H^a_{\operatorname{cr} }(X_{{{\mathcal O}_{\overline{K}}},n})\{-b\} \]

as the composition

\[ \alpha^n_{ab}(x):= \psi_n\overline{c}_{b,2b-a}^{\operatorname{syn}}(j^*)^{-1} D(d,m,b,2b-a)(\overline {c}_{b,2b-a}^{\operatorname{\acute{e}t} })^{-1}(D(d,m,b,2b-a)x), \]

where $\psi _n$ is the natural projection

\[ \psi_n: H^a_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}^{\prime}_n(b))\to H^a_{\operatorname{cr} }(X_{{{\mathcal O}_{\overline{K}}},n}). \]

Here $(\overline {c}_{b,2b-a}^{\operatorname {\acute {e}t} })^{-1}(D(d,m,b,2b-a)x)$ is any element in the preimage of $D(d,m,b,2b-a)x$ (by Proposition 2.14, $D(d,m,b,2b-a)x$ lies in the image of $\overline {c}_{b,2b-a}^{\operatorname {\acute {e}t} }$). By Proposition 2.14, any ambiguity in that definition comes from a class of $y$ such that $D(d,m,b,2b-a)[y]=[z]$, $z\in F^{b+1}_{\gamma }K_{2b-a}(X_{\overline {K}},\mathbf {Z}/p^n)$, and this ambiguity we killed by twisting the definition of $\alpha ^n_{ab}$ by a factor of $D(d,m,b,2b-a)$.

Define the morphism

\[ \alpha_{ab}:H^a_{\operatorname{\acute{e}t} }(X_{\overline{K}},\mathbf{Q}_p(b))\to H^a_{\operatorname{cr} }(X_{0})\otimes_{W(k)} \mathbf{B}_{\operatorname{cr} }\{-b \} \]

as the composition of $\mathbf {Q}\otimes {\mathrel{\mathop{\kern0pt\longleftarrow}\limits_n^{\lim }}} \alpha ^n_{ab}$ with the Kato–Messing isomorphism $h_{\operatorname {cr} }: H^a_{\operatorname {cr} }(X_{{{\mathcal O} _{\overline {K}}}})_{\mathbf {Q}}\simeq H^a_{\operatorname {cr} }(X_{0})\otimes _{W(k)} \mathbf {B}^+_{\operatorname {cr} }$ and the division by $D(d,m,b,2b-a)^2$.

The following theorem generalizes our [Reference NiziołNiz08, Theorem 3.8] from schemes to finite simplicial schemes.

Theorem 3.6 Let $X$ be any proper smooth finite simplicial ${{\mathcal O} _K}$-scheme. Assume that $X\simeq \operatorname {sk} _mX$ and let $d=\max _{s\leq m}\dim X_m$. Then, assuming that $b\geq 2d+2$, the functorial Galois-equivariant morphism

\[ \alpha_{ab}: H^a_{\operatorname{\acute{e}t} }(X_{\overline{K}},\mathbf{Q}_p(b)) \otimes_{\mathbf{Q}_p} \mathbf{B}_{\operatorname{cr} } \to H^a_{\operatorname{cr} }(X_{0})\otimes_{W(k)} \mathbf{B}_{\operatorname{cr} }\{-b \} \]

is an isomorphism. Moreover, the map $\alpha _{ab}$ preserves the Frobenius, is compatible with products and Tate twists, and, after extension to $\mathbf {B}_{\operatorname {dR} }$, induces an isomorphism of filtrations.

Proof. We argue by induction on $m\geq 0$. The case $m=0$ is treated by [Reference NiziołNiz08, Theorem 3.8]. Assume that our theorem is true for $m-1$. To show it for $m$, consider the homotopy cofiber sequence

\[ \operatorname{sk} _{m-1}X_{{\mathcal O}_{\overline{K}}}\to \operatorname{sk} _mX_{{\mathcal O}_{\overline{K}}}\to \operatorname{sk} _{m}X_{{\mathcal O}_{\overline{K}}}/\operatorname{sk} _{m-1}X_{{\mathcal O}_{\overline{K}}} \]

and apply the period morphisms $\alpha _{*,*}$ to it. We get the following map of sequences.

Here we put $H^*_{\operatorname {\acute {e}t} }(T,b)=H^{*}_{\operatorname {\acute {e}t} }(T_{\overline {K}},\mathbf {Q}_p(b))\otimes \mathbf {B}_{\operatorname {cr} }$, $H^*_{\operatorname {cr} }(T,b)= H^{*}_{\operatorname {cr} }(T)\otimes \mathbf {B}_{\operatorname {cr} }\{-b \}$. And we defined

\begin{gather*} H^{*}_{\operatorname{\acute{e}t} }(X^{\prime}_{m},b) =H^{*}_{\operatorname{\acute{e}t} }(X_{m},b)\cap\ker s_0^*\cap\cdots\cap\ker s_{m-1}^*,\\ H^{*}_{\operatorname{cr} }(X^{\prime}_{m,0},b) =H^{*}_{\operatorname{cr} }(X_{m,0},b)\cap\ker s_0^*\cap\cdots\cap\ker s_{m-1}^*, \end{gather*}

where each $s_i: X_{m-1}\to X_{m}$ is a degeneracy map. The horizontal sequences are exact by functoriality and finiteness of the étale and crystalline cohomologies. By the inductive hypothesis, we have the isomorphisms shown in the diagram. Hence, the period morphism

\[ \alpha_{ab}: H^a_{\operatorname{\acute{e}t} }(\operatorname{sk} _mX_{\overline{K}},\mathbf{Q}_p(b)) \otimes_{\mathbf{Q}_p} \mathbf{B}_{\operatorname{cr} } \to H^a_{\operatorname{cr} }(\operatorname{sk} _mX_{0})\otimes_{W(k)} \mathbf{B}_{\operatorname{cr} }\{-b \} \]

is an isomorphism. Since $H^a_{\operatorname {\acute {e}t} }(\operatorname {sk} _mX_{\overline {K}},\mathbf {Q}_p(b))\stackrel {\sim }{\to }H^a_{\operatorname {\acute {e}t} }(X_{\overline {K}},\mathbf {Q}_p(b))$ and $H^a_{\operatorname {cr} }(\operatorname {sk} _mX_{0})\stackrel {\sim }{\to }H^a_{\operatorname {cr} }(X_{0})$, this proves the first claim of the theorem.

We will now check that the morphism $\alpha _{ab}$ is compatible with products. This follows from the fact that the morphism $h_{\operatorname {cr} }$ is compatible with products and from the following lemma.

