2.1 Definition and basic properties

One can naturally introduce the following truncated analogue of Hodge-properness condition.

Remark 2.2. In other words, a smooth qcqs Artin stack $\mathcal{X}$ is d-Hodge-proper if

\begin{align*}\tau^{\leqslant d-j}R\Gamma(\mathcal X,\wedge^j\mathbb L_{\mathcal X/R})\in \mathrm{Coh}(R)\end{align*}

for any $j\geqslant 0$ .

The d-Hodge-properness condition is much more flexible than just Hodge-properness. In particular, there are many more schemes that are Hodge-proper up to some degree, but are not themselves proper.

Proof. Let $j\colon U\rightarrow X$ be the embedding. Since X is Cohen–Macaulay and Z has codimension $d+2$ we have $R^0j_*\mathcal O_U=\mathcal O_X$ and $R^ij_* \mathcal O_U=0$ for $0\lt i\leqslant d$ . From [Sta26, Tag 0BLT] it follows that for $0\leqslant i\leqslant d$ the sheaf $R^ij_* \mathcal F$ is coherent when $\mathcal F$ is finite locally free. In other words, $R^{\leqslant d}j_*\mathcal F\in \mathrm{Coh}(X)$ . Taking $\mathcal F=\Omega^k_{U/R}$ and applying Lemma 2.8 to the global sections functor $R\Gamma\colon \mathrm{QCoh}(X) \rightarrow D(\mathrm{Mod}_{R})$ (which is left t-exact) and $M=Rj_* \Omega^k_{U/R}$ we get that for any $k\geqslant 0$

Since X is proper over R we get $\tau^{\leqslant d}R\Gamma(U, \Omega^k_{U/R})\in \mathrm{Coh}(R)$ for any $k\geqslant 0$ . We are done by Remark 2.2.

The property that is actually used by us in the proofs is even slightly weaker, though completely analogous in spirit.

Remark 2.6. In other words, a smooth qcqs Artin stack $\mathcal{X}$ is d-de Rham-proper if

\begin{align*}\tau^{\leqslant d}R\Gamma_{\mathrm{dR}}(\mathcal X/R)\in \mathrm{Coh}(R).\end{align*}

The following is immediate.

Proof. By [Reference Kubrak and PrikhodkoKP22, Corollary 1.1.6(2)] one has a convergent (Hodge-to-de Rham) spectral sequence

\begin{align*} E_1^{i,j}=H^j(\mathcal X,\wedge^i\mathbb L_{\mathcal X/R})\Rightarrow H^{i+j}_{\mathrm{dR}}(\mathcal X/R), \end{align*}

from which we see that if $\mathcal{X}$ is d-Hodge-proper, then $H^i_{\mathrm{dR}}(\mathcal X/R)$ is finitely generated over R for $i\leqslant d$ .

As we show, the following simple lemma can be used to control the ‘degree of coherence’ of complexes under certain operations.

Lemma 2.8. Let $\mathscr C$ and $\mathscr D$ be stable $\infty$ -categories endowed with t-structures. Let $F\colon \mathscr C \rightarrow \mathscr D$ be an exact functor which is left t-exact up to a shift by s.Footnote ³ Then for any $d\in \mathbb Z$ and any $M\in \mathscr C$ one has a natural equivalence

1. (1) Let $R\rightarrow R'$ be a faithfully flat map of Noetherian rings. Then $\mathcal{X}$ is d-Hodge-proper (respectively, d-de Rham-proper) if and only if the base change $\mathcal X_{R'}\colon = \mathcal X \times_{\operatorname{Spec} R} \operatorname{Spec} R'$ is.

2. (2) Let $R\rightarrow R'$ be a map of Noetherian rings that has Tor-amplitude $[-s,0]$ . Then the base change $\mathcal X_{R'}$ is $(d-s)$ -Hodge proper (respectively, $(d-s)$ -de Rham-proper).

Proof. Consider the projection $q\colon \mathcal X_{R'} \rightarrow \mathcal X$ . Since $\mathcal{X}$ is smooth (and qcqs) and $R\rightarrow R'$ is of finite Tor-amplitude we can apply base change to get an equivalence

By base change for cotangent complex (and its exterior powers, see [Reference Kubrak and PrikhodkoKP24, Corollary A.2.48]) we also have $q^*(\wedge^i\mathbb L_{\mathcal X/R})\simeq \wedge^i\mathbb L_{\mathcal X_{R'}/R'}$ . Similarly, for de Rham cohomology we have

In point (1), by flatness of R’, $\tau^{\leqslant d-i}R\Gamma(\mathcal X_{R'},\wedge^i\mathbb L_{\mathcal X_{R'}/R'})\simeq \tau^{\leqslant d-i}R\Gamma(\mathcal X, \wedge^i\mathbb L_{\mathcal X/R})\otimes_R R'$ for any $i\geqslant 0$ and we are done by Lemma 2.10 below. The proof for de Rham cohomology is analogous. For point (2), Lemma 2.8 (applied to the tensor product $-\otimes_R R'$ and $M=R\Gamma(\mathcal X, \wedge^i\mathbb L_{\mathcal X/R})$ ) gives us an equivalence

for any $i\geqslant 0$ and, similarly, also

Since $\tau^{\leqslant d-i}R\Gamma(\mathcal X, \wedge^i\mathbb L_{\mathcal X/R})\in \mathrm{Coh}(R)$ for any $i\geqslant 0$ by d-Hodge-properness of $\mathcal{X}$ we have $\tau^{\leqslant d-i}R\Gamma(\mathcal X, \wedge^i\mathbb L_{\mathcal X/R})\otimes_R R'\in \mathrm{Coh}(R')$ (see, e.g., the argument in [Reference Kubrak and PrikhodkoKP24, Lemma 1.1.6]), which then also holds for the $(d-s-i)$ th truncation. This shows that $\mathcal X_{R'}$ is $(d-s)$ -Hodge-proper. The rest of the proof for the de Rham-proper condition is similar.

3.1 de Rham cohomology over $B_{\mathrm{dR}}$

In this section we show that the ‘de Rham cohomology over $B_{\mathrm{dR}}$ ’ of the Raynaud generic fiber of a d-Hodge proper stack agrees with the algebraic de Rham cohomology up to degree $d-1$ . We describe in detail the notion of (p,t)-completed tensor product $-\widehat{\otimes}_{\mathcal O_K} B_{\mathrm{dR}}$ and show that it agrees with that in [Reference Diao, Lan, Liu and ZhuDLL⁺23] in the cases of our interest (see Remark 3.8).

Let K be a complete discretely valued p-adic field with a perfect residue field k. As usual, we let $K_0:=W(k)[{1}/{p}]\subset K$ and $C:=\widehat{\overline{K}}$ . Let $e\colon = [K:K_0]$ be the ramification index. Recall Fontaine’s infinitesimal period ring ${A_{\mathrm{inf}}}:={A_{\mathrm{inf}}}(\mathcal O_C)$ and Fontaine’s map $\theta\colon {A_{\mathrm{inf}}} \twoheadrightarrow \mathcal O_{C}$ (for notation, see, e.g., [Reference Kubrak and PrikhodkoKP24, Appendix B]). As a slight variation of the construction we also consider a period ring $A_{\mathrm{inf}, K} \colon = \mathcal O_K \otimes_{W(k)} {A_{\mathrm{inf}}}$ . Let $B_{{\mathrm{dR}},K}^+\colon = (A_{\mathrm{inf}, K}[{1}/{p}])^\wedge_{\ker \theta}$ be the corresponding analogue of $B_{\mathrm{dR}}^+$ . Note that the natural map $\mathcal O_K\otimes_{W(k)}B_{\mathrm{dR}}^+ \rightarrow B_{{\mathrm{dR}},K}^+$ is an equivalence. Indeed, both sides are t-adically complete $B_{\mathrm{dR}}^+$ -modules and the map becomes identity modulo $(t)\simeq \ker \theta\subset B_{\mathrm{dR}}$ .

Recall that the embedding $\overline K \hookrightarrow C\simeq B_{\mathrm{dR}}^+/(t)$ lifts canonically to a map $\overline K\dashrightarrow B_{\mathrm{dR}}^+$ since $B_{\mathrm{dR}}^+$ is a strictly Henselian local ring. One then has a canonical map $B_{{\mathrm{dR}},K}^+\simeq \mathcal O_K\otimes_{W(k)}B_{\mathrm{dR}}^+\rightarrow B_{\mathrm{dR}}^+$ which is the tensor product of the identity map $B_{\mathrm{dR}}^+\xrightarrow{{\mathrm{id}}}B_{\mathrm{dR}}^+$ and the restriction of $\overline K\rightarrow B_{\mathrm{dR}}^+$ to $\mathcal O_K$ .

Remark 3.1. Note that the ring $B_{{\mathrm{dR}},K}^+$ is, in fact, isomorphic to the product of e copies of $B_{\mathrm{dR}}^+$ . Indeed, since $B_{\mathrm{dR}}^+$ is a $\overline K$ -algebra and $\mathcal O_K\otimes_{W(k)}\overline K\simeq \overline K^{\oplus e}$ , one has isomorphisms

The map $B_{{\mathrm{dR}},K}^+\rightarrow B_{{\mathrm{dR}}}^+$ in these terms is just the projection on one of the components. Note that this map is flat.

The following construction plays a key role for this paper. Below, we also consider the composition $B_{{\mathrm{dR}},K}^+\rightarrow B_{{\mathrm{dR}}}^+\rightarrow B_{\mathrm{dR}}$ (recall that $B_{\mathrm{dR}}\colon = B_{\mathrm{dR}}^+[{1}/{t}]$ ).

Construction 3.2. Let M be a complex of $\mathcal O_K$ -modules. We define the (p,t)-completed tensor products $M\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt} B_{\mathrm{dR}}^+$ and $M\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt} B_{\mathrm{dR}}$ as follows:

\begin{gather*}M\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt} B_{\mathrm{dR}}^+ \colon = \bigg((M\otimes_{W(k)} {A_{\mathrm{inf}}} )_p^\wedge\bigg[\frac{1}{p}\bigg]\bigg)_{\ker \theta}^\wedge\otimes_{B_{{\mathrm{dR}},K}^+}B_{\mathrm{dR}}^+,\\ M\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt} B_{\mathrm{dR}} \colon = \bigg((M\otimes_{W(k)} {A_{\mathrm{inf}}} )_p^\wedge\bigg[\frac{1}{p}\bigg]\bigg)_{\ker \theta}^\wedge\otimes_{B_{{\mathrm{dR}},K}^+}B_{\mathrm{dR}}. \end{gather*}

The following lemma is useful later.

Proof. Note that the functor $M\mapsto (M\mathbin{\widehat{\otimes}}_{W(k)}{A_{\mathrm{inf}}})[{1}/{p}]/\xi^k$ is left t-exact up to a shift by 1. Indeed, for $k=1$ we have $M\mapsto (M\mathbin{\widehat{\otimes}}_{W(k)}\mathcal O_C)[{1}/{p}]$ , which is the composition of tensor product $-\otimes_{W(k)}\mathcal O_C$ (which is t-exact since $\mathcal O_C$ is p-torsion free), derived p-adic completion (which is left t-exact up to a shift by 1) and localization (which is t-exact). To get the statement for a general k one can argue by induction using the fiber sequence

Then $M\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt} B_{\mathrm{dR}}$ is obtained by first taking limit of $(M\mathbin{\widehat{\otimes}}_{W(k)}{A_{\mathrm{inf}}})[{1}/{p}]/\xi^k$ over k and then the tensor product $-\otimes_{B_{{\mathrm{dR}},K}^+}B_{\mathrm{dR}}$ (which is t-exact by Remark 3.1). Since limits are left t-exact this gives the claim.

Remark 3.5. The map $\theta$ and the natural embedding $\mathcal O_K \hookrightarrow \mathcal O_{C}$ agree on W(k) and produce a natural morphism

One can show that there is also another formula for $M\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt}B_{\mathrm{dR}}^+$ in terms of $\ker \theta_K$ -adic completion

\begin{align*}M\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt} B^+_{\mathrm{dR}} \colon = \bigg((M\otimes_{W(k)} {A_{\mathrm{inf}}} )_p^\wedge\bigg[\frac{1}{p}\bigg]\bigg)_{\ker \theta_K}^\wedge.\end{align*}

Now we can define ‘Hodge’ and ‘de Rham’ cohomology of the Raynaud generic fiber ‘over $B_{\mathrm{dR}}$ ’.

Construction 3.6. Let U be a smooth affine scheme over $\mathcal O_K$ . We define

\begin{align*}R\Gamma(\widehat U_K, \Omega^i\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt} B_{\mathrm{dR}}) \colon = \Omega^i_U\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt} B_{\mathrm{dR}} \quad\text{and}\quad R\Gamma_{\mathrm{dR}}(\widehat U_K /B_{\mathrm{dR}}) \colon = \Omega^\bullet_{U, {\mathrm{dR}}} \mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt} B_{\mathrm{dR}}.\end{align*}

We call $R\Gamma_{\mathrm{dR}}(\widehat U_K /B_{\mathrm{dR}})$ the de Rham cohomology of $U_K$ over $B_{\mathrm{dR}}$ . We extend this definition to all smooth Artin $\mathcal O_K$ -stacks via the right Kan extension. That is, for a smooth Artin stack $\mathcal O_K$ -stack $\mathcal{X}$ we have

\begin{gather*}R\Gamma(\widehat{\mathcal X}_K, \Omega^i\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt} B_{\mathrm{dR}}) \simeq \lim_{U \in \mathrm{Aff}^{\mathrm{sm}, {\mathrm{op}}}_{/\mathcal X}} R\Gamma(\widehat U_K, \Omega^i\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt} B_{\mathrm{dR}}),\\R\Gamma_{\mathrm{dR}}(\widehat{\mathcal X}_K / B_{\mathrm{dR}}) \simeq \lim_{U \in \mathrm{Aff}^{\mathrm{sm}, {\mathrm{op}}}_{/\mathcal X}} R\Gamma_{\mathrm{dR}}(\widehat U_K / B_{\mathrm{dR}}).\end{gather*}

One also has an obvious version with $B_{\mathrm{dR}}^+$ instead of $B_{\mathrm{dR}}$ .

Remark 3.8. We point out that the definitions in Construction 3.6 agree with those in [Reference Liu and ZhuLZ17, § 3.1] and [Reference Diao, Lan, Liu and ZhuDLL⁺23, Definition 3.1.1] in the case of Raynaud generic fiber of a smooth scheme. Indeed, let $X=\mathrm{Spa}(R,R^+)$ be a smooth affinoid adic space over $\mathrm{Spa}(K,\mathcal O_K).$ The complex $R\Gamma(X_{\mathrm{an}}, \Omega^i_X\mathbin{\widehat{\otimes}} B_{\mathrm{dR}}^+)$ (in the notation of [Reference Liu and ZhuLZ17, § 3.1]) is, by definition, given by the limit over n of the p-completed tensor products $\Omega^i_X\mathbin{\widehat{\otimes}}_K{\kern-1pt} B_{\mathrm{dR}}^+/t^n$ . Let $X=\widehat{U}_K$ be the Raynaud generic fiber of a smooth affine $\mathcal O_K$ -scheme $U\colon = \operatorname{Spec} A$ : in this case $(R,R^+)\simeq (A^\wedge_p[{1}/{p}],A^\wedge_p)$ . We construct the natural isomorphisms

the analogous isomorphisms $\Omega^i_U\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt}B_{\mathrm{dR}} \xrightarrow{\sim}\Omega^i_X\mathbin{\widehat{\otimes}}B_{\mathrm{dR}}$ and $\Omega^i_{U,{\mathrm{dR}}}\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt}B_{\mathrm{dR}}\xrightarrow{\sim} \Omega^i_{X,{\mathrm{dR}}}\mathbin{\widehat{\otimes}}B_{\mathrm{dR}}$ then follow by inverting t and considering the Hodge filtration correspondingly. Note that, by smoothness, $\Omega^i_U$ is a perfect A-module, and since $\Omega^i_X\simeq \Omega^i_U\otimes_A R$ , it is enough to consider the case $i=0$ ; namely, of the structure sheaf. This reduces to the existence of a natural equivalence

(3.1)

for any $n\geqslant 0$ .

Since $R^+\simeq A^\wedge_p$ we have an equivalence

On the other hand, the tensor product on the right is defined by choosing suitable rings of definition in R, K, $B_{\mathrm{dR}}^+/t^n$ , tensoring them up, p-completing the result and inverting p after. Let us take $R^+\subset R$ , $\mathcal O_K\subset K$ and $\mathcal O_K\cdot {A_{\mathrm{inf}}}/\xi^n \subset B^+_{\mathrm{dR}}/t^n$ ; then we have

\begin{align*} R\mathbin{\widehat{\otimes}}_K{\kern-1pt} B_{\mathrm{dR}}^+/t^n\simeq (R^+\otimes_{\mathcal O_K} (\mathcal O_K\cdot {A_{\mathrm{inf}}}/\xi^n))^\wedge_p\bigg[\frac{1}{p}\bigg]. \end{align*}

On the other hand, the left-hand side can be rewritten as

\begin{align*} ((R^+ {\otimes}_{W(k)}({A_{\mathrm{inf}}}/\xi^n))^\wedge_p \otimes_{\mathcal O_K\otimes_{W(k)}{A_{\mathrm{inf}}}/\xi^n} (\mathcal O_K\cdot {A_{\mathrm{inf}}}/\xi^n))\bigg[\frac{1}{p}\bigg]. \end{align*}

Rewriting $(R^+ {\otimes}_{W(k)}({A_{\mathrm{inf}}}/\xi^n)$ as $R^+\otimes_{\mathcal O_K} (\mathcal O_K \otimes_{W(k)} ({A_{\mathrm{inf}}}/\xi^n))$ one sees that both expressions agree before inverting p and modulo $p^n$ . Thus, it is just enough to see that the tensor product

\begin{align*} (R^+ {\otimes}_{W(k)}({A_{\mathrm{inf}}}/\xi^n))^\wedge_p \otimes_{\mathcal O_K\otimes_{W(k)}{A_{\mathrm{inf}}}/\xi^n} (\mathcal O_K\cdot {A_{\mathrm{inf}}}/\xi^n) \end{align*}

is already (classically) p-complete. This follows from $ \mathcal O_K\cdot {A_{\mathrm{inf}}}/\xi^n$ being a finitely presented $\mathcal O_K\otimes_{W(k)}{A_{\mathrm{inf}}}/\xi^n$ -module.Footnote ⁴

We need the following basic result about $B_{\mathrm{dR}}$ -cohomology.

1. (1) The natural map

   
   is an equivalence.

2. (2) So is the natural map .

3. (3) If, in addition, $\mathcal{X}$ is d-de Rham-proper over $\mathcal O_K$ , then the natural map

   
   induces an isomorphism $H^i_{\mathrm{dR}}(X_K/K)\otimes_K B_{\mathrm{dR}}\simeq H^i_{\mathrm{dR}}(\widehat {\mathcal X}_K/B_{\mathrm{dR}})$ for $i\leqslant d-1$ and an embedding $H^d_{\mathrm{dR}}(X_K/K)\otimes_K B_{\mathrm{dR}}\hookrightarrow H^d_{\mathrm{dR}}(\widehat {\mathcal X}_K/B_{\mathrm{dR}})$ .

Finally, part (3) follows from part (2) and applying Lemma 3.12 to $M=R\Gamma_{\mathrm{dR}}(\mathcal X/\mathcal O_K)$ .

Remark 3.10. If $\mathcal{X}$ is Hodge-proper over $\mathcal O_K$ , then it is d-Hodge and, consequently, d-de Rham proper for any $d\geqslant 0$ . Thus, from Proposition 3.9(3) we get that the map

is a quasi-isomorphism.

Lemma 3.11. Let $M^\bullet$ be a co-simplicial diagram of uniformly bounded below complexes of $\mathcal O_K$ -modules. Then the natural map

is an equivalences.

Lemma 3.12. Let $M \in D(\mathrm{Mod}_{\mathcal O_K})$ be such that $\tau^{\leqslant d} M\in \mathrm{Coh}(\mathcal O_K)$ . Then the natural map

induces an isomorphism $H^i(M)\otimes_{\mathcal O_K}B_{\mathrm{dR}}\simeq H^i(M\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt} B_{\mathrm{dR}})$ for $i\leqslant d-1$ and an embedding $H^d(M)\otimes_{\mathcal O_K}B_{\mathrm{dR}}\hookrightarrow H^d(M\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt} B_{\mathrm{dR}})$ .

Proof. The map in question is an equivalence for $M = \mathcal O_K$ (Remark 3.3), hence it is also so for all perfect (or equivalently, coherent) $\mathcal O_K$ -modules. This applies in particular to $\tau^{\leqslant d}(M)$ . Recall that $-\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt}B_{\mathrm{dR}}$ is left t-exact up to a shift by 1, while $-\otimes_{\mathcal O_K}B_{\mathrm{dR}}$ is just left t-exact since $B_{\mathrm{dR}}$ is p-torsion free. From Lemma 2.8 we get a commutative square

which shows that the low horizontal map is an equivalence. By passing to cohomology, this gives the statement for $i\leqslant d-1$ .

Note also that for any i (from left t-exactness of $-\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt}B_{\mathrm{dR}}$ up to a shift by 1) we have a short exact sequence

\begin{align*}0 \rightarrow H^0(H^i(M)\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt} B_{\mathrm{dR}}) \rightarrow H^i(M\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt} B_{\mathrm{dR}}) \rightarrow H^{-1}( H^{i+1}(M)\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt} B_{\mathrm{dR}}) \rightarrow 0.\end{align*}

By the assumption, for $i=d$ the module $H^d(M)$ is still finitely generated, so $H^d(M)\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt} B_{\mathrm{dR}}\simeq H^d(M)\otimes_{\mathcal O_K} B_{\mathrm{dR}}$ , and the short exact sequence gives that $H^d(M)\otimes_{\mathcal O_K}B_{\mathrm{dR}}\hookrightarrow H^d(M\mathbin{\widehat{\otimes}}_{\mathcal O_K}{\kern-1.5pt} B_{\mathrm{dR}})$ .

3.2 Log period sheaves and $B_{\mathrm{dR}}$ -comparison map

Let X be a smooth adic space over $\mathrm{Spa}(K,\mathcal O_K)$ . Let $i\colon D \rightarrow X$ be a normal crossings divisor (see [Reference Diao, Lan, Liu and ZhuDLL⁺19, Example 2.3.17] for the definition). The divisor D endows X with a natural structure of a (smooth) log adic space (see [Reference Diao, Lan, Liu and ZhuDLL⁺23, Example 2.1.2]). We denote by $X_{D,{\acute{\mathrm{e}}\mathrm{t}}}$ , $X_{D,\mathrm{prok}\acute{\mathrm{e}}\mathrm{t}}$ the associated Kummer étale (see [Reference Diao, Lan, Liu and ZhuDLL⁺19, § 4.1]) and pro-Kummer étale (see [Reference Diao, Lan, Liu and ZhuDLL⁺19, Definition 5.1.2]) sites correspondingly.

Remark 3.14. Let $U\colon = X\setminus D$ be the complement to D and let $j\colon U \rightarrow X$ be the natural embedding. Pull-back of D to U is empty, $U_{\varnothing,{\acute{\mathrm{e}}\mathrm{t}}}\simeq U_{\acute{\mathrm{e}}\mathrm{t}}$ , and thus one has a natural (derived) push-forward functor $j_{*,{\acute{\mathrm{e}}\mathrm{t}}}\colon \mathcal{S}\mathrm{hv}(U_{\acute{\mathrm{e}}\mathrm{t}},\mathbb Z/n)\rightarrow \mathcal{S}\mathrm{hv}(X_{D,{\acute{\mathrm{e}}\mathrm{t}}},\mathbb Z/n)$ between the derived categories of sheaves of $\mathbb Z/n$ -modules (for any $n\geqslant 0$ ). By [Reference Diao, Lan, Liu and ZhuDLL⁺19, Lemma 4.6.5] for the constant sheaf $\mathbb Z/n\in \mathcal{S}\mathrm{hv}(U_{\acute{\mathrm{e}}\mathrm{t}},\mathbb Z/n)$ one has an equivalence $j_{*,{\acute{\mathrm{e}}\mathrm{t}}} \mathbb Z/n \xrightarrow{\sim} \mathbb Z/n$ ; this then induces an equivalence

for the global sections. Taking $n=p^k$ and passing to the limit over k one also gets an equivalence

(3.2)

In [Reference Diao, Lan, Liu and ZhuDLL⁺23] the period sheaves $\mathbb B_{\mathrm{dR}}^+$ , $\mathbb B_{\mathrm{dR}}$ , $\mathcal O\mathbb B_{{\mathrm{dR}},\log}^+$ , $\mathcal O\mathbb B_{{\mathrm{dR}},\log}$ on $X_{D,\mathrm{prok}\acute{\mathrm{e}}\mathrm{t}}$ were introduced. While the definition of $\mathbb B_{\mathrm{dR}}^+$ , $\mathbb B_{\mathrm{dR}}$ is essentially the same as in [Reference ScholzeSch13] (see [Reference Diao, Lan, Liu and ZhuDLL⁺23, Definition 2.2.3]), the definition of $\mathcal O\mathbb B_{{\mathrm{dR}},\log}^+$ , $\mathcal O\mathbb B_{{\mathrm{dR}},\log}$ is more involved and takes into account the log structure in a more significant way (see [Reference Diao, Lan, Liu and ZhuDLL⁺23, Definition 2.2.10]). Let also $\Omega^{1}_{X,D,\log}$ be the sheaf of log differential 1-forms on X (see [Reference Diao, Lan, Liu and ZhuDLL⁺19, Definition 3.2.25]). Similarly, one defines $\Omega^{j}_{X,D,\log}\colon = \wedge^j_{\mathcal O_X}\Omega^{1}_{X,D,\log}$ for all $j\geqslant 0$ (see [Reference Diao, Lan, Liu and ZhuDLL⁺19, Definition 3.2.29]).

We denote by $\mu\colon X_{D,\mathrm{prok}\acute{\mathrm{e}}\mathrm{t}} \rightarrow X_{\mathrm{an}}$ the natural map of sites. We also consider the map $\mu'\colon (X_C)_{D,\mathrm{prok}\acute{\mathrm{e}}\mathrm{t}}\simeq (X_{D,\mathrm{prok}\acute{\mathrm{e}}\mathrm{t}})_{/X_C} \rightarrow X_{\mathrm{an}}$ where $X_C\colon = X\times_K C$ is the base change of X to $C\colon = \widehat{\overline K}$ . For brevity we continue to denote the pull-backs $\mu^*\Omega^{j}_{X,D,\log}$ and $\mu'^*\Omega^{j}_{X,D,\log}$ by $\Omega^{j}_{X,D,\log}$ .

The sheaf $\mathcal O \mathbb B_{{\mathrm{dR}},\log}^+$ on $X_{D,\mathrm{prok}\acute{\mathrm{e}}\mathrm{t}}$ has a natural log connection $\nabla\colon \mathcal O \mathbb B_{{\mathrm{dR}}}^+ \rightarrow \mathcal O \mathbb B_{{\mathrm{dR}}}^+\otimes_{\mathcal O_X} \Omega^1_{X,D,\log}$ (see [Reference Diao, Lan, Liu and ZhuDLL⁺23, 2.2.15]) which then extends to a log connection $\nabla\colon \mathcal O \mathbb B_{{\mathrm{dR}},\log} \rightarrow \mathcal O \mathbb B_{{\mathrm{dR}},\log}\otimes_{\mathcal O_X} \Omega^1_{X,D,\log}$ on $\mathcal O \mathbb B_{{\mathrm{dR}},\log}$ (see [Reference Diao, Lan, Liu and ZhuDLL⁺23, 2.2.17]). We then have the following.

Proposition (The Poincarà lemma [Reference Diao, Lan, Liu and ZhuDLL⁺23, Corollary 2.4.2]). The complex

\begin{align*}0\rightarrow \mathbb B_{\mathrm{dR}} \rightarrow \mathcal O \mathbb B_{{\mathrm{dR}},\log} \xrightarrow{\nabla} \mathcal O \mathbb B_{{\mathrm{dR}},\log}\otimes_{\mathcal O_X} \Omega^1_{X,D,\log} \xrightarrow{\nabla} \mathcal O \mathbb B_{{\mathrm{dR}}} \otimes_{\mathcal O_X} \Omega^2_{X,D,\log} \xrightarrow{\nabla} \cdots\end{align*}

of sheaves on $X_{D,\mathrm{prok}\acute{\mathrm{e}}\mathrm{t}}$ is acyclic.

This gives a quasi-isomorphism $\mathbb B_{\mathrm{dR}}\xrightarrow{\sim} DR_{\log}(\mathcal O \mathbb B_{{\mathrm{dR}},\log},\nabla)$ in the derived categories of sheaves on $X_{D,\mathrm{prok}\acute{\mathrm{e}}\mathrm{t}}$ and $(X_C)_{D,\mathrm{prok}\acute{\mathrm{e}}\mathrm{t}}$ (here $DR_{\log}$ denotes the log de Rham complex).

One can also explicitly describe the push-forwards with respect to the map $\mu'\colon (X_C)_{D,\mathrm{prok}\acute{\mathrm{e}}\mathrm{t}} \rightarrow X_{\mathrm{an}}$ of the individual terms of the de Rham complex $DR_{\log}(\mathcal O \mathbb B_{{\mathrm{dR}},\log},\nabla)$ . Namely, recall the sheaf $\mathcal O_X\mathbin{\widehat{\otimes}} B_{\mathrm{dR}}$ ([Reference Diao, Lan, Liu and ZhuDLL⁺23, Definition 3.1.1], see also the discussion in Remark 3.8) and consider $\Omega^j_{X,D,\log}\mathbin{\widehat{\otimes}} B_{\mathrm{dR}}\colon = \Omega^j_{X,D,\log}\otimes_{\mathcal O_X} \mathcal O_X\mathbin{\widehat{\otimes}} B_{\mathrm{dR}}$ (these sheaves are called $\Omega^j_{X,\log}\mathbin{\widehat{\otimes}} B_{\mathrm{dR}}$ in [Reference Diao, Lan, Liu and ZhuDLL⁺23, Definition 3.1.6]). Let $d_{ B_{\mathrm{dR}}}\colon \mathcal O_X\mathbin{\widehat{\otimes}} B_{\mathrm{dR}} \rightarrow \Omega^1_{X,D,\log}\mathbin{\widehat{\otimes}} B_{\mathrm{dR}}$ be the map induced by the natural log connection $d\colon = \mathcal O_X \rightarrow \Omega^1_{X,D,\log}$ on $\mathcal O_X$ and let

\begin{align*} \Omega^\bullet_{X,D,\log {\mathrm{dR}}}\mathbin{\widehat{\otimes}} B_{\mathrm{dR}}\colon = \mathcal O_X\mathbin{\widehat{\otimes}} B_{\mathrm{dR}} \rightarrow \Omega^1_{X,D,\log}\mathbin{\widehat{\otimes}} B_{\mathrm{dR}} \rightarrow \Omega^2_{X,D,\log}\mathbin{\widehat{\otimes}} B_{\mathrm{dR}} \rightarrow \cdots \end{align*}

be the corresponding log de Rham complex.

Returning to the previous discussion, one has an equivalence

(see [Reference Diao, Lan, Liu and ZhuDLL⁺23, Lemma 3.3.2]), which, by the projection formula, also gives

\begin{align*}\Omega^j_{X,D,\log}\mathbin{\widehat{\otimes}} B_{\mathrm{dR}} \xrightarrow{\sim} R\mu'_*(\mathcal O \mathbb B_{{\mathrm{dR}},\log}\otimes \Omega^j_{X,D,\log})\end{align*}

for any $j\geqslant 0$ . These isomorphisms then ultimately lead to equivalences

(3.3)

We can now pass to the construction of the $B_{\mathrm{dR}}$ -comparison map.

Construction 3.17. Let X, $i\colon D\hookrightarrow X$ be as in the beginning of this section and let $U\colon = X {\kern-1pt}\setminus{\kern-1pt}D$ . Let $R\Gamma_{\acute{\mathrm{e}}\mathrm{t}}(U_C,\mathbb Q_p)\colon = R\Gamma_{\acute{\mathrm{e}}\mathrm{t}}(U_C,\mathbb Z_p)[{1}/{p}]$ and $R\Gamma((X_C)_{D,{\acute{\mathrm{e}}\mathrm{t}}},\mathbb Q_p)\colon = R\Gamma((X_C)_{D,{\acute{\mathrm{e}}\mathrm{t}}},\mathbb Z_p)[{1}/{p}]$ . By (3.2) we have an equivalence

\begin{align*}R\Gamma_{\acute{\mathrm{e}}\mathrm{t}}(U_C,\mathbb Q_p)\xrightarrow{\sim} R\Gamma((X_C)_{D,{\acute{\mathrm{e}}\mathrm{t}}},\mathbb Q_p).\end{align*}

Having this (with a further identification $R\Gamma((X_C)_{D,{\acute{\mathrm{e}}\mathrm{t}}},\mathbb Z_p)\simeq R\Gamma((X_C)_{D,\mathrm{prok}\acute{\mathrm{e}}\mathrm{t}},\mathbb Z_p)$ ), the map $\mathbb Z_p \rightarrow \mathbb B_{\mathrm{dR}}$ of sheaves on $X_{D,\mathrm{prok}\acute{\mathrm{e}}\mathrm{t}}$ induces a natural map

\begin{align*}R\Gamma_{\acute{\mathrm{e}}\mathrm{t}}(U_C,\mathbb Q_p)\otimes_{\mathbb Q_p}B_{\mathrm{dR}} \longrightarrow R\Gamma(X_{D,\mathrm{prok}\acute{\mathrm{e}}\mathrm{t}},\mathbb B_{\mathrm{dR}}).\end{align*}

Composing further with the equivalence $R\Gamma(X_{D,\mathrm{prok}\acute{\mathrm{e}}\mathrm{t}},\mathbb B_{\mathrm{dR}}) \xrightarrow{\sim} R\Gamma(X_{\mathrm{an}}, \Omega^\bullet_{X,D,\log {\mathrm{dR}}}\mathbin{\widehat{\otimes}} B_{\mathrm{dR}})$ coming from (3.3) we get a natural map

\begin{align*}\Theta_{X,D}\colon R\Gamma_{\acute{\mathrm{e}}\mathrm{t}}(U_C,\mathbb Q_p)\otimes_{\mathbb Q_p}B_{\mathrm{dR}} \longrightarrow R\Gamma(X_{\mathrm{an}}, \Omega^\bullet_{X,D,\log {\mathrm{dR}}}\mathbin{\widehat{\otimes}} B_{\mathrm{dR}})\end{align*}

which we call the $B_{\mathrm{dR}}$ -comparison map.

Example 3.18. By [Reference Diao, Lan, Liu and ZhuDLL⁺23, Theorem 3.2.3(3)], if X is proper the map $\Theta_{X,D}$ is an equivalence for any normal crossings divisor D. Moreover, by [Reference Diao, Lan, Liu and ZhuDLL⁺23, Lemma 3.6.2] one can make an identification

\begin{align*} R\Gamma_{\log {\mathrm{dR}}}(X/K)\otimes_{K}B_{\mathrm{dR}} \xrightarrow{\sim} R\Gamma(X, \Omega^\bullet_{X,D,\log {\mathrm{dR}}}\mathbin{\widehat{\otimes}} B_{\mathrm{dR}}). \end{align*}

In the case of the analytification $X^{\mathrm{an}}, D^{\mathrm{an}}$ of a smooth proper K-scheme X with a normal crossings divisor D, one can make further natural identifications

\begin{align*} R\Gamma_{\log {\mathrm{dR}}}(X^{\mathrm{an}}/K)\xrightarrow[\mathrm{GAGA}]{\sim} R\Gamma_{\log {\mathrm{dR}}}(X/K) \xrightarrow{\sim} R\Gamma_{\mathrm{dR}}(U/K), \end{align*}

where the first equivalence is induced by GAGA and the second is induced by the restriction of the algebraic log de Rham complex (as a complex of sheaves) to $U\colon = X\setminus D$ .

4.1 The key commutative square

Let U be a scheme over $\mathcal O_K$ . Let $U_{K}\colon = U\times_{\mathcal O_K} K$ and $\widehat U_K$ be the algebraic and Raynaud generic fibers of U. We have a natural map $\psi_U\colon\widehat U_K\rightarrow U_K^{\mathrm{an}}$ , where $U_K^{\mathrm{an}}$ is the analytification of $U_K$ (for a more detailed discussion of this situation, see [Reference Kubrak and PrikhodkoKP24, § 3.3]).

In the following we also need to consider the auxiliary data of a compactification X of $U_K$ by a simple normal crossings divisor D. To clarify, by this we mean a smooth proper K-scheme X with a simple normal crossings divisor $D \hookrightarrow X$ and an isomorphism $\varepsilon\colon U_K\xrightarrow{\sim }X\setminus D$ . Let $\mathrm{Comp}(U_K)_{{\mathrm{nc}}}$ be the category of such compactifications $(X,D,\varepsilon)$ . An important property of $\mathrm{Comp}(U_K)_{{\mathrm{nc}}}$ is that it is cofiltered.

Let $X_1,X_2$ be a pair of compactifications of S by normal crossings divisors. Let $Z^\prime$ be the closure of S in $X_1 \times X_2$ . Resolving the singularities as above we find a $Z\twoheadrightarrow Z'$ which is an isomorphism over S and such that the complement $Z\setminus S$ is a simple normal crossings divisor. By construction, such Z maps both to $X_1$ and $X_2$ (using projections $X_1 \leftarrow Z'\rightarrow X_2$ ).

Let now $f,g\colon X_1 \rightrightarrows X_2$ be a pair of parallel arrows in $\mathrm{Comp}(S)_{{\mathrm{nc}}}$ . Consider the closure $Z^\prime$ of the diagonal embedding $S \hookrightarrow X_1\times_{X_2} X_1$ . Let $Z\twoheadrightarrow Z^\prime$ be a resolution of singularities as above. By construction there is a map $e\colon Z \to Y$ such that $f\circ e = g\circ e$ .

In the next construction for an affine smooth $\mathcal O_K$ -scheme U we construct the following natural commutative square.

(4.1)

This is the key ingredient in the proof of local acyclicity.

4.2 The map $\Upsilon_{-,\mathbb Q_p}$ for d-Hodge-proper stacks

In this section we establish $\mathbb Q_p$ -local acyclicity of a Hodge-proper stack over $\mathcal O_K$ . First we extend the square (4.1) to smooth Artin $\mathcal O_K$ -stacks.

Proposition 4.5. Let $\mathcal{X}$ be a smooth qcqs Artin stack over $\mathcal O_K$ . Then there is a commutative square of the form

(4.3)

where the left vertical map is an equivalence.

Proposition 4.6. Let $\mathcal{X}$ be a smooth d-de Rham-proper stack over $\mathcal O_K$ . Then the map $\Upsilon_{\mathcal X,\mathbb Q_p}$ induces an embedding

for $i\leqslant d$ .

We get the following result for Hodge-proper stacks as an immediate consequence.

Theorem 4.8. Let $\mathcal{X}$ be a smooth Hodge-proper stack over $\mathcal O_K$ . Then the natural map

\begin{align*}\Upsilon_{\mathcal X,\mathbb Q_p}\colon R\Gamma_{\acute{\mathrm{e}}\mathrm{t}}(\mathcal X_{C}, \mathbb Q_p) \longrightarrow R\Gamma_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_{C}, \mathbb Q_p)\end{align*}

is an equivalence.

Proof. Since the left vertical arrow in (4.3) is an isomorphism we have $\dim_{\mathbb Q_p} H^i_{\acute{\mathrm{e}}\mathrm{t}}(\mathcal X_{C}, \mathbb Q_p)=\dim_K H^i_{\mathrm{dR}}(\mathcal X_K/K)$ . Moreover, since $\mathcal{X}$ is Hodge-proper, we have

\begin{align*}\dim_K H^i_{\mathrm{dR}}(\mathcal X_K/K)=\dim_{\mathbb Q_p} H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_{C}, \mathbb Q_p)\end{align*}

by [Reference Kubrak and PrikhodkoKP24, Proposition 4.3.14]. Thus, $\dim H^i_{\acute{\mathrm{e}}\mathrm{t}}(\mathcal X_{C}, \mathbb Q_p)=\dim H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_{C}, \mathbb Q_p)$ for any $i\geqslant 0$ . On the other hand, a Hodge-proper stack is d-de Rham-proper for any d, and thus by Proposition 4.6 $\Upsilon_{\mathcal X,\mathbb Q_p}$ is injective on cohomology. So $\Upsilon_{\mathcal X,\mathbb Q_p}$ induces an isomorphism on cohomology, and thus is a quasi-isomorphism.

Remark 4.10. More generally, using [Reference Kubrak and PrikhodkoKP24, Proposition 4.3.35 and Theorem 4.3.38], from Theorem 4.8 we obtain for a Hodge-proper stack $\mathcal{X}$ over $\mathcal O_K$ all the comparisons that one usually has in p-adic Hodge theory. Namely, for any $n\geqslant 0$ we get a $(\varphi,G_K)$ -equivariant isomorphism

\begin{align*} H^n_{\acute{\mathrm{e}}\mathrm{t}}({\mathcal X}_{{C}}, \mathbb Q_p)\otimes_{\mathbb Q_p} B_{\mathrm{crys}} \simeq H^n_{{\mathrm{crys}}}({\mathcal X}_k/W(k))\bigg[\frac{1}{p}\bigg]\otimes_{W(k)[{1}/{p}]} B_{\mathrm{crys}}, \end{align*}

a filtered $G_K$ -equivariant isomorphism

\begin{align*} H^n_{\acute{\mathrm{e}}\mathrm{t}}({\mathcal X}_{{C}}, \mathbb Q_p)\otimes_{\mathbb Q_p} B_{\mathrm{dR}} \xrightarrow{\sim} H^n_{\mathrm{dR}}(\mathcal X/K)\otimes_K B_{\mathrm{dR}} \end{align*}

and the ( $G_K$ -equivariant) Hodge–Tate decomposition

\begin{align*} H^n_{\acute{\mathrm{e}}\mathrm{t}}({\mathcal X}_{{C}},\mathbb Q_p)\otimes_{\mathbb Q_p}{C} \simeq \bigoplus_{i+j=n} H^j({\mathcal X}_{K},\wedge^i\mathbb L_{\mathcal X/K})\otimes_{K}{C}(-i), \end{align*}

where ${C}(-i)$ denotes the $-i$ th Tate twist.

4.3 Example: the complement to a complete intersection in $\mathbb P^n$ .

Here we would like to illustrate the difference between d-de Rham-properness and d-Hodge-properness on a particular example.

Namely, let $Z\hookrightarrow \mathbb P^n_{\mathcal O_K}$ be a complete intersection given by $f_1=f_2=\cdots= f_d=0$ , where each $f_i$ is a homogeneous polynomial of degree $d_i$ . We also assume that Z is flat over $\mathcal O_K$ . Let $\mathcal X\colon = \mathbb P^n_{\mathcal O_K}\!\backslash Z$ . By Lemma 2.4, $\mathcal{X}$ is $(d-2)$ -Hodge-proper over $\mathcal O_K$ . We claim that:

– $\mathcal{X}$ is not $(d-1)$ -Hodge-proper;

– $\mathcal{X}$ is still $(d-1)$ -de Rham proper.

Denote by $j\colon \mathcal X \rightarrow \mathbb P^n_{\mathcal O_K}$ the natural embedding. Also, let $\mathcal I_Z\subset \mathcal O_{\mathbb P^n}$ be the sheaf of ideals defining Z. It is generated by the images of maps $\alpha_{i}\colon \mathcal O_{\mathbb P^n}(-d_i)\rightarrow \mathcal O_{\mathbb P^n}$ given by $f_i\in H^0(\mathbb P^n, \mathcal O_{\mathbb P^n}(d_i))$ . Since Z is a complete intersection, the sheaf $\mathcal O_Z\colon = \mathcal O_{\mathbb P^n}/\mathcal I_Z$ has a free Koszul-type resolution, namely

\begin{align*}{\mathrm{Kos}_{\mathbb P^n}^*}(\alpha_1,\ldots,\alpha_d)\colon = \bigotimes_{i=1}^d (\mathcal O_{\mathbb P^n}(-d_i)\xrightarrow{\alpha_{i}} \mathcal O_{\mathbb P^n})\simeq \mathcal O_Z,\end{align*}

where $\mathcal O_{\mathbb P^n}(-d_i)\xrightarrow{\alpha_{i}} \mathcal O_{\mathbb P^n}$ is considered as a two-term complex concentrated in cohomological degrees 0 and $-1$ . For a d-tuple $\underline s\colon = (s_1,\ldots,s_d)\in \mathbb N^d$ denote by $Z^{[\underline s]}\subset \mathbb P^n_{\mathcal O_K}$ the subscheme defined by $f_1^{s_1}=f_2^{s_2}=\cdots=f_d^{s_d}=0$ . The subscheme $Z^{[\underline s]}$ is still a complete intersection, and we have

\begin{align*}{\mathrm{Kos}}_{\mathbb P^n}^*(\alpha_1^{s_1},\ldots,\alpha_d^{s_d})\simeq \mathcal O_{Z^{[\underline s]}},\end{align*}

where $\alpha_{i}^{s_i}\colon \mathcal O_{\mathbb P^n}(-s_id_i)\rightarrow \mathcal O_{\mathbb P^n}$ is the map induced by $f_i^{s_i}$ . Sheaves $\mathcal O_{Z^{[\underline s]}}$ , as well as the Koszul complexesFootnote ⁶ ${\mathrm{Kos}}_{\mathbb P^n}^*(\alpha_1^{s_1},\ldots,\alpha_i^{s_i},\ldots, \alpha_d^{s_d})$ naturally form an inverse system (via the termwise partial ordering on $\mathbb N^d$ ). Also note that one has the following formula for the dual complex:

\begin{align*}{\mathrm{Kos}}_{\mathbb P^n}^*(\alpha_1^{s_1},\ldots, \alpha_d^{s_d})^{\vee}\simeq \mathrm{Kos}_{\mathbb P^n}^*(\alpha_1^{s_1},\ldots, \alpha_d^{s_d})(s_1d_1+\cdots +s_dd_d)[-d]\end{align*}

(this follows from the isomorphism of complexes $(\mathcal O_{\mathbb P^n}(-s_id_i)\rightarrow \mathcal O_{\mathbb P^n})^\vee \simeq (\mathcal O_{\mathbb P^n}(-s_id_i)\rightarrow \mathcal O_{\mathbb P^n})(s_id_i)[-1]$ ).

One has a fiber sequence

\begin{align*}\underline{R\Gamma}_Z(\mathcal O_{\mathbb P^n})\rightarrow \mathcal O_{\mathbb P^n} \rightarrow Rj_*\mathcal O_{\mathcal X},\end{align*}

where $\underline{R\Gamma}_Z(\mathcal O_{\mathbb P^n})$ is the (sheaf of) local cohomology. We get that $H^i(\mathcal X,\mathcal O_{\mathcal X})$ is finitely generated if and only if $H^{i+1}(\mathbb P^n,\underline{R\Gamma}_Z(\mathcal O_{\mathbb P^n}))$ is. Thus, to show that $\mathcal{X}$ is not $(d-1)$ -Hodge-proper it is enough to show that $H^d(\mathbb P^n,\underline{R\Gamma}_Z(\mathcal O_{\mathbb P^n}))$ is not finitely generated.

The local cohomology $\underline{R\Gamma}_Z(\mathcal O_{\mathbb P^n})$ is given explicitly by

\begin{align*}\underline{R\Gamma}_Z(\mathcal O_{\mathbb P^n})\simeq\mathop{\operatorname{colim}}\limits_{\underline{s}\in \mathbb N^d}\mathrm{RHom}_{\mathcal O_{\mathbb P^n}}(\mathcal O_{Z^{[\underline s]}},\mathcal O_{\mathbb P^n})\end{align*}

(see, e.g., [Sta26, Tag 0956]). Naturally, one can use Koszul resolutions to compute RHom: one sees that

\begin{align*}\mathrm{RHom}_{\mathcal O_{\mathbb P^n}}(\mathcal O_{Z^{[\underline s]}},\mathcal O_{\mathbb P^n})\simeq{\mathrm{Kos}}_{\mathbb P^n}^*(\alpha_1^{s_1},\ldots, \alpha_d^{s_d})^{\vee}\simeq \mathcal O_{Z^{[\underline s]}}(s_1d_1+\cdots +s_dd_d)[-d].\end{align*}

In this way

\begin{align*}\underline{R\Gamma}_Z(\mathcal O_{\mathbb P^n})\simeq\mathop{\operatorname{colim}}\limits_{\underline{s}\in \mathbb N^d} \mathcal O_{Z^{[\underline s]}}(s_1d_1+\cdots +s_dd_d)[-d].\end{align*}

In particular, $\tau^{\lt d} \underline{R\Gamma}_Z(\mathcal O_{\mathbb P^n})\simeq 0$ .

Note that $\mathcal O_{Z^{[\underline s]}}$ has a finite filtrationFootnote ⁷ by a poset $P_{\underline s}^\lt \colon = \{\underline t\in\mathbb N^d_{\geqslant 0}\mid \underline t\lt \underline s\}$ with the associated graded given by $\bigoplus_{\underline t\in P_{\underline s}} \mathcal O_Z(-t_1d_1-\cdots -t_dd_d)$ . For the twist $\mathcal O_{Z^{[\underline s]}}(s_1d_1+\cdots +s_dd_d)$ this gives a filtration with the associated graded $\bigoplus_{\underline t\in P_{\underline s}} \mathcal O_Z((t_1+1)d_1+\cdots +(t_d+1)d_d)$ . Moreover, for $\underline s'\gt \underline s$ the corresponding map on the associated graded

\begin{align*}\bigoplus_{\underline t\in P_{\underline s}} \mathcal O_Z((t_1+1)d_1+\cdots +(t_d+1)d_d) \longrightarrow \bigoplus_{\underline t\in P_{\underline s'}} \mathcal O_Z((t_1+1)d_1+\cdots +(t_d+1)d_d)\end{align*}

is the embedding induced by $P_{\underline s}\hookrightarrow P_{\underline s'}$ . This way one sees that for the cokernel $K_{\underline s,\underline s'}$ of the above map has an associated graded given by direct sum of line bundles $\mathcal O_Z(\ell)$ with $\ell\gt \min_i(s_id_i)$ ; moreover, as $s'\rightarrow \infty$ the number of these line bundles grows to $\infty$ as well. Thus, for a fixed $\underline s\gg 0$ and $\underline s'\rightarrow \infty$ we have $\mathrm{rk}_{\mathcal O_K}H^0(\mathbb P^n, K_{\underline s,\underline s'})\rightarrow \infty $ . Using the exact sequence

\begin{align*}0\rightarrow H^0\bigg(\mathcal O_{Z^{[\underline s]}}\bigg(\sum_is_id_i\bigg)\!\bigg)\rightarrow H^0\bigg(\mathcal O_{Z^{[\underline s']}}\bigg(\sum_i s_i'd_i\bigg)\!\bigg)\rightarrow H^0(K_{\underline s,\underline s'})\rightarrow H^1\bigg(\mathcal O_{Z^{[\underline s]}}\bigg( \sum_i s_id_i\bigg)\!\bigg)\end{align*}

one sees that

\begin{align*}H^d(\underline R\Gamma_Z(\mathcal O_{\mathbb P^n}))\simeq H^0\bigg(\mathop{\operatorname{colim}}\limits_{\underline s'\in\mathbb N^d}\mathcal O_{Z^{[\underline s']}}\bigg(\sum_is_i'd_i\bigg)\!\bigg)\simeq\mathop{\operatorname{colim}}\limits_{\underline s'\in \mathbb N^d}H^0\bigg(\mathcal O_{Z^{[\underline s']}}\bigg(\sum_is_i'd_i\bigg)\!\bigg)\end{align*}

has infinite rank over $\mathcal O_K$ , and so is not finitely generated.

We now consider the de Rham cohomology of $\mathcal{X}$ . We need to show that $H^{d-1}_{\mathrm{dR}}(\mathcal X/\mathcal O_K)$ is finitely generated over $\mathcal O_K$ . For any $k\geqslant 0$ by projection formula we have

\begin{equation*} Rj_*\Omega^k_{\mathcal X}\simeq Rj_*j^*\Omega^k_{\mathbb P^n}\simeq Rj_*\mathcal O_{\mathcal X}\otimes_{\mathcal O_{\mathbb P^n}} \Omega^k_{\mathbb P^n}.\end{equation*}

Since $\tau^{\lt d}\underline{R\Gamma}_Z(\mathcal O_{\mathbb P^n})=0$ we have that $\tau^{\lt d-1}Rj_*\mathcal O_{\mathcal X}\simeq \mathcal O_{\mathbb P^n}$ and, consequently $\tau^{\lt d-1}Rj_*\Omega^k_{\mathcal X}\simeq \Omega^k_{\mathbb P^n}$ . In particular, for $i\gt 0$ and $j\lt d-1$ we get that $H^{i,j}(\mathcal X/\mathcal O_K)\simeq H^{i,j}(\mathbb P^n/\mathcal O_K)$ . In this way, on the first page of the Hodge-to-de Rham spectral sequence there is a single Hodge cohomology of total degree $d-1$ that is infinitely generated, namely $H^{0,d-1}(\mathcal X/\mathcal O_K)\colon = H^{d-1}(\mathcal X,\mathcal O_{\mathcal X})$ . Let $E_\infty^{0,d-1}\subset H^{0,d-1}(\mathcal X/\mathcal O_K)$ be the corresponding term on the infinite page; it would be enough to show that $E_\infty^{0,d-1}$ is finitely generated. However, we identified $H^{0,d-1}(\mathcal X/\mathcal O_K)$ with $H^d(\underline R\Gamma_Z(\mathcal O_{\mathbb P^n}))\simeq \operatorname{colim}_{\underline s'\in \mathbb N^d} H^0(\mathcal O_{Z^{[\underline s']}}(\textstyle \sum_i s_i'd_i))$ , and so it is p-torsion free since we assumed Z to be flat over $\mathcal O_K$ . Consequently, so is $E_\infty^{0,d-1}$ and thus it will be enough to show that it is finitely generated after inverting p. Note that the p-localization of the Hodge-to-de Rham spectral sequence or $\mathcal{X}$ exactly gives that for $\mathcal X_K$ . Since over a field of characteristic 0 via excision we have $H^i_{\mathrm{dR}}(\mathcal X_K/K)\simeq H^{i}_{\mathrm{dR}}(\mathbb P_K^n/K)$ for $i\lt 2d$ , we get that $H^{d-1}_{\mathrm{dR}}(\mathcal X_K/K)$ , and $E_\infty^{0,d-1}[{1}/{p}]$ in particular, are finite-dimensional.

5.1 Coherence properties of various cohomology

Let R be a (classical) Noetherian ring, and let $I=(x_1,\ldots,x_n)\subset R$ be an ideal generated by a regular sequence (so that $R/I\simeq \bigotimes_{i=1}^n \mathrm{cofib}(R\xrightarrow{\cdot x_i}R)$ ). Assume also that R is I-adically complete (in the classical sense).

Here comes the first application.

Corollary 5.2. Let $\mathcal{X}$ be a d-de Rham proper stack over a perfect field k of characteristic p. Then

\begin{align*} \tau^{\leqslant d}(R\Gamma_{\mathrm{crys}}(\mathcal X/W(k)))\in \mathrm{Coh}(W(k)). \end{align*}

Proof. By de Rham and crystalline comparisons [Reference Kubrak and PrikhodkoKP24, Propositions 2.3.20 and 2.5.7], we have

\begin{align*}R\Gamma_{\mathrm{crys}}(\mathcal X/W(k))\otimes_{W(k)} k\simeq R\Gamma_{\mathrm{dR}}(\mathcal X/k).\end{align*}

The rest then follows from Lemma 5.1, since $R\Gamma_{\mathrm{crys}}(\mathcal X/W(k))$ is derived p-complete.

Remark 5.3. We note that $M\colon = \tau^{\geqslant d+1}R\Gamma_{\mathrm{crys}}(\mathcal X/W(k))[d+1]$ also satisfies the following finiteness condition:

\begin{align*} \tau^{\lt 0}([M/p])\in \mathrm{Coh}(k). \end{align*}

More generally, given a stack $\mathcal{X}$ over $\mathcal O_K$ one can establish similar properties for the prismatic cohomology (over the Breuil–Kisin prism). Here, let K be a discretely valued complete field extension of $\mathbb Q_p$ with a perfect residue field k. For a choice of uniformizer $\pi\in \mathcal O_K$ we have the corresponding Breuil–Kisin prism $(\mathfrak S, E(u))=(W(k)[[u]], E(u))$ with $\mathfrak S/E(u) =\mathcal O_K$ . Recall [Reference Kubrak and PrikhodkoKP24, Definition 2.2.1] that for a prestack $\mathcal{X}$ over $\mathcal O_K$ we have the prismatic cohomology complex $R\Gamma_{{{\Delta}}}(\mathcal X/\mathfrak S)\in D(\mathrm{Mod}_{\mathfrak S})$ , as well as its twisted version $R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\in D(\mathrm{Mod}_{\mathfrak S})$ , which, in the case of the Breuil–Kisin prism, is also identified with the ‘Frobenius twist’ $\varphi_{\mathfrak S}^*R\Gamma_{{{\Delta}}}(\mathcal X/\mathfrak S)$ . The complexes $R\Gamma_{{{\Delta}}}(\mathcal X/\mathfrak S)$ , $R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)$ are derived (p,u)-adically complete and one has a natural ‘Frobenius map’

\begin{align*}\varphi_{{{\Delta}}}\colon R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S) \longrightarrow R\Gamma_{{{{\Delta}}}}(\mathcal X/\mathfrak S),\end{align*}

which becomes an equivalence after inverting E(u) in the case $\mathcal{X}$ is a smooth Artin stack [Reference Kubrak and PrikhodkoKP24, Remark 2.2.16].

Corollary 5.4. Let $\mathcal{X}$ be a smooth Artin stack over $\mathcal O_K$ . If the reduction $\mathcal X_k$ is d-de Rham proper over k, then both

\begin{align*} \tau^{\leqslant d}R\Gamma_{{{{\Delta}}}}(\mathcal X/\mathfrak S)\in \mathrm{Coh}(\mathfrak S) \quad \text{and} \quad \tau^{\leqslant d}R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\in \mathrm{Coh}(\mathfrak S). \end{align*}

Proof. The map $\varphi_{\mathfrak S}\colon \mathfrak S \rightarrow \mathfrak S$ is faithfully flat, so by Lemma 2.10 it is enough to show that $\tau^{\leqslant d}R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\in \mathrm{Coh}(\mathfrak S)$ . The claim then follows from crystalline comparison [Reference Kubrak and PrikhodkoKP24, Remark 2.5.10]

\begin{align*}R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\otimes_{\mathfrak S}\mathfrak S/u\simeq R\Gamma_{\mathrm{crys}}(\mathcal X_k/W(k))\end{align*}

and Corollary 5.2.

Remark 5.5. By Corollary 5.4, we get that if the reduction $\mathcal X_k$ is d-de Rham proper, the $\mathfrak S$ -modules $H^i_{{{\Delta}}}(\mathcal X/\mathfrak S)$ are finitely generated if $i\leqslant d$ . The isomorphisms

\begin{align*} \varphi_{{{\Delta}}}\bigg[\frac{1}{E}\bigg]\colon \varphi_{\mathfrak S}^*H^i_{{{\Delta}}}(\mathcal X/\mathfrak S)\bigg[\frac{1}{E}\bigg] \xrightarrow{\sim} H^i_{{{\Delta}}}(\mathcal X/\mathfrak S)\bigg[\frac{1}{E}\bigg] \end{align*}

then endow each of them with a Breuil–Kisin module structure. Just given the existence of this structure, we get that $H^i_{{{\Delta}}}(\mathcal X/\mathfrak S)[{1}/{p}]$ is a finite free $\mathfrak S[{1}/{p}]$ -module for $i\leqslant d$ (see, e.g., [Reference Bhatt, Morrow and ScholzeBMS18, Proposition 4.3]). In fact, we show that (under some torsion-free assumptions) the corresponding lattice in a crystalline Galois representation is described by étale cohomology of the Raynaud generic fiber (see Remark 5.25).

Remark 5.6. The d-de Rham properness of $\mathcal X_k$ also implies some important properties of prismatic cohomology in degrees higher than d, that turn out to be rather crucial later. Namely, even though $M\colon = \tau^{\geqslant d+1}R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)[d+1]$ is not an object of $\mathrm{Coh}(\mathfrak S)$ , it is still derived (p,u)-complete and satisfies the following finiteness condition: namely, $\tau^{\lt 0}[M/(p,u)]\in \mathrm{Coh}(k)$ . Indeed, let $C\colon = R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)[d+1]$ ; we have $M\simeq \tau^{\geqslant 0}(C)$ . Note that by our assumptions on $\mathcal X_k$ and de Rham comparison we have $\tau^{\lt 0}([C/(p,u)])\in \mathrm{Coh}(k)$ . We have a fiber sequence $ \tau^{\lt 0}C \rightarrow C \rightarrow M $ which gives a fiber sequence

\begin{align*} [\tau^{\lt 0}C/(p,u)]\rightarrow [C/(p,u)] \rightarrow [M/(p,u)]. \end{align*}

We have that $\tau^{\lt 0}C$ is coherent by Corollary 5.4, and so it follows that $\tau^{\lt 0}([M/(p,u)])\in \mathrm{Coh}(k)$ .

Consequently, applying $[-/(p,u)]$ to the fiber sequence $H^0(M)[0] \rightarrow M \rightarrow \tau^{\gt 0}M$ and considering the corresponding long exact sequence of cohomology we get that $H^{-2}([H^0(M)/(p,u)])\simeq H^{-2}([M/(p,u)])$ while there is also an embedding $H^{-1}([H^0(M)/(p,u)])\hookrightarrow H^{-1}([M/(p,u)])$ , and so both are finite-dimensional vector spaces over k. Recalling the definition of M we get from this that the (p,u)-torsion $H^{d+1}_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)[p][u]$ is a finite-dimensional k-vector space.

We record a part of the discussion in Remark 5.6 as Lemma 5.7 for future reference.

Lemma 5.7. Let $\mathcal{X}$ be a smooth Artin stack over $\mathcal O_K$ such that the reduction $\mathcal X_k$ is d-de Rham proper over k. Then $M\colon = \tau^{\geqslant d+1}R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)[d+1]$ satisfies

\begin{align*} \tau^{\lt 0}([M/(p,u)])\in \mathrm{Coh}(k). \end{align*}

Let R be a p-bounded p-complete ring. The following notation is convenient.

Definition 5.8. We define the p-completed Hodge cohomology $R\Gamma_{\mathrm H}(\widehat{\mathcal X}/R)$ of an Artin R-stack $\mathcal{X}$ as the derived p-completion $R\Gamma_{\mathrm H}({\mathcal X}/R)^\wedge_p$ . We also denote

\begin{align*}H^{j,i}(\widehat{\mathcal X}/R)\colon = H^i(R\Gamma(\mathcal X,\wedge^j\mathbb L_{\mathcal X/R})^\wedge_p).\end{align*}

Similarly, if $\mathcal{X}$ is smooth over R we define the p-completed de Rham cohomology

\begin{align*}R\Gamma_{\mathrm{dR}}(\widehat{\mathcal X}/R)\colon = R\Gamma_{\mathrm{dR}}(\mathcal X/R)^\wedge_p.\end{align*}

Naturally, we have coherence results for $R\Gamma_{\mathrm H}(\widehat{\mathcal X}/R)$ and $R\Gamma_{\mathrm{dR}}(\widehat{\mathcal X}/R)$ implied by d-Hodge and d-de Rham-properness of the mod p reduction $\mathcal X_{R/p}$ of $\mathcal{X}$ .

1. (1) If the reduction $\mathcal X_{R/p}$ is d-Hodge-proper over $R/p$ we have

   \begin{align*}\tau^{\leqslant d}R\Gamma_{\mathrm H}(\widehat{\mathcal X}/R)\in \mathrm{Coh}(R).\end{align*}
   Equivalently, $H^{j,i}(\widehat{\mathcal X}/R)$ is a finitely generated R-module for any $i+j\leqslant d $ .

2. (2) If the reduction $\mathcal X_{R/p}$ is d-de Rham-proper over $R/p$ we have

   \begin{align*}\tau^{\leqslant d}R\Gamma_{\mathrm{dR}}(\widehat{\mathcal X}/R)\in \mathrm{Coh}(R).\end{align*}

Proof. Since p is a non-zero divisor in R, we have $R/p\simeq \mathrm{cofib}(R\xrightarrow{\cdot p}R)$ . Thus, for any $M\in D(\mathrm{Mod}_{R})$ and the definition of derived p completion we have $M^\wedge_p\otimes_R R/p\simeq M\otimes_R R/p$ . In addition, $R/p$ is of Tor-amplitude $[-1,0]$ as an R-module. This, together with base change for Hodge and de Rham cohomology [Reference Kubrak and PrikhodkoKP22, Proposition 1.1.8], leads to equivalences

\begin{align*} R\Gamma_{\mathrm H}(\widehat{\mathcal X}/R)\otimes_R R/p \xrightarrow{\sim} R\Gamma_{\mathrm H}({\mathcal X}_{R/p}/(R/p)), \quad R\Gamