2. Recollections

Let us first recall the construction of $ H^i_{\text{crys}}(X/\mathbb{B}^+_{\text{dR}})$ . The original construction in [ Reference Bhatt, Morrow and ScholzeBMS18 ] is to construct an explicit complex for each small affinoid X, and then glue. In [ Reference GuoGuo21a ], Guo defined an infinitesimal site and reconstruct $ H^i_{\text{crys}}(X/\mathbb{B}^+_{\text{dR}})$ as the cohomology of the structure sheaf on this site. Further, the $\mathbb{B}^+_{\text{dR}}$ -prismatic site is introduced in [ Reference GuoGuo21b ], which unifies the previous constructions. We will use the formulation of $\mathbb{B}^+_{\text{dR}}$ -prismatic site for convenience, but the original construction of [ Reference Bhatt, Morrow and ScholzeBMS18 ] can also be invoked here.

For our purpose, we only need to know that $ H^i_{\text{crys}}(X/\mathbb{B}^+_{\text{dR}})$ can be computed as the cohomology of a sheaf of complexes

\begin{equation*}L\eta_{\xi} \Delta_{X/\mathbb{B}^+_{\text{dR}}}\end{equation*}

on the étale site $X_{\acute{\text{e}}\text{t}}$ of X, where $L\eta_{\xi}$ is the decalage functor with respect to $\xi$ as defined in [ Reference Bhatt, Morrow and ScholzeBMS18 , section 6], and $\Delta_{X/\mathbb{B}^+_{\text{dR}}}$ is the sheaf of complexesFootnote 5 of $\mathbb{B}^+_{\text{dR}}$ -modules on $X_{\acute{\text{e}}\text{t}}$ which sends every affinoid to its derived $\mathbb{B}^+_{\text{dR}}$ -prismatic cohomology (with respect to the structure sheaf) as defined in [ Reference GuoGuo21b , section 2]. In other words, we have

(1) \begin{equation} H^i_{\text{crys}}(X/\mathbb{B}^+_{\text{dR}})\cong R^i\Gamma(X_{\acute{\text{e}}\text{t}},L\eta_{\xi} \Delta_{X/\mathbb{B}^+_{\text{dR}}}),\end{equation}

which is [ Reference GuoGuo21b , theorem 6·0·1].

It is proved in [ Reference GuoGuo21b , theorem 5·1·1] and the discussion before it that we have a canonical quasi-isomorphism

(2) \begin{equation} L\eta_{\xi} \Delta_{X/\mathbb{B}^+_{\text{dR}}} \otimes^L_{\mathbb{B}^+_{\text{dR}}} \mathbb{B}^+_{\text{dR}}/\xi\cong H^{\bullet} (\Delta_{X/\mathbb{B}^+_{\text{dR}}} /\xi)\cong \Omega_X^{\bullet},\end{equation}

where the first isomorphism is [ Reference Bhatt, Morrow and ScholzeBMS18 , proposition 6·12], and $\Omega_X^{\bullet}$ is the de Rham complex of X.

Moreover, there is a natural quasi-isomorphism

(3) \begin{equation} R\Gamma(X_{\acute{\text{e}}\text{t}}, \Delta_{X/\mathbb{B}^+_{\text{dR}}})\cong R\Gamma(X_{\text{pro}\acute{\text{e}}\text{t}}, \mathbb{B}^+_{\text{dR}})\cong R\Gamma(X_{\acute{\text{e}}\text{t}}, \mathbb{Q}_p) \otimes_{\mathbb{Q}_p} \mathbb{B}^+_{\text{dR}},\end{equation}

where $\mathbb{B}^+_{\text{dR}}$ is, by abuse of notation, the $\mathbb{B}^+_{\text{dR}}$ -period sheaf on $X_{\text{pro}\acute{\text{e}}\text{t}}$ as defined in [ Reference ScholzeSch13 , definition 6·1]. The first isomorphism is [ Reference GuoGuo21b , theorem 7·2·1], the second is in the proof of [ Reference ScholzeSch13 , theorem 8·4]. The canonical comparison morphism

\begin{equation*} H^i_{\text{crys}}(X/\mathbb{B}^+_{\text{dR}}) \longrightarrow H^{i}_{\acute{\text{e}}\text{t}}(X,\mathbb{Q}_p) \otimes_{\mathbb{Q}_p} \mathbb{B}_{\text{dR}}^+\end{equation*}

is induced from the canonical map

\begin{equation*}\iota:L\eta_{\xi} \Delta_{X/\mathbb{B}^+_{\text{dR}}}\longrightarrow \Delta_{X/\mathbb{B}^+_{\text{dR}}}\end{equation*}

together with the identification. Note that $\iota$ exists because $\Delta_{X/\mathbb{B}^+_{\text{dR}}} \in D_{\geq 0}$ and $ H^0(\Delta_{X/\mathbb{B}^+_{\text{dR}}} )$ is $\xi$ -torsion-free.

Lastly, it follows from the proof of [ Reference GuoGuo21b , theorem 7·2·1] that we have a natural quasi-isomorphism

(4) \begin{equation} \overline{\Delta}_{X/\mathbb{B}^+_{\text{dR}}} := \Delta_{X/\mathbb{B}^+_{\text{dR}}} \otimes^L_{\mathbb{B}^+_{\text{dR}}} \mathbb{B}^+_{\text{dR}}/\xi\cong R\nu_{*} \hat{\mathcal{O}}_X\end{equation}

where $\nu: X_{\text{pro}\acute{\text{e}}\text{t}} \rightarrow X_{\acute{\text{e}}\text{t}}$ and $\hat{\mathcal{O}}_X$ are as in the introduction.

3. A refinement of $L\eta$ -functor

In this section, we construct a refinement of the decalage functor $L\eta$ introduced in [ Reference Bhatt, Morrow and ScholzeBMS18 ], which is a special case of the construction in [ Reference Berthelot and OgusBO78 , definition 8·6] with p replaced by $\mathcal{I}$ . We work in the same setting as [ Reference Bhatt, Morrow and ScholzeBMS18 ].

Let $(T, \mathcal{O}_T)$ be a ringed topos, and $D(\mathcal{O}_T)$ the derived category of $\mathcal{O}_T$ -modules. Let $\mathcal{I} \subset \mathcal{O}_T$ be an invertible ideal sheaf.

Recall that a complex $K^{\bullet}$ of $\mathcal{O}_T$ -modules is said to be $\mathcal{I}$ -torsion-free if the canonical map $\mathcal{I}\otimes K^i \rightarrow K^i$ is injective for every i, and we can define a complex $(\eta_{\mathcal{I}} K)^{\bullet}$ with terms

\begin{equation*}(\eta_{\mathcal{I}} K )^i= \{ x \in K^i | dx \in \mathcal{I} K^{i+1} \} \otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes i},\end{equation*}

where $K^{\bullet}$ is a $\mathcal{I}$ -torsion-free complex, and there is a natural differential map making $\eta_{\mathcal{I}} K^{\bullet} $ a complex. By [ Reference Bhatt, Morrow and ScholzeBMS18 , lemma 6·4], we have

\begin{equation*} H^i(\eta_{\mathcal{I}} K^{\bullet}) \cong ( H^i(K^{\bullet})/ H^i(K^{\bullet})[\mathcal{I}]) \otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes i},\end{equation*}

so we can derive the construction to obtain a functor

\begin{equation*}L\eta_{\mathcal{I}}: D(\mathcal{O}_T) \longrightarrow D(\mathcal{O}_T).\end{equation*}

Definition 3·1. Let $m \in \mathbb{Z}$ , and $K^{\bullet }$ an $\mathcal{I}$ -torsion-free complex of $\mathcal{O}_T$ -modules. Define a new complex $(\eta_{\mathcal{I},m} K)^{\bullet}$ with terms

\begin{equation*}(\eta_{\mathcal{I},m} K)^i =\begin{cases} (\eta_{\mathcal{I}}K)^i & i \geq m \\ K^i \otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes m} & i < m. \end{cases}\end{equation*}

The differential is inherited from that of $\eta_{\mathcal{I}}K$ (resp. $K^{\bullet}$ ( $\otimes \text{Id}_{\mathcal{I}^{\otimes m}}$ )) for $i \geq m$ (resp. $i< m-1$ ). For $i=m-1$ , the differential

\begin{equation*}d: K^{m-1} \otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes m} \longrightarrow (\eta_{\mathcal{I}}K)^m = \{ x \in K^m | dx \in \mathcal{I} K^{m+1} \} \otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes m}\end{equation*}

is defined to be $d \otimes \text{Id}_{\mathcal{I}^{\otimes m}}$ .

We first compute the cohomology of $\eta_{\mathcal{I},m} K^{\bullet}$ .

Lemma 3·2. Let $K^{\bullet}$ be an $\mathcal{I}$ -torsion-free complex, then we have a natural isomorphism

\begin{equation*} H^i(\eta_{\mathcal{I},m} K^{\bullet}) \cong\begin{cases} H^i(\eta_{\mathcal{I}} K^{\bullet}) & i > m \\ H^i( K^{\bullet}) \otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes m} & i \leq m. \end{cases}\end{equation*}

We can now dervie the $\eta_{\mathcal{I},m}$ -construction to obtain a functor

\begin{equation*}L\eta_{\mathcal{I},m}: D(\mathcal{O}_T) \longrightarrow D(\mathcal{O}_T).\end{equation*}

The most important property of $L\eta_{\mathcal{I},m}$ is that it forms a filtration. In other words, for every $\mathcal{I}$ -torsion-free complex $K^{\bullet}$ , we have a canonical filtration

\begin{equation*}\cdots\subset\eta_{\mathcal{I},m+1} K^{\bullet} \subset \eta_{\mathcal{I},m} K^{\bullet}\subset \cdots.\end{equation*}

The inclusion $\eta_{\mathcal{I},m+1} K^{\bullet} \subset \eta_{\mathcal{I},m} K^{\bullet} $ at degrees $i\geq m+1$ (resp. $i\leq m-1$ ) is the identity of $(\eta_{\mathcal{I}} K^{\bullet})^i$ (resp. $\text{id}_{K^i \otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes m}} \otimes (\mathcal{I} \hookrightarrow \mathcal{O}_T) $ ). For degree m, it is given by the canonical inclusion

\begin{equation*}(K^{m} \otimes\mathcal{I} )\otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes m} \longrightarrow (\eta_{\mathcal{I}}K)^m = \{ x \in K^m | dx \in \mathcal{I} K^{m+1} \} \otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes m}\end{equation*}

induced by $(\mathcal{I} \hookrightarrow \mathcal{O}_T )\otimes \text{id}_{\mathcal{I}^{\otimes m}}$ . We can summarise the information as in the diagram

We can compute the graded pieces of the filtration.

Lemma 3·3. Let $K^{\bullet}$ be an $\mathcal{I}$ -torsion-free complex, we have a canonical isomorphism

\begin{equation*}\eta_{\mathcal{I},m} K^{\bullet}/ \eta_{\mathcal{I},m+1} K^{\bullet}\cong\tau_{\leq m} (K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{O}_T/\mathcal{I}) \otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes m}.\end{equation*}

Proof. This is obvious except for degree m. We compute that

\begin{equation*}(\eta_{\mathcal{I}}K)^{m}/(K^m \otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes m+1} )\cong\frac{\{ x \in K^m | dx \in \mathcal{I} K^{m+1} \}}{\mathcal{I} K^m}\otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes m}\end{equation*}

\begin{equation*}\cong\{ \bar{x} \in K^m \otimes_{\mathcal{O}_T} \mathcal{O}_T/\mathcal{I} \ | \ d\bar{x} = 0 \in K^{m+1} \otimes_{\mathcal{O}_T} \mathcal{O}_T/\mathcal{I}\}\otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes m},\end{equation*}

but this is exactly the degree m part of $\tau_{\leq m} (K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{O}_T/\mathcal{I}) \otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes m}$ by definition of $\tau_{\leq m} $ .

A good way to package the information of $\eta_{\mathcal{I},m}$ -construction is to view $\{ L \eta_{\mathcal{I},m} \}_{m \in \mathbb{Z}}$ as a functor

\begin{equation*}L \eta_{\mathcal{I},\bullet} : D(\mathcal{O}_T)\longrightarrow FD(\mathcal{O}_T)\end{equation*}

from the derived category $D(\mathcal{O}_T)$ to the filtered derived category of $\mathcal{O}_T$ . Moreover, the graded quotient functor can be identified as

\begin{equation*}\text{Gr}^m(L \eta_{\mathcal{I},\bullet} ({-})) \cong\tau_{\leq m} ({-}\;\! \otimes^L_{\mathcal{O}_T} \mathcal{O}_T/\mathcal{I}) \otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes m}.\end{equation*}

The next result we want to establish is the behaviour of $L \eta_{\mathcal{I},\bullet}$ under the quotient by $\mathcal{I}$ . Recall as in [ Reference Bhatt, Morrow and ScholzeBMS18 , proposition 6·12] we have a canonical quasi-isomorphism

\begin{equation*}(L \eta_{\mathcal{I}} K )\otimes^L_{\mathcal{O}_T} \mathcal{O}_T/\mathcal{I} \cong H^{\bullet}( K/\mathcal{I}),\end{equation*}

where $ H^{\bullet}( K/\mathcal{I})$ is the complex whose degree i th term is $ H^i(K \otimes^L_{\mathcal{O}_T} \mathcal{O}_T/\mathcal{I}) \otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes i}$ and the differential is given by the Bockstein map with respect to

\begin{equation*}0 \longrightarrow \mathcal{I}/\mathcal{I}^2 \longrightarrow \mathcal{O}_T/\mathcal{I}^2 \longrightarrow \mathcal{O}_T/\mathcal{I} \longrightarrow 0.\end{equation*}

We will prove a slight refinement of the result for $L \eta_{\mathcal{I},\bullet}$ .

Let $K^{\bullet}$ be a $\mathcal{I}$ -torsion-free complex, we first observe that there is a canonical filtration

\begin{equation*}\eta_{\mathcal{I},m} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{I} \subset \eta_{\mathcal{I},m+1} K^{\bullet}\subset\eta_{\mathcal{I},m} K^{\bullet},\end{equation*}

which in particular explains why $\eta_{\mathcal{I},m} K^{\bullet}/\eta_{\mathcal{I},m+1} K^{\bullet}$ is an $\mathcal{O}_T / \mathcal{I}$ -complex. We have identified the second graded piece, and the first graded piece can also be computed as follows.

Proposition 3·4. Let $K^{\bullet}$ be an $\mathcal{I}$ -torsion-free complex, then we have a canonical quasi-isomorphism

\begin{equation*}\eta_{\mathcal{I},m+1} K^{\bullet}/\eta_{\mathcal{I},m} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{I}\cong F_{m+1} H^{\bullet}( K/\mathcal{I}),\end{equation*}

where $ F_{m+1}$ is the stupid filtration of complexes, i.e.

\begin{equation*}( F_{m+1} C^{\bullet})^i =\begin{cases} C^i & i \geq m+1 \\ 0 & i \leq m. \end{cases}\end{equation*}

Proof. We see immediately from the definition that the degree $\geq m+1$ (resp. $\leq m-1$ ) part of the left-hand side is $(\eta_{\mathcal{I}} K^{\bullet})^i \otimes_{\mathcal{O}_T} \mathcal{O}_T/\mathcal{I}$ (resp. 0). It remains to identify the degree m-part of the LHS.

By definition

\begin{equation*}(\eta_{\mathcal{I},m+1} K^{\bullet}/\eta_{\mathcal{I},m} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{I})^m =\frac{K^m }{ \{ x \in K^m | dx \in \mathcal{I} K^{m+1} \} }\otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes m+1},\end{equation*}

which is identified through the differential with

\begin{equation*}B^{m+1}(K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{O}_T / \mathcal{I}) \otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes m+1},\end{equation*}

where $B^{m+1}(K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{O}_T / \mathcal{I})$ is the image of $(K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{O}_T / \mathcal{I})^m$ inside $(K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{O}_T / \mathcal{I})^{m+1}$ .

We first prove that the m th cohomology of the LHS is trivial. This is equivalent to the differential map

\begin{equation*}\frac{K^m }{ \{ x \in K^m | dx \in \mathcal{I} K^{m+1} \} }\otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes m+1}\longrightarrow (\eta_{\mathcal{I}} K^{\bullet})^{m+1} \otimes_{\mathcal{O}_T} \mathcal{O}_T/\mathcal{I}\end{equation*}

being injective. But this is clear, if $x \in K^m$ is mapped to $\mathcal{I} \{ y \in K^{m+1} | dy \in \mathcal{I} K^{m+2} \} $ , namely $dx \in \mathcal{I} \{ y \in K^{m+1} | dy \in \mathcal{I} K^{m+2} \} \subset \mathcal{I} K^{m+1}$ , proving that $\bar{x}$ is zero in the left-hand side.

We can now proceed exactly as in the proof of [ Reference Bhatt, Morrow and ScholzeBMS18 , proposition 6·12]. We have canonical maps

\begin{equation*} (\eta_{\mathcal{I}} K^{\bullet})^{i} \otimes_{\mathcal{O}_T} \mathcal{O}_T/\mathcal{I} \longrightarrow Z^i(K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{O}_T/\mathcal{I}) \otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes i} \end{equation*}

which induces the quasi-isomorphism

\begin{equation*}\eta_{\mathcal{I}} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{O}_T/\mathcal{I}\cong H^{\bullet} (K/\mathcal{I}).\end{equation*}

We have a commutative diagram

where the differential on the right-hand side is induced from the Bockstein map, except the second to the lowest one, which is the canonical inclusion.

It induces a quasi-isomorphism in degrees $\geq m+2$ by [ Reference Bhatt, Morrow and ScholzeBMS18 , proposition 6·12], and we have seen that the degree m cohomology of both sides are trivial. Thus it remains to show that it induces a quasi-isomorphism in degree $m+1$ , this is the same as the proof of $loc.cit.$ .

We first prove that the map on $m+1$ th cohomology is surjective. Let $\bar{x}\in Z^{m+1}(K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{O}_T/\mathcal{I})$ be a cocycle under Bockstein map, then by definition there exists a lift $x \in K^{m+1}$ of $\bar{x}$ together with $y \in \mathcal{I} K^{m+1}$ such that $dx \equiv dy \ \text{mod} \ \mathcal{I}^2 K^{m+2}$ . This implies that $x \in \{ x \in K^{m+1} | dx \in \mathcal{I} K^{m+2} \} $ . Moreover, it tells us that $dx \in \mathcal{I} \{ x \in K^{m+2} | dx \in \mathcal{I} K^{m+3} \} $ , which means x defines a cocycle on the LHS that maps to $\bar{x}$ .

Now we show that the map on cohomology is injective. Let $x \in \{ x \in K^{m+1} | dx \in \mathcal{I} K^{m+2} \} $ whose reduction in $\{ x \in K^{m+1} | dx \in \mathcal{I} K^{m+2} \}/ \mathcal{I} \{ x \in K^{m+1} | dx \in \mathcal{I} K^{m+2} \}$ is a cocycle on the LHS. Moreover, we assume that its reduction $\bar{x} \in K^{m+1}/\mathcal{I}$ lies in $B^{m+1}(K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{O}_T / \mathcal{I})$ . By the isomorphism of the lower horizontal map, there exists $y \in K^m$ such that $dy \equiv x \ \text{mod} \ \mathcal{I} K^{m+1}$ . The cocycle condition implies that $dx \in \mathcal{I}^2 \{ x \in K^{m+2} | dx \in \mathcal{I} K^{m+3} \}$ , but this implies that $dy \equiv x \ \text{mod} \ \mathcal{I} \{ x \in K^{m+1} | dx \in \mathcal{I} K^{m+2} \}$ , i.e. x defines a trivial cohomological class on the LHS.

Corollary 3·5. Let $K^{\bullet}$ be a $\mathcal{I}$ -torsion-free complex, then

\begin{equation*} H^i(\eta_{\mathcal{I},m} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{O}_T /\mathcal{I} )\cong\begin{cases} H^i( H^{\bullet}(K/\mathcal{I})) & i \geq m+1 \\ Z^m( H^{\bullet}(K/\mathcal{I})) & i=m \\ H^i(K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{O}_T/\mathcal{I}) \otimes_{\mathcal{O}_T} \mathcal{I}^{\otimes m} & i \leq m-1. \end{cases}\end{equation*}

Moreover, the connecting morphism

\begin{equation*}\eta_{\mathcal{I},m} K^{\bullet}/\eta_{\mathcal{I},m+1} K^{\bullet}\longrightarrow \eta_{\mathcal{I},m+1} K^{\bullet}/(\eta_{\mathcal{I},m} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{I} )[1]\end{equation*}

of the distinguished triangle

\begin{equation*}\eta_{\mathcal{I},m+1} K^{\bullet}/(\eta_{\mathcal{I},m} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{I} )\longrightarrow \eta_{\mathcal{I},m} K^{\bullet}/(\eta_{\mathcal{I},m} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{I} )\longrightarrow \eta_{\mathcal{I},m} K^{\bullet}/\eta_{\mathcal{I},m+1} K^{\bullet}\longrightarrow [1]\end{equation*}

factorises as

where $\beta$ is the differential of the complex $ H^{\bullet}(K/\mathcal{I})$ , namely the Bockstein map.

Lastly, the long exact sequence associated to the distinguished triangle gives

\begin{equation*}0 \longrightarrow H^m(\eta_{\mathcal{I},m} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{O}_T /\mathcal{I})\longrightarrow H^m(\eta_{\mathcal{I},m} K^{\bullet}/\eta_{\mathcal{I},m+1} K^{\bullet})\end{equation*}

\begin{equation*}\longrightarrow H^{m+1}( \eta_{\mathcal{I},m+1} K^{\bullet}/(\eta_{\mathcal{I},m} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{I} ))\longrightarrow H^{m+1}( \eta_{\mathcal{I},m} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{O}_T /\mathcal{I})\longrightarrow 0,\end{equation*}

which is identified with

\begin{equation*}0 \longrightarrow Z^m( H^{\bullet}(K/\mathcal{I}))\longrightarrow H^{\bullet}(K/\mathcal{I})^m\overset{\beta}{\longrightarrow }Z^{m+1}( H^{\bullet}(K/\mathcal{I}))\longrightarrow H^{m+1}( H^{\bullet}(K/\mathcal{I}))\longrightarrow 0.\end{equation*}

Proof. The statement follows immediately from the long exact sequence associated to the distinguished triangle together with Proposition 3·4 and Lemma 3·3, once we have identified the connecting morphism.

The factorisation of the connecting morphism follows from an easy t-structure argument, by noting that the source (resp. target) lies in $D_{\leq m}(\mathcal{O}_T)$ (resp. $D_{\geq m}(\mathcal{O}_T)$ ). We now prove that the factorisation $\beta$ is the Bockstein differential map. This follows easily from a diagram chasing. We write down the diagram of the distinguished triangle

Given $\bar{x} \in Z^m(K^{\bullet}/\mathcal{I}K^{\bullet})$ representing an element of $ H^{\bullet}(K/\mathcal{I})^m$ , we choose a lift $x \in \{ x \in K^m | dx \in \mathcal{I} K^{m+1} \}$ of it. Then $\beta(\bar{x})$ is represented by dx viewed as an element of $ \mathcal{I} K^{m+1} / \mathcal{I} \{ x \in K^{m+1} | dx \in \mathcal{I} K^{m+2} \}$ , but this clearly also represents the Bockstein map of $\bar{x}$ .

We have another natural inclusion

\begin{equation*}\eta_{\mathcal{I},m+1} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{I}\subset\eta_{\mathcal{I},m} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{I} \subset \eta_{\mathcal{I},m+1} K^{\bullet}\end{equation*}

giving rise to the distinguished triangle

\begin{equation*}(\eta_{\mathcal{I},m} K^{\bullet}/\eta_{\mathcal{I},m+1} K^{\bullet}) (1)\longrightarrow \eta_{\mathcal{I},m+1} K^{\bullet}/(\eta_{\mathcal{I},m+1} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{I} )\longrightarrow \eta_{\mathcal{I},m+1} K^{\bullet}/(\eta_{\mathcal{I},m} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{I} )\end{equation*}

with connecting morphism

\begin{equation*} F_{m+1} H^{\bullet}(K/\mathcal{I})\cong\eta_{\mathcal{I},m+1} K^{\bullet}/(\eta_{\mathcal{I},m} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{I} )\longrightarrow (\eta_{\mathcal{I},m} K^{\bullet}/\eta_{\mathcal{I},m+1} K^{\bullet}) (1)[1]\end{equation*}

\begin{equation*}\cong\tau_{\leq m} (K^{\bullet} /\mathcal{I}) (m+1) [1],\end{equation*}

which has to be zero since the source and target sit in different cohomological degrees. Thus we have a splitting

\begin{equation*}\eta_{\mathcal{I},m+1} K^{\bullet}/(\eta_{\mathcal{I},m+1} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{I} )\cong(\eta_{\mathcal{I},m} K^{\bullet}/\eta_{\mathcal{I},m+1} K^{\bullet}) (1)\oplus\eta_{\mathcal{I},m+1} K^{\bullet}/(\eta_{\mathcal{I},m} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{I} ).\end{equation*}

Then Proposition 3·4 and Lemma 3·3 identify the direct summand.

Corollary 3·6. Let $K^{\bullet}$ be a $\mathcal{I}$ -torsion-free complex, we have an identification

\begin{equation*}\eta_{\mathcal{I},m+1} K^{\bullet}/(\eta_{\mathcal{I},m+1} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{I} )\cong(\tau_{\leq m} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{O}_T/ \mathcal{I}) (m+1)\oplus F_{m+1} H^{\bullet} (K^{\bullet}/\mathcal{I}),\end{equation*}

which splits the distinguished triangle

\begin{equation*}(\tau_{\leq m} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{O}_T/ \mathcal{I}) (m+1)\longrightarrow \eta_{\mathcal{I},m+1} K^{\bullet}/(\eta_{\mathcal{I},m+1} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{I} )\longrightarrow F_{m+1} H^{\bullet} (K^{\bullet}/\mathcal{I})\longrightarrow [1].\end{equation*}

Moreover, the distinguished triangle

\begin{equation*}\eta_{\mathcal{I},m+1} K^{\bullet}/(\eta_{\mathcal{I},m} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{I} )\longrightarrow \eta_{\mathcal{I},m} K^{\bullet}/(\eta_{\mathcal{I},m} K^{\bullet} \otimes_{\mathcal{O}_T} \mathcal{I} )\longrightarrow \eta_{\mathcal{I},m} K^{\bullet}/\eta_{\mathcal{I},m+1} K^{\bullet}\longrightarrow [1]\end{equation*}

is compatible with the splitting in the sense that we have a commutative diagram

where the the arrows a and b are the canonical map corresponding to the Hodge filtration and standard truncated filtration respectively, while c and d are graded quotient maps corresponding to them.

4. Proof

We now start the proof of the theorem. We first observe that the Bialynicki–Birula type construction of Fil $_m$ is fundamentally on cohomology groups, whereas the Hodge–Tate filtration, in the classical construction, originates from a filtration on complexes. Thus a natural way to proceed is to upgrade $\text{Fil}_m$ to a filtration on complexes and then compare the two filtration on the derived category level. This is achieved by the $L\eta_{\xi,\bullet}$ -operation introduced in the previous section. In some sense, we need to have a suitably derived Bialynicki–Birula construction.

Let $m \in \mathbb{Z}_{\geq 0}$ , and we consider

\begin{equation*}L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}}\end{equation*}

which is naturally a subcomplex of $L\eta_{\xi}\Delta_{X/\mathbb{B}^+_{\text{dR}}}$ . By Lemma 3·3, we have

\begin{equation*}Gr^m \Delta_{X/\mathbb{B}^+_{\text{dR}}}:=L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}}/L\eta_{\xi,m+1} {\Delta_{X/\mathbb{B}^+_{\text{dR}}}}^{6}\cong\tau_{\leq m}\overline{ \Delta}_{X/\mathbb{B}^+_{\text{dR}}} (m)\cong\tau_{\leq m} R\nu_* \hat{\mathcal{O}}_X (m),\end{equation*}

which implies that under the canonical map

\begin{equation*}L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}}\longrightarrow \xi^m \Delta_{X/\mathbb{B}^+_{\text{dR}}}\longrightarrow \overline{ \Delta}_{X/\mathbb{B}^+_{\text{dR}}}(m),\end{equation*}

Footnote 6 the image of $ H^i(X, L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}})$ inside

\begin{equation*} H^i(X,\overline{ \Delta}_{X/\mathbb{B}^+_{\text{dR}}})(m)\cong H^{i}_{\acute{\text{e}}\text{t}}(X,\mathbb{Q}_p) \otimes_{\mathbb{Q}_p} \mathbb{C} (m)\end{equation*}

is exactly (m th Tate twist of) the m th Hodge–Tate filtration.

We observe that there is a natural map

\begin{equation*} H^i(X, L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}})\longrightarrow H^i(X, L\eta_{\xi} \Delta_{X/\mathbb{B}^+_{\text{dR}}})\cap\xi^m H^i(X, \Delta_{X/\mathbb{B}^+_{\text{dR}}} ),\end{equation*}

and we want to prove that it induces an identification of $ H^i(X, L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}})$ with

\begin{equation*} H^i(X, L\eta_{\xi} \Delta_{X/\mathbb{B}^+_{\text{dR}}})\cap\xi^m H^i(X, \Delta_{X/\mathbb{B}^+_{\text{dR}}} )\cong H^i_{\text{crys}}(X/\mathbb{B}^+_{\text{dR}}) \cap\xi^m( H^{i}_{\acute{\text{e}}\text{t}}(X,\mathbb{Q}_p) \otimes_{\mathbb{Q}_p} \mathbb{B}_{\text{dR}}^+)),\end{equation*}

which completes the proof.

First proof. We prove that $ H^i(X, L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}})$ is $\xi$ -torsion-free by descending induction on m. The base case is when m is big enough so we have $L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}} = \Delta_{X/\mathbb{B}^+_{\text{dR}}}(m)$ , in which case the torsion-freeness follows from the primitive comparison theorem

\begin{equation*} H^i(X, L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}}) \cong H^i(X,\Delta_{X/\mathbb{B}^+_{\text{dR}}}(m))\cong H^i_{\acute{\text{e}}\text{t}}(X, \mathbb{Q}_p) \otimes_{\mathbb{Q}_p} \mathbb{B}^+_{\text{dR}}(m).\end{equation*}

Now assume that $ H^i(X, L\eta_{\xi,m+1} \Delta_{X/\mathbb{B}^+_{\text{dR}}})$ is $\xi$ -torsion-free, we want to prove that the same is true for $ H^i(X, L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}})$ . Using the long exact sequence associated to

\begin{equation*} L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}} \overset{\xi}{\longrightarrow }L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}}\longrightarrow L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}}/ \xi L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}}\longrightarrow [1],\end{equation*}

it is enough to prove that the natural map

\begin{equation*} H^i(X, L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}})\longrightarrow H^i(X, L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}}/ \xi L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}})\end{equation*}

is surjective.

We have the commutative diagram

corresponding to the map of distinguished triangles

We note that the map a is surjective by our induction hypothesis. Namely, the connecting map

\begin{equation*} H^{i} (X, Gr^m \Delta_{X/\mathbb{B}^+_{\text{dR}}})\longrightarrow H^{i+1}(X, L\eta_{\xi,m+1} \Delta_{X/\mathbb{B}^+_{\text{dR}}})\end{equation*}

is 0 since the right-hand side is torsion free by induction hypothesis.

We have that h is surjective as well. This is again because of our inductive hypothesis that $ H^{i+1}(X, L\eta_{\xi,m+1} \Delta_{X/\mathbb{B}^+_{\text{dR}}})$ is torsion-free, so the natural map

\begin{equation*}p: H^{i}(X, L\eta_{\xi,m+1} \Delta_{X/\mathbb{B}^+_{\text{dR}}})\longrightarrow H^{i}(X, L\eta_{\xi,m+1} \Delta_{X/\mathbb{B}^+_{\text{dR}}}/\xi L\eta_{\xi,m+1} \Delta_{X/\mathbb{B}^+_{\text{dR}}})\end{equation*}

is surjective, but Corollary 3·6 implies that

\begin{equation*} H^{i}(X, L\eta_{\xi,m+1} /\xi L\eta_{\xi,m+1} \Delta_{X/\mathbb{B}^+_{\text{dR}}})\cong H^i(X, Gr^m \Delta_{X/\mathbb{B}^+_{\text{dR}}}) (1)\oplus H^i(X,L\eta_{\xi,m+1} /\xi L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}})\end{equation*}

composing the projection to second factor with p implies that h is surjective.

Now we prove that f is surjective, which completes the proof. Given $x \in H^i(X, L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}}/\xi L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}})$ , we can find $y \in H^i(X, L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}})$ such that $d(f(y)) = a(y) = d(x)$ by surjectivity of a. Thus $x-f(y) \in \text{Ker}(d)=\text{Im}(b)$ , so there exists $w \in H^i(X,L\eta_{\xi,m+1} \Delta_{X/\mathbb{B}^+_{\text{dR}}}/\xi L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}})$ such that $b(w) =x-f(y)$ , the surjectivity of h tells us that $w= h(z)$ for $z \in H^i(X,L\eta_{\xi,m+1} \Delta_{X/\mathbb{B}^+_{\text{dR}}})$ , which implies that

\begin{equation*}x= f(y)+b(h(z)) = f(y + g(z)),\end{equation*}

proving the surjectivity of f.

Second proof. We prove that $ H^i(X, L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}})$ is $\xi$ -torsion-free by ascending induction on m. The base case $m=0$ is [ Reference Bhatt, Morrow and ScholzeBMS18 , theorem 13·19] since

\begin{equation*}L\eta_{\xi,0} \Delta_{X/\mathbb{B}^+_{\text{dR}}}= L\eta_{\xi} \Delta_{X/\mathbb{B}^+_{\text{dR}}},\end{equation*}

which follows from $\Delta_{X/\mathbb{B}^+_{\text{dR}}} \in D_{\geq 0}$ and $\xi$ -torsion-freeness of $ H^0(\Delta_{X/\mathbb{B}^+_{\text{dR}}})$ .

Now assume that $ H^i(X, L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}})$ is $\xi$ -torsion-free, we want to prove that the same is true for $ H^i(X, L\eta_{\xi,m+1} \Delta_{X/\mathbb{B}^+_{\text{dR}}})$ . It is enough to prove that the natural map

\begin{equation*} H^i(X, L\eta_{\xi,m+1} \Delta_{X/\mathbb{B}^+_{\text{dR}}})\longrightarrow H^i(X, L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}})\end{equation*}

is injective. Using the long exact sequence associated to the distinguished triangle

(5) \begin{equation} L\eta_{\xi,m+1} \Delta_{X/\mathbb{B}^+_{\text{dR}}} \longrightarrow L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}}\longrightarrow Gr^m \Delta_{X/\mathbb{B}^+_{\text{dR}}}\longrightarrow [1],\end{equation}

it is enough to show that

\begin{equation*} H^{i-1} (X, Gr^m \Delta_{X/\mathbb{B}^+_{\text{dR}}})\longrightarrow H^i(X, L\eta_{\xi,m+1} \Delta_{X/\mathbb{B}^+_{\text{dR}}})\end{equation*}

is 0. Note that the $i=0$ case is trivial, as the left–hand side cohomology group is 0.

We have the commutative diagram

where the middle row and column are exact, being part of the long exact sequence associated to distinguished triangles. The first row is the connecting morphism as specified in Corollary 3·5, and the square is commutative since it is induced by the morphism of distinguished triangles

The arrow b is injective by our inductive hypothesis, so a is injective as well.

We know from (2) and Corollary 3·5 that $\beta$ factorises as

where $Z^{m+1}\Omega_X^{\bullet}$ is the sheaf of closed $m+1$ -forms, and d is the usual differential of de Rham complexes. By the degeneration of Hodge–de Rham spectral sequence proved in [ Reference Bhatt, Morrow and ScholzeBMS18 , theorem 13·3], we know that

\begin{equation*} H^{i-1}(X, \Omega_X^m) \overset{d}{\longrightarrow } H^{i-1}(X, \Omega_X^{m+1})\end{equation*}

is 0, and we claim that this implies that the composition

\begin{equation*} H^{i{-}1}(X, \Omega_X^m[{-}m]) \overset{d}{\longrightarrow } H^{i{-}1}(X, Z^{m{+}1}\Omega_X^{\bullet}[{-}m])\longrightarrow H^i(X, F_{m{+}1} \Omega_X^{\bullet}) \cong H^{i{-}1}(X, F_{m{+}1} \Omega_X^{\bullet}[1])\end{equation*}

is 0, so $\beta$ is 0 as well.

Indeed, filtering both $ \Omega_X^m[-m]$ and $ F_{m+1} \Omega_X^{\bullet}[1]$ by the Hodge filtration, we see that the map on the first graded piece is $ H^{i-1}(X, \Omega_X^m[-m]) \overset{d}{\longrightarrow } H^{i-1}(X, \Omega_X^{m+1}[-m])$ which we have seen to be 0. The map on other graded pieces are also 0, since $\Omega_X^m[-m]$ has only one non-zero graded piece, proving the claim.

Now let $x \in H^{i-1} (X, Gr^m \Delta_{X/\mathbb{B}^+_{\text{dR}}})$ , then $f(x) \in \text{Ker}(h)$ since $\beta =0$ , so we have $f(x) = a(y)$ for some $y \in H^i(X, \xi L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}})$ . Then $b(y)=g(a(y))=g(f(x))=0$ , which implies that $y=0$ by injectivity of b, so $f(x)=a(y)=0$ . Hence we have $f=0$ , finishing the induction.

The proposition tells us that long exact sequence assoicated to the distinguished triangle (5) splits into short exact sequences

\begin{equation*}0 \longrightarrow H^i(X, L\eta_{\xi,m+1} \Delta_{X/\mathbb{B}^+_{\text{dR}}})\longrightarrow H^i(X, L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}})\longrightarrow H^i(X, \tau_{\leq m}\overline{ \Delta}_{X/\mathbb{B}^+_{\text{dR}}}) (m)\longrightarrow 0,\end{equation*}

indeed, $ H^i(X, \tau_{\leq m}\overline{ \Delta}_{X/\mathbb{B}^+_{\text{dR}}}) (m)$ is $\xi$ -torsion which admits no non-zero connecting morphism to the $\xi$ -torsion-free module $ H^{i+1}(X, L\eta_{\xi,m+1} \Delta_{X/\mathbb{B}^+_{\text{dR}}}) $ . In particular,

\begin{equation*} H^i(X, L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}}) \longrightarrow H^i(X, L\eta_{\xi,0} \Delta_{X/\mathbb{B}^+_{\text{dR}}}) = H^i(X, L\eta_{\xi} \Delta_{X/\mathbb{B}^+_{\text{dR}}}),\end{equation*}

so the canonical map

\begin{equation*} H^i(X, L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}})\longrightarrow H^i(X, L\eta_{\xi} \Delta_{X/\mathbb{B}^+_{\text{dR}}})\cap\xi^m H^i(X, \Delta_{X/\mathbb{B}^+_{\text{dR}}} )\end{equation*}

is injective. We now prove that it is also surjective.

We proceed by induction on m. The base case $m=0$ is clear, as both sides are $ H^i(X, L\eta_{\xi} \Delta_{X/\mathbb{B}^+_{\text{dR}}})$ . Assume that the map is surjective for m, we prove it is also surjective for $m+1$ . We consider the commutative diagram

where the two vertical sequences are exact, the middle horizontal arrow is an isomorphism by our inductive hypothesis, and the bottom horizontal arrow is injective by the degeneration of Hodge–Tate spectral sequence ([ Reference Bhatt, Morrow and ScholzeBMS18 , theorem 13·3]). Now an easy diagram chasing proves that the first horizontal arrow is surjective, finishing the induction.

5. Miscellany

We document an interesting byproduct in our treatment of Hodge–Tate filtration, namely we can give a conceptual explanation why the degeneration of Hodge–Tate spectral sequences is equivalent to that of Hodge–de Rham spectral sequences. The claim is clear in the proof of [ Reference Bhatt, Morrow and ScholzeBMS18 , theorem 13·3], which is by dimension counting. Note that the torsion-freeness of $ H^i_{\text{crys}}(X/\mathbb{B}^+_{\text{dR}})$ is used in $loc.cit.$ as a bridge between the dimension of the de Rham cohomology and étale cohomology.

Proposition 5·1. Let X be a proper smooth rigid analytic variety over a complete algebraically closed non-archimedean field $\mathbb{C}$ of mixed characteristic p, the degeneration of the Hodge–Tate spectral sequence

\begin{equation*}E_2^{p,q} = H^{p}(X,\Omega_{X}^q)({-}\;\!q) \ \ \Longrightarrow H^{p+q}_{\acute{\text{e}}\text{t}}(X,\mathbb{Q}_p) \otimes_{\mathbb{Q}_p} \mathbb{C}\end{equation*}

is equivalent to the degeneration of Hodge–de Rham spectral sequence

\begin{equation*}E_1^{p,q}= H^{q}(X,\Omega_{X}^p) \Longrightarrow H^{p+q}(X,\Omega_X^{\bullet}).\end{equation*}

More precisely, in the natural diagram

where f and g are the canonical maps corresponding to the truncation filtration and Hodge filtration respectively, we have that

\begin{equation*} Coker(f) = Coker(g) \subset H^{i}(X,\Omega_{X}^m[-m]) \end{equation*}

as subspaces of $ H^{i}(X,\Omega_{X}^m[-m])$ .

Proof. We know that the Hodge–Tate spectral sequence is induced from the filtration $ H^i(X, \tau_{\leq m}\overline{\Delta}_{X/\mathbb{B}^+_{\text{dR}}})$ corresponding to standard truncation, and the Hodge–de Rham spectral sequence is induced from the Hodge filtration $ H^{i}(X,F_m\Omega_{X}^{\bullet})$ . It is then clear that the equivalence of degeneration is implied by the claim on f and g.

We now prove the claim. We have a commutative diagram

where the vertical and horizontal sequences are both short exact sequences, which follows from Proposition 4·1. The factorisation h is induced by the canonical map

\begin{equation*}L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}}/\xi L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}}\longrightarrow L\eta_{\xi,m} \Delta_{X/\mathbb{B}^+_{\text{dR}}}/L\eta_{\xi,m+1} \Delta_{X/\mathbb{B}^+_{\text{dR}}}\cong\tau_{\leq m}\overline{ \Delta}_{X/\mathbb{B}^+_{\text{dR}}} (m),\end{equation*}

and h is surjective because it is a factorisation of a surjective map.

Now Corollary 3·6 gives us a commutative diagram

and the claim in the proposition follows immediately.