Lemma 3.7 Let $x\in H^a(X_{\overline {K}},\mathbf {Z}/p^n(b))$, $y\in H^{\rm c}(X_{\overline {K}},\mathbf {Z}/p^n(e))$, $2b-a>2$, $2e-c>2$, and $p^n\geq 5$. Set $K(b,e)=-(b+e-1)!/((b-1)!(e-1)!)$. Then (assuming that all the indices are in the valid range)

\begin{align*} &K(b,e)D(d,m,b,2b-a)^2 D(d,m,e,2e-c)^2 \alpha^n_{a+c, b+e}(x\cup y)\\ &\quad =K(b,e)D(d,m,b+e,2b+2e-a-c)^2\alpha^n_{ab}(x)\cup \alpha^n_{ce}(y). \end{align*}

The claim about Tate twists follows from the following computation.

Lemma 3.8 Let $p^n\geq 5$ and $b\geq 2d+2$. We have the following relationship between Tate twists:

\[ (-b)D(d,m,b,2b-a)^2\alpha^n_{a,b+1}(\zeta_n x) =(-b)D(d,m,b+1,2b+2-a)^2\alpha^n_{ab}(x)t. \]

Proof. This follows just as in [Reference NiziołNiz08, Lemma 3.6] from Lemma 3.7 and the fact that $\overline {c}_{1,2}^{\operatorname {\acute {e}t} }(\beta _n)=\zeta _n$ and $\overline {c}_{1,2}^{\operatorname {syn}}(\tilde {\beta }_n)=t$ (see [Reference NiziołNiz98, Lemma 4.1]). Here $\beta _n\in K_2(\overline {K},\mathbf {Z}/p^n)$ and $\tilde {\beta }_n\in K_2({\mathcal O} _{\overline {K}},\mathbf {Z}/p^n)$ are the Bott elements associated to $\zeta _n$.

Now, to prove the claim about filtrations, first we evoke Lemma 2.5 that yields compatibility of the period morphism with filtrations and then we note that it suffices to prove the analog of our claim for the associated grading, i.e., that, for $i\in \mathbf {Z}$, the induced map

\[ \alpha_{ab}: H^a_{\operatorname{\acute{e}t} }(X_{\overline{K}},\mathbf{Q}_p(b)) \otimes_{\mathbf{Q}_p} C(i) {\to} \bigoplus_{j\in\mathbf{Z}}H^{a-j}(X_{K},\Omega^j_{X_K/K})\otimes_{K} C(i+b-j) \]

is an isomorphism. But this can be proved by an analogous argument to the one we used to prove the first claim of the theorem.

Example 3.9 Rational crystalline conjecture for cohomology with compact support Again, as a special case consider a smooth proper scheme $X$ over ${{\mathcal O} _K}$ with a divisor $D$. We assume $D$ to have relative simple normal crossings and all the irreducible components smooth over ${{\mathcal O} _K}$. Let $U$ denote the complement of $D$ in $X$ and $d$ be the relative dimension of $X$. Equip $X$ with the log-structure induced by $D$. Consider the simplicial scheme $C(X,D):=\operatorname {cofiber}(\widetilde {D}_{{\scriptscriptstyle \bullet }}\stackrel {i}{\to } X),$ where all the schemes have trivial log-structure. We have $C(X,D)\simeq \operatorname {sk} _mC(X,D)$, where $m$ is the number of irreducible components of $D$. Applying the above constructions to $C(X,D)$, we obtain the following basic diagram.

Recall that we have

\[ H^{a}_{\operatorname{\acute{e}t} ,c}(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}^{\prime}_n(b)) \simeq H^{a}_{\operatorname{\acute{e}t} }(C(X_{{\mathcal O}_{\overline{K}}},D_{{\mathcal O}_{\overline{K}}}),{\mathcal{S}}^{\prime}_n(b)). \]

From this, we get a Galois-equivariant map

\[ \alpha_{ab}: H^a_{\operatorname{\acute{e}t} ,c}(U_{\overline{K}},\mathbf{Q}(b)) \to H^a_{\operatorname{cr} ,c}(X_0)\otimes \mathbf{B}_{\operatorname{cr} }\{-b\} \]

and the following corollary of Theorem 3.6.

Corollary 3.10 The Galois-equivariant morphism

\[ \alpha_{ab}:H^a_{\operatorname{\acute{e}t} ,c}(U_{\overline{K}},\mathbf{Q}_p(b))\otimes \mathbf{B}_{\operatorname{cr} } \to H^a_{\operatorname{cr} ,c}(X_0)\otimes \mathbf{B}_{\operatorname{cr} }\{-b\} \]

is an isomorphism for $b\geq 2d+2$. Moreover, the map $\alpha _{ab}$ preserves the Frobenius, is compatible with products and Tate twists, and, after extension to $\mathbf {B}_{\operatorname {dR} }$, induces an isomorphism of filtrations.

4.1.1 The case of schemes

Recall the following formulation of the semistable conjecture of Fontaine and Jannsen.

Conjecture 4.1 Semistable conjecture Let $X$ be a proper, log-smooth, fine, and saturated ${\mathcal O} _K^{\times }$-log-scheme with Cartier-type reduction. There exists a natural $\mathbf {B}_{\operatorname {st}}$-linear Galois-equivariant period isomorphism

\[ \alpha_{i}:H^i_{\operatorname{\acute{e}t} }(X_{\overline{K},\operatorname{tr}},{\mathbf{Q} }_p) \otimes_{{\mathbf{Q} }_p} \mathbf{B}_{\operatorname{st}} \stackrel{\sim}{\to} H^i_{\operatorname{HK} }(X)\otimes_{F} \mathbf{B}_{\operatorname{st}} \]

that preserves the Frobenius and the monodromy operators and, after extension to $\mathbf {B}_{\operatorname {dR} }$, induces a filtered isomorphism

\[ \alpha_{i}:H^i_{\operatorname{\acute{e}t} }(X_{\overline{K},\operatorname{tr}},{\mathbf{Q} }_p) \otimes_{{\mathbf{Q} }_p} \mathbf{B}_{\operatorname{dR} } \stackrel{\sim}{\to} H^i_{\operatorname{dR} }(X_K)\otimes_{K} \mathbf{B}_{\operatorname{dR} }. \]

This conjecture was proved, possibly under additional assumptions, by Kato [Reference KatoKat94a], Tsuji [Reference TsujiTsu99a, Reference TsujiTsu03], Yamashita [Reference YamashitaYam11], Faltings [Reference FaltingsFal02], Nizioł [Reference NiziołNiz08], and Beilinson [Reference BeilinsonBei13]. It was generalized to formal schemes by Colmez and Nizioł [Reference Colmez and NiziołCN17] and by Česnavičius and Koshikawa [Reference Česnavičius and KoshikawaČK19] (who generalized the proof of the crystalline conjecture by Bhatt, Morrow and Scholze [Reference Bhatt, Morrow and ScholzeBMS18]) in the case when there is no horizontal divisor.

Let $r\geq 0$. For a period isomorphism $\alpha _i$ as above, we define its twist

\[ \alpha_{i,r}: H^i_{\operatorname{\acute{e}t} }(X_{\overline{K},\operatorname{tr}},{\mathbf{Q} }_p(r)) \otimes_{{\mathbf{Q} }_p} \mathbf{B}_{\operatorname{st}} \to H^i_{\operatorname{HK} }(X)\otimes_{F} \mathbf{B}_{\operatorname{st}}\{-r\} \]

as $\alpha _{i,r}:=t^r\alpha _{i}\epsilon ^{-r}$. Clearly, it is an isomorphism. It follows from Conjecture 4.1 that we can recover the étale cohomology with the Galois action from the Hyodo–Kato cohomology:

(4.2)\begin{equation} \alpha_{i,r}: H^i_{\operatorname{\acute{e}t} }(X_{\overline{K},\operatorname{tr}},{\mathbf{Q}}_p(r))\stackrel{\sim}{\to} (H^i_{\operatorname{HK} }(X)\otimes_{F}\mathbf{B}_{\operatorname{st}})^{N=0,\phi=p^r}\cap F^r(H^i_{\operatorname{dR} }(X_K)\otimes_K\mathbf{B}_{\operatorname{dR} }). \end{equation}

For $r\geq i$, by Lemma 2.9, the right-hand side is isomorphic to $H^i_{\operatorname {\acute {e}t} }(X_{{\mathcal O} _{\overline {K}}},{\mathcal {S}}'(r))_{\mathbf {Q}}$, i.e., there exists a natural isomorphism

\[ H^i_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}'(r))_{\mathbf{Q}}\stackrel{\sim}{\to} (H^i_{\operatorname{HK} }(X)\otimes_{F}\mathbf{B}_{\operatorname{st}})^{N=0,\phi=p^r}\cap F^r(H^i_{\operatorname{dR} }(X_K)\otimes_K\mathbf{B}_{\operatorname{dR} }). \]

We will denote by

\[ \tilde{\alpha}_{i,r}:H^i_{\operatorname{\acute{e}t} }(X_{\overline{K},\operatorname{tr}},{\mathbf{Q}}_p(r))\stackrel{\sim}{\to} H^i_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}'(r))_{\mathbf{Q}} \]

the induced isomorphism and call it the syntomic period isomorphism.

The following lemma is immediate.

4.1.2 The case of simplicial schemes

The above discussion carries over to finite simplicial schemes. That is, we assume that we have a period isomorphism $\alpha _i$ as in Conjecture 4.1 but for a finite simplicial scheme $X$ with components as in Conjecture 4.1. It then yields an isomorphism $\alpha _{i,r}$ as in (4.2) for $i\leq r$. We will need the following analog of Lemma 2.9.

Lemma 4.4 Let $r\geq i$. There exists a natural isomorphism

\[ H^i_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}'(r))_{\mathbf{Q}}\stackrel{\sim}{\to} (H^i_{\operatorname{HK} }(X)\otimes_{F}\mathbf{B}_{\operatorname{st}})^{N=0,\phi=p^r}\cap F^r(H^i_{\operatorname{dR} }(X_K)\otimes_K\mathbf{B}_{\operatorname{dR} }). \]

Proof. By functoriality of all the maps involved, the proof of Lemma 2.9 yields a quasi-isomorphism

We have natural isomorphisms

\begin{gather*} H^i((\mathrm{R}\Gamma_{\operatorname{HK} }(X)\otimes_F{\mathbf{B}}^+_{\operatorname{st}})^{N=0,\phi=p^r}) \simeq (H^i_{\operatorname{HK} }(X)\otimes_F{\mathbf{B}}^+_{\operatorname{st}})^{N=0,\phi=p^r},\\ H^i((\mathrm{R}\Gamma_{\operatorname{dR} }(X_K) \otimes_K\mathbf{B}^+_{\operatorname{dR} })/F^r) \simeq (H^i_{\operatorname{dR} }(X_K) \otimes_K\mathbf{B}^+_{\operatorname{dR} })/F^r. \end{gather*}

The first isomorphism holds because $H^i_{\operatorname {HK} }(X)\otimes _F{\mathbf {B}}^+_{\operatorname {st}}$ is a $(\phi ,N)$-module (see [Reference Nekovář and NiziołNN16, proof of Corollary 3.25] for an argument) and the second one because we have a degeneration of the Hodge–de Rham spectral sequence for $X$. This yields a natural long exact sequence

\begin{align*} (H^{i-1}_{\operatorname{HK} }(X)\otimes_F{\mathbf{B}}^+_{\operatorname{st}})^{N=0,\phi=p^r} & {\buildrel {\rho_{\pi}\otimes\iota} \over \longrightarrow} (H^{i-1}_{\operatorname{dR} }(X_K) \otimes_K\mathbf{B}^+_{\operatorname{dR} })/F^r {\buildrel {\partial} \over \longrightarrow} H^i_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}'(r))_{\mathbf{Q}}\\ & \to (H^i_{\operatorname{HK} }(X)\otimes_F{\mathbf{B}}^+_{\operatorname{st}})^{N=0,\phi=p^r} {\buildrel {\rho_{\pi}\otimes\iota} \over \longrightarrow} (H^i_{\operatorname{dR} }(X_K) \otimes_K\mathbf{B}^+_{\operatorname{dR} })/F^r. \end{align*}

It suffices thus to show that, for $i\leq r$, the map $\partial$ in the above exact sequence is zero. Or that the map

\[ (H^{i-1}_{\operatorname{HK} }(X)\otimes_{F}\mathbf{B}_{\operatorname{st}})^{N=0,\phi=p^r} {\buildrel {\rho_{\pi}\otimes\iota_{\pi}} \over \longrightarrow} (H^{i-1}_{\operatorname{dR} }(X_K)\otimes_K\mathbf{B}_{\operatorname{dR} })/F^r \]

is surjective. But this follows from the fact that the pair $H^{i-1}_{\operatorname {HK} }(X),H^{i-1}_{\operatorname {dR} }(X_K)$ is an admissible filtered $(\phi ,N)$-module such that $F^{r}H^{i-1}_{\operatorname {dR} }(X_K)=0$ (see [Reference Colmez and NiziołCN17, Proposition 5.20]).

As above, we will denote by

\[ \tilde{\alpha}_{i,r}:H^i_{\operatorname{\acute{e}t} }(X_{\overline{K},\operatorname{tr}},{\mathbf{Q}}_p(r))\stackrel{\sim}{\to} H^i_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}'(r))_{\mathbf{Q}} \]

the induced isomorphism and call it the syntomic period isomorphism. Again, the following lemma is immediate.

4.2.1 A $K$-theoretical uniqueness criterion

We will prove now a uniqueness criterion for period morphisms that generalizes the one stated in [Reference NiziołNiz09]. Let $X$ be a proper scheme over ${\mathcal O} _K$ with semistable reduction and of pure relative dimension $d$. Let $i:D\hookrightarrow X$ be the horizontal divisor and set $U=X\setminus D$. Equip $X$ with the log-structure induced by $D$ and the special fiber.

Proposition 4.6 Let $r\geq 2d+2$. There exists a unique semistable period morphism

\[ \tilde{\alpha}_{i,r}: H^{i}_{\operatorname{\acute{e}t} ,c}(U_{\overline{K}},{\mathbf{Q}}_p(r))\to H^{i}_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},S'(r)(D))_{\mathbf{Q}} \]

that makes the diagram from § 3.2 commute.

4.2.2 Comparison of period morphisms for cohomology with compact support

The comparison morphisms of Faltings [Reference Faltings and IgusaFal89, Reference FaltingsFal02] and Tsuji [Reference TsujiTsu99a] extend easily to finite simplicial schemes. This was done explicitly in [Reference KisinKis02, Reference TsujiTsu98]. In particular, they extend to cohomology with compact support. We will show in this section that they are equal to the period morphisms constructed in § 3. We will use for that the uniqueness criterion for period morphisms stated above. We will do the computations just for cohomology with compact support in the semistable case. The arguments in other cases are analogous.

Theorem 4.7 (i) There exists a unique natural $p$-adic period isomorphism

\[ \alpha_i: H^i_{\operatorname{\acute{e}t} ,c}(U_{\overline{K}},{\mathbf{Q}}_p)\otimes \mathbf{B}_{\operatorname{st}}\stackrel{\sim}{\to} H^i_{\operatorname{cr} ,c}(X_0/W(k)^0)\otimes_{W(k)} \mathbf{B}_{\operatorname{st}} \]

such that:

1. (a) $\alpha _i$ is $\mathbf {B}_{\operatorname {st}}$-linear, Galois equivariant, and compatible with Frobenius;

2. (b) $\alpha _i$, extended to $\mathbf {B}_{dR}$, induces a filtered isomorphism

   \[ \alpha^{\operatorname{dR} }_i: H^i_{\operatorname{\acute{e}t} ,c}(U_{\overline{K}},{\mathbf{Q}}_p)\otimes \mathbf{B}_{\operatorname{dR} }\stackrel{\sim}{\to} H^i_{\operatorname{dR} ,c}(X_K)\otimes_{K} \mathbf{B}_{\operatorname{dR} }; \]

3. (c) $\alpha _i$ is compatible with the étale and syntomic higher Chern classes from $p$-adic $K$-theory.

(ii) The period morphisms of Faltings, Tsuji, and Nizioł are equal.Footnote ⁷

Proof. The first claim follows from Proposition 4.6 and Lemma 4.5.

For the second claim, choose $r$ such that $r\geq 2d+2$ and $r\geq i$. It suffices to show that the Faltings, Tsuji, and Nizioł period morphisms $\alpha ^{F}_{i,r}$, $\alpha ^{T}_{i,r}$, and $\alpha ^{N}_{i,r}$

\[ \alpha^*_{i,r}: H^i_{\operatorname{\acute{e}t} ,c}(U_{\overline{K}},{\mathbf{Q}}_p(r))\otimes \mathbf{B}_{\operatorname{st}}\stackrel{\sim}{\to} H^i_{\operatorname{cr} ,c}(X_0/W(k)^0)\otimes _{W(k)}\mathbf{B}_{\operatorname{st}}\{-r\}, \]

and their de Rham analogs are equal. For that, apply the first claim. The needed compatibility of the period morphism with higher $p$-adic Chern classes is clear in the case of the map $\alpha ^{N}_{i,r}$ and was proved in [Reference NiziołNiz09, Corollaries 4.14 and 5.9] for the other two maps. These corollaries are stated for proper log-schemes but their proofs carry over to the case of finite simplicial schemes (with the same properties).

4.3.1 Tsuji period morphism

We will briefly discuss the period morphism used by Tsuji. Let $X$ be a log-smooth log-scheme over ${\mathcal O} _K^{\times }$. Recall that Fontaine and Messing, and Kato have defined natural period morphisms on the étale site of $X_0$ [Reference Fontaine, Messing and RibetFM87, Reference TsujiTsu98]

\[ \beta_{r}^{\rm T}: {\mathcal{S}}_n(r) \rightarrow i^*\mathrm{R} j_*{\mathbf{Z}}/p^n(r)',\quad r\geq 0, \]

where $i:X_0\hookrightarrow X, j: X_{K,\operatorname {tr}}\hookrightarrow X$ are the natural immersions. Here we set ${\mathbf {Z}}/p^n(r)^{\prime }:=(1/(p^aa!){\mathbf {Z}}_p(r))\otimes {\mathbf {Z}}/p^n$, where $a$ is the largest integer $\leq r/(p-1)$. Recall that we have the fundamental exact sequence [Reference TsujiTsu99a, Theorem 1.2.4]

\[ 0\to {\mathbf{Z}}/p^n(r)^{\prime}\to J_{\operatorname{cr} ,n}^{\langle r\rangle } {\buildrel {1-\phi_r} \over \longrightarrow} \mathbf{A}_{\operatorname{cr} ,n}\to 0, \]

where

\[ J_n^{\langle r\rangle }:= \{x\in J_{n+s}^{[r]}\mid \phi(x)\in p^r\mathbf{A}_{\operatorname{cr} ,n+s}\}/p^n \]

for some $s\geq r$.

The above period morphisms were used to prove the following comparison theorem.

Theorem 4.8 (Tsuji [Tsu99a, 3.3.4, Theorem 3.4.4]) (i) Let $X$ be a semistable scheme over ${\mathcal O} _K$ or a finite base change of such a scheme. Then, for any $0\leq i\leq r$, the kernel and cokernel of the period morphism

\[ \beta_{r}^{\rm T}: {\mathcal{H}}^i({\mathcal{S}}_n(r)_{\overline{X}}) \rightarrow \overline{i}^*\mathrm{R}^i\overline{j}_*{\mathbf{Z}}/p^n(r)'_{X_{\overline{K},\operatorname{tr}}} \]

are annihilated by $p^N$ for an integer $N$ which depends only on $p$, $r$, and $i$. Here $\overline {i}$ and $\overline {j}$ are extensions of $i$ and $j$ to $\overline {X}:=X_{{\mathcal O} _{\overline {K}}}$.

(ii) Assume moreover that $X$ is proper. Then, for any $0\leq i\leq r$, the induced morphism

\[ H^i_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}(r))_{\mathbf{Q}}\to H^i_{\operatorname{\acute{e}t} }(X_{\overline{K}, \operatorname{tr}},\mathbf{Q}_p(r)) \]

is an isomorphism.

For a proper semistable scheme $X$ over ${\mathcal O} _K$ and $r\geq i$, the modulo $p^n$ and rational semistable Tsuji period morphisms are defined as

(4.9)\begin{equation} \begin{gathered} \beta_{r,n}^{\rm T}: \mathrm{R}\Gamma_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}^{\prime}_n(r)) {\buildrel {\operatorname{can}} \over \longrightarrow} \mathrm{R}\Gamma_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}_n(r)) {\buildrel {\beta^{\rm T}_r} \over \longrightarrow} \mathrm{R}\Gamma_{\operatorname{\acute{e}t} }(X_{\overline{K},\operatorname{tr}},\mathbf{Z}/p^n(r)^{\prime}),\\ \beta_{r}^{\rm T}: \mathrm{R}\Gamma_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}^{\prime}(r))_{\mathbf{Q}} {\buildrel {\operatorname{can}} \over \longrightarrow} \mathrm{R}\Gamma_{\operatorname{\acute{e}t} }(X_{{\mathcal O}_{\overline{K}}},{\mathcal{S}}(r))_{\mathbf{Q}} {\buildrel {\beta^{\rm T}_r} \over \longrightarrow} \nonumber \mathrm{R}\Gamma_{\operatorname{\acute{e}t} }(X_{\overline{K},\operatorname{tr}},{\mathbf{Q}}_p(r))\stackrel{ p^{-r}}{\to} \mathrm{R}\Gamma_{\operatorname{\acute{e}t} }(X_{\overline{K},\operatorname{tr}},{\mathbf{Q}}_p(r)). \end{gathered} \end{equation}

By Theorem 4.8, $\beta _{r}^{\rm T}$ is a quasi-isomorphism after truncation at $\tau _{\leq r}$.

The Tsuji period morphism

\[ \alpha^{\rm T}_{i,r}: H^i_{\operatorname{\acute{e}t} }(X_{\overline{K},\operatorname{tr}},\mathbf{Q}_p(r)){\to} (H^i_{\operatorname{HK} }(X)\otimes_F\mathbf{B}_{\operatorname{st}})^{N=0,\phi=p^r} \]

is defined by composing the above morphism with the map $h_{\pi }$ (and changing $\widehat {\mathbf {B}}_{\operatorname {st}}$ to $\mathbf {B}_{\operatorname {st}}$).

4.3.2 Beilinson comparison theorem

In [Reference BeilinsonBei13], Beilinson proved the following comparison theorem.

Theorem 4.10 Semistable conjecture [Bei13] Let $X$ be a proper semistable scheme over ${\mathcal O} _K$ endowed with its canonical log-structure. There exists a natural $\mathbf {B}_{\operatorname {st}}$-linear Galois-equivariant period isomorphismFootnote ⁸

\[ \alpha^B_{h,i}:H^i_{\operatorname{\acute{e}t} }(X_{\overline{K},\operatorname{tr}},{\mathbf{Q} }_p) \otimes_{{\mathbf{Q} }_p} \mathbf{B}_{\operatorname{st}} \stackrel{\sim}{\to} H^{B,i}_{\operatorname{HK} }(X)\otimes_{F} \mathbf{B}_{\operatorname{st}} \]

that preserves the Frobenius and the monodromy operators and, after extension to $\mathbf {B}_{\operatorname {dR} }$, induces a filtered isomorphism

\[ \alpha^B_{i}:H^i_{\operatorname{\acute{e}t} }(X_{\overline{K},\operatorname{tr}},{\mathbf{Q} }_p) \otimes_{{\mathbf{Q} }_p} \mathbf{B}_{\operatorname{dR} } \stackrel{\sim}{\to} H^i_{\operatorname{dR} }(X_K)\otimes_{K} \mathbf{B}_{\operatorname{dR} }. \]

We added the subscript $h$ (for $h$-topology) to underscore the different formulation from Theorem 4.1. Here $H^{B,i}_{\operatorname {HK} }(X)$ is the Beilinson–Hyodo–Kato cohomology [Reference BeilinsonBei13, 1.16.1] and the base change to the de Rham comparison uses the Beilinson–Hyodo–Kato isomorphism

\[ \rho^B: H^{B,i}_{\operatorname{HK} }(X)\otimes_FK\stackrel{\sim}{\to} H^i_{\operatorname{dR} }(X_K) \]

as well as the canonical map $\iota _p:\mathbf {B}_{\operatorname {st}}\to \mathbf {B}_{\operatorname {dR} }$ [Reference Nekovář and NiziołNN16, § 2.1]. A priori, these Hyodo–Kato-type constructions are not the same as the original ones (for one thing, they are independent of the choice of the uniformizer $\pi$; in fact, they should be seen, in a sense that can be made precise, as associated to the canonical choice of $p$). However, the two constructions are related by a natural quasi-isomorphism, i.e., there is a natural map $\kappa$ that makes the following diagram commute [Reference Nekovář and NiziołNN16, (31)].

4.3.3 Beilinson equivalence of topoi

To describe the Beilinson period morphism, we will need to work with $h$-topology on the generic fiber. Beilinson has shown that $h$-topology has a base consisting of semistable schemes. We will review his result briefly.

For a field $K$, let ${\mathcal {V}}ar_K$ denote the category of varieties over $K$. We will equip it with $h$-topology (see [Reference BeilinsonBei12, 2.3]), i.e., the coarsest topology finer than the Zariski and proper topologies.Footnote ⁹ We note that the $h$-topology is finer than the étale topology. It is generated by the pretopology whose coverings are finite families of maps $\{Y_i\to X\}$ such that $Y:=\coprod Y_i\to X$ is a universal topological epimorphism (i.e., a subset of $X$ is Zariski open if and only if its preimage in $Y$ is open). We denote by $\mathcal {V}ar_{K,h},X_h$, $X\in \mathcal {V}ar_{K}$, the corresponding $h$-sites.

Let $K$ be now as in § 2. An arithmetic pair over $K$ is an open embedding $j:U\hookrightarrow \overline {U}$ with dense image of a $K$-variety $U$ into a reduced proper flat $V$-scheme $\overline {U}$. A morphism $(U,\overline {U})\to (T,\overline {T})$ of pairs is a map $\overline {U}\to \overline {T}$ which sends $U$ to $T$. In the case that the pairs represent log-regular schemes, this is the same as a map of log-schemes. For a pair $(U,\overline {U})$, we set $V_U:=\Gamma (\overline {U},{\mathcal O} _{\overline {U}})$ and $K_U:=\Gamma (\overline {U}_K,{\mathcal O} _{\overline {U}})$. The ring $K_U$ is a product of several finite extensions of $K$ (labeled by the connected components of $\overline {U}$) and, if $\overline {U}$ is normal, $V_U$ is the product of the corresponding rings of integers.

A semistable pair over $K$ [Reference BeilinsonBei12, 2.2] is a pair of schemes $(U,\overline {U})$ over $(K,V)$ such that:

1. (i) $\overline {U}$ is regular and proper over $V$;

2. (ii) $\overline {U}\setminus U$ is a divisor with normal crossings on $\overline {U}$;

3. (iii) the closed fiber $\overline {U}_0$ of $\overline {U}$ is reduced and its irreducible components are regular.

A closed fiber is taken over the closed points of $V_U$. We will think of semistable pairs as log-schemes equipped with log-structure given by the divisor $\overline {U}\setminus U$. The closed fiber $\overline {U}_0$ has the induced log-structure.

A semistable pair over $\overline {K}$ [Reference BeilinsonBei12, 2.2] is a pair of connected schemes $(T,\overline {T})$ over $(\overline {K},\overline {V})$ such that there exist a semistable pair $(U,\overline {U})$ over $K$ and a $\overline {K}$-point $\alpha : K_U\to \overline {K}$ such that $(T,\overline {T})$ is isomorphic to the base change $(U_{\overline {K}},\overline {U}_{\overline {V}})$. We will denote by ${\mathcal {P}}_{\overline {K}}^{\rm ss}$ the category of semistable pairs over $\overline {K}$.

Let, for just a moment, $K$ be any field of characteristic 0. A geometric pair over $K$ is a pair $(U,\overline {U})$ of varieties over $K$ such that $\overline {U}$ is proper and $U\subset \overline {U}$ is open and dense. We say that the pair $(U,\overline {U})$ is an nc-pair if $\overline {U}$ is regular and $\overline {U}\setminus U$ is a divisor with normal crossings in $\overline {U}$; it is a strict nc-pair if the irreducible components of $U\setminus \overline {U}$ are regular. A morphism of pairs $f:(U_1,\overline {U}_1)\to (U,\overline {U})$ is a map $\overline {U}_1\to \overline {U}$ that sends $U_1$ to $U$. We denote the category of nc-pairs over $K$ by ${\mathcal {P}}_K^{\rm nc}$.

For the category ${\mathcal {P}}_{\overline {K}}^{\rm ss}$ mentioned above, let $\gamma : (U,\overline {U})\to U$ denote the forgetful functor. Beilinson proved [Reference BeilinsonBei12, 2.5] that the category $({\mathcal {P}}_{\overline {K}}^{\rm ss},\gamma )$ forms a base for $\mathcal {V}ar_{\overline {K},h}$. This implies that $\gamma$ induces an equivalence of the topoi

\[ \gamma: {\rm Shv}_h({\mathcal{P}}_{\overline{K}}^{\rm ss})\stackrel{\sim}{\to} {\rm Shv}_h(\mathcal{V}ar_{\overline{K}}). \]

Similarly for the categories ${\mathcal {P}}_{K}^{\rm ss}$ and ${\mathcal {P}}_K^{\rm nc}$ (and the category $\mathcal {V}ar_{K}$).

4.3.4 Definitions of cohomology sheaves

We will now recall briefly the definition of geometric syntomic cohomology, i.e., syntomic cohomology over $\overline {K}$, from [Reference Nekovář and NiziołNN16], and the related cohomologies from [Reference BeilinsonBei13].

(i) Absolute crystalline cohomology. For $(U,\overline {U})\in {\mathcal {P}}^{\rm ss}_{\overline {K}}$, $r\geq 0$, we have the absolute crystalline cohomology complexes and their completions

\begin{align*} \mathrm{R}\Gamma_{\operatorname{cr} }(U,\overline{U},{\mathcal{J}}^{[r]})_n&:=\mathrm{R}\Gamma_{\operatorname{cr} }(\overline{U}_{n,\operatorname{\acute{e}t} },\mathrm{R} u_*{\mathcal{J}}^{[r]}),\\ \mathrm{R}\Gamma_{\operatorname{cr} }(U,\overline{U},{\mathcal{J}}^{[r]})&:= \operatorname{holim} _n\mathrm{R}\Gamma_{\operatorname{cr} }(U,\overline{U},{\mathcal{J}}^{[r]})_n,\\ \mathrm{R}\Gamma_{\operatorname{cr} }(U,\overline{U},{\mathcal{J}}^{[r]})_\mathbf{Q}&:=\mathrm{R}\Gamma_{\operatorname{cr} }(U,\overline{U},{\mathcal{J}}^{[r]})\otimes\mathbf{Q}_p, \end{align*}

where $u: \overline {U}_{n,\operatorname {cr} }\to \overline {U}_{n,\operatorname {\acute {e}t} }$ is the natural projection. The complex $\mathrm {R}\Gamma _{\operatorname {cr} }(U,\overline {U})$ is a perfect $\mathbf {A}_{\operatorname {cr} }$-complex and

\[ \mathrm{R}\Gamma_{\operatorname{cr} }(U,\overline{U})_n\simeq \mathrm{R}\Gamma_{\operatorname{cr} }(U,\overline{U})\otimes^{L}_{\mathbf{A}_{\operatorname{cr} }}{\mathbf{A}_{\operatorname{cr} }}/p^n\simeq \mathrm{R}\Gamma_{\operatorname{cr} }(U,\overline{U})\otimes^{L}{\mathbf{Z}}/p^n. \]

In general, we have $\mathrm {R}\Gamma _{\operatorname {cr} }(U,\overline {U},{\mathcal {J}}^{[r]})_n\simeq \mathrm {R}\Gamma _{\operatorname {cr} }(U,\overline {U},{\mathcal {J}}^{[r]})\otimes ^{L}{\mathbf {Z}}/p^n$. Moreover, by [Reference TsujiTsu99a, 1.6.3 and 1.6.4],

\[ J^{[r]}_{\operatorname{cr} }=\mathrm{R}\Gamma_{\operatorname{cr} }(\operatorname{Spec} (\overline{K}),\operatorname{Spec} (\overline{V}),{\mathcal{J}}^{[r]}). \]

The absolute crystalline cohomology complexes are filtered $E_{\infty }$ algebras over $\mathbf {A}_{\operatorname {cr} ,n}$, $\mathbf {A}_{\operatorname {cr} }$, or $\mathbf {A}_{\operatorname {cr} ,\mathbf {Q}}$, respectively. Moreover, the rational ones are filtered commutative dg algebras.

Let ${\mathcal {J}}^{[r]}_{\operatorname {cr} }$ and ${\mathcal {A}}_{\operatorname {cr} }$ be the $h$-sheafifications on $\mathcal {V}ar_{\overline {K}}$ of the presheaves sending $(U,\overline {U})\in {\mathcal {P}}^{\rm ss}_{\overline {K}}$ to $\mathrm {R}\Gamma _{\operatorname {cr} }(U,\overline {U},{\mathcal {J}}^{[r]})$ and $\mathrm {R}\Gamma _{\operatorname {cr} }(U,\overline {U})$, respectively. Let ${\mathcal {J}}^{[r]}_{\operatorname {cr} ,n}$ and ${\mathcal {A}}_{\operatorname {cr} ,n}$ denote the $h$-sheafifications of the mod-$p^n$ versions of the respective presheaves; and let ${\mathcal {J}}^{[r]}_{\operatorname {cr} ,\mathbf {Q}}$ and ${\mathcal {A}}_{\operatorname {cr} ,\mathbf {Q}}$ be the $h$-sheafifications of the rational versions of the same presheaves.

For $X\in \mathcal {V}ar_{\overline {K}}$, set $\mathrm {R}\Gamma _{\operatorname {cr} }(X_h):=\mathrm {R}\Gamma (X_h,{\mathcal {A}}_{\operatorname {cr} })$. It is a filtered (by $\mathrm {R}\Gamma (X_h,{\mathcal {J}}^{[r]}_{\operatorname {cr} })$, $r\geq 0$) $E_{\infty }$ $\mathbf {A}_{\operatorname {cr} }$-algebra equipped with the Frobenius action $\phi$. The Galois group $G_K$ acts on ${\mathcal {V}}ar_{\overline {K}}$ and it acts on $X\mapsto \mathrm {R}\Gamma _{\operatorname {cr} }(X_h)$ by transport of structure. If $X$ is defined over $K$, then $G_K$ acts naturally on $\mathrm {R}\Gamma _{\operatorname {cr} }(X_h)$.

(ii) Geometric syntomic cohomology. For $r\geq 0$, the mod-$p^n$, completed, and rational syntomic complexes $\mathrm {R}\Gamma _{\operatorname {syn}}(U,\overline {U},r)_n$, $\mathrm {R}\Gamma _{\operatorname {syn}}(U,\overline {U},r)$, and $\mathrm {R}\Gamma _{\operatorname {syn}}(U,\overline {U},r)_\mathbf {Q}$ are defined by the formulas

\begin{align*} \mathrm{R}\Gamma_{\operatorname{syn}}(U,\overline{U},r)_n& := [\mathrm{R}\Gamma_{\operatorname{cr} }(U,\overline{U},{\mathcal{J}}^{[r]})_n {\buildrel {p^r-\phi} \over \longrightarrow} \mathrm{R}\Gamma_{\operatorname{cr} }(U,\overline{U})_n)],\\ \mathrm{R}\Gamma_{\operatorname{syn}}(U,\overline{U},r)& :=\operatorname{holim} _n\mathrm{R}\Gamma_{\operatorname{syn}}(U,\overline{U},r)_n,\\ \mathrm{R}\Gamma_{\operatorname{syn}}(U,\overline{U},r)_{\mathbf{Q}}& := [\mathrm{R}\Gamma_{\operatorname{cr} }(U,\overline{U},{\mathcal{J}}^{[r]})_\mathbf{Q} {\buildrel {1-\phi_r} \over \longrightarrow} \mathrm{R}\Gamma_{\operatorname{cr} }(U,\overline{U})_\mathbf{Q})]. \end{align*}

We have $\mathrm {R}\Gamma _{\operatorname {syn}}(U,\overline {U},r)_n\simeq \mathrm {R}\Gamma _{\operatorname {syn}}(U,\overline {U},r)\otimes ^{L}{\mathbf {Z}}/p^n$. Let ${\mathcal {S}}^{\prime }(r)$ be the $h$-sheafification on $\mathcal {V}ar_{\overline {K}}$ of the presheaf sending $(U,\overline {U})\in {\mathcal {P}}^{\rm ss}_{\overline {K}}$ to $\mathrm {R}\Gamma _{\operatorname {syn}}(U,\overline {U},r)$. Let ${\mathcal {S}}^{\prime }_n(r)$ and ${\mathcal {S}}^{\prime }(r)_{\mathbf {Q}}$ denote the $h$-sheafifications of the mod-$p^n$ and the rational versions of the same presheaf, respectively.

For $r\geq 0$, set $\mathrm {R}\Gamma _{\operatorname {syn}}(X_h,r)_n=\mathrm {R}\Gamma (X_h,{\mathcal {S}}^{\prime }_n(r))$, $\mathrm {R}\Gamma _{\operatorname {syn}}(X_h,r):=\mathrm {R}\Gamma (X_h,{\mathcal {S}}^{\prime }(r)_\mathbf {Q})$. We have

\begin{align*} \mathrm{R}\Gamma_{\operatorname{syn}}(X_h,r)_n &\simeq [\mathrm{R}\Gamma(X_h,{\mathcal{J}}^{[r]}_{\operatorname{cr} ,n})\stackrel{p^r-\phi}{\longrightarrow}\mathrm{R}\Gamma(X_h,{\mathcal{A}}_{\operatorname{cr} ,n})],\\ \mathrm{R}\Gamma_{\operatorname{syn}}(X_h,r) &\simeq [\mathrm{R}\Gamma(X_h,{\mathcal{J}}^{[r]}_{\operatorname{cr} ,{\mathbf{Q}}})\stackrel{1-\phi_r}{\longrightarrow}\mathrm{R}\Gamma(X_h,{\mathcal{A}}_{\operatorname{cr} ,{\mathbf{Q}}})]. \end{align*}

The direct sum $\bigoplus _{r\geq 0}\mathrm {R}\Gamma _{\operatorname {syn}}(X_h,r)$ is a graded $E_{\infty }$ algebra over ${\mathbf {Z}}_p$.

(iii) de Rham cohomology. Consider the presheaf $(U,\overline {U})\mapsto \mathrm {R}\Gamma _{\operatorname {dR} }(U,\overline {U}):=\mathrm {R}\Gamma (\overline {U},\Omega ^{\scriptscriptstyle \bullet }_{(U,\overline {U})})$ of filtered dg $K$-algebras on ${\mathcal {P}}^{\rm nc}_K$. Let ${\mathcal {A}}_{\operatorname {dR} }$ be its $h$-sheafification. It is a sheaf of filtered $K$-algebras on $\mathcal {V}ar_K$. For $X\in \mathcal {V}ar_K$, we have Deligne's de Rham complex of $X$ equipped with Deligne's Hodge filtration: $\mathrm {R}\Gamma _{\operatorname {dR} }(X_h):=\mathrm {R}\Gamma (X_h,{\mathcal {A}}_{\operatorname {dR} })$.

(iv) Beilinson–Hyodo–Kato cohomology. Let ${\mathcal {A}}^B_{\operatorname {HK} }$ be the $h$-sheafification of the presheaf of (arithmetic) Beilinson–Hyodo–Kato cohomology $(U,\overline {U})\mapsto \mathrm {R}\Gamma ^B_{\operatorname {HK} }(U,\overline {U})_{\mathbf {Q}}$ on ${\mathcal {P}}_{K}^{\rm ss}$; this is an $h$-sheaf of $E_{\infty }$ $F$-algebras on ${\mathcal {V}}ar_{K}$ equipped with a $\phi$-action and a derivation $N$ such that $N\phi =p\phi N$. For $X\in {\mathcal {V}}ar_{K}$, set $\mathrm {R}\Gamma ^B_{\operatorname {HK} }(X_h):=\mathrm {R}\Gamma (X_h,{\mathcal {A}}^B_{\operatorname {HK} })$.

Let ${\mathcal {A}}^B_{\operatorname {HK} }$ be the $h$-sheafification of the presheaf $(U,\overline {U})\mapsto \mathrm {R}\Gamma ^B_{\operatorname {HK} }(U,\overline {U})_{\mathbf {Q}}$ of (geometric) Beilinson–Hyodo–Kato cohomology on ${\mathcal {P}}^{\rm ss}_{\overline {K}}$. This is an $h$-sheaf of $E_{\infty }$ $F^{\rm nr}$-algebras, where $F^{\rm nr}$ is the maximal unramified extension of $F$, equipped with a $\phi$-action and locally nilpotent derivation $N$ such that $N\phi =p\phi N$. For $X\in \mathcal {V}ar_{\overline {K}}$, set $\mathrm {R}\Gamma ^B_{\operatorname {HK} }(X_h):=\mathrm {R}\Gamma (X_h,{\mathcal {A}}^B_{\operatorname {HK} }).$ We have the Beilnson–Hyodo–Kato quasi-isomorphism

\[ \rho_h^B: \mathrm{R}\Gamma_{\operatorname{HK} }^B(X_h)\otimes_{F^{\rm nr}}\overline{K}\stackrel{\sim}{\to} \mathrm{R}\Gamma_{\operatorname{dR} }(X_h). \]

(v) Comparison statements. The $h$-topology definitions of cohomology are often compatible with the original definitions.

Lemma 4.11 We have the following comparison statements.

(i) For $(U,\overline {U})\in {\mathcal {P}}^{\rm nc}_L$, $L=K,\overline {K}$, the canonical map $\mathrm {R}\Gamma _{\operatorname {dR} }(U,\overline {U})\stackrel {\sim }{\to } \mathrm {R}\Gamma _{\operatorname {dR} }(U_h)$ is a filtered quasi-isomorphism [Reference BeilinsonBei12, 2.4].

(ii) For any $(U,\overline {U})\in {\mathcal {P}}^{\rm ss}_{\overline {K}}$, $r\geq 0$, the canonical maps

\[ \mathrm{R}\Gamma_{\operatorname{cr} }(U,\overline{U},{\mathcal{J}}^{[r]})_{\mathbf{Q}}\stackrel{\sim}{\to} \mathrm{R}\Gamma(U_h,{\mathcal{J}}^{[r]}_{\operatorname{cr} })_{\mathbf{Q}},\quad \mathrm{R}\Gamma^B_{\operatorname{HK} }(U,\overline{U})\stackrel{\sim}{\to}\mathrm{R}\Gamma^B_{\operatorname{HK} }(U_h) \]

are quasi-isomorphisms (see [Reference BeilinsonBei13, 2.4] and [Reference Nekovář and NiziołNN16, Proposition 3.21]). In particular,

\[ \mathrm{R}\Gamma_{\operatorname{syn}}(U,\overline{U},r)\stackrel{\sim}{\to}\mathrm{R}\Gamma_{\operatorname{syn}}(U_h,r). \]

(iii) For any arithmetic pair $(U,\overline {U})$ that is fine, log-smooth over ${\mathcal O} _K^{\times }$, and of Cartier type, the canonical map

\[ \mathrm{R}\Gamma^B_{\operatorname{HK} }(U,\overline{U})\stackrel{\sim}{\to}\mathrm{R}\Gamma^B_{\operatorname{HK} }(U_h) \]

is a quasi-isomorphism [Reference Nekovář and NiziołNN16, Proposition 3.18].

4.3.5 Poincaré lemma

We will recall the Poincaré lemma of Beilinson [Reference BeilinsonBei13] and its syntomic cohomology version [Reference Nekovář and NiziołNN16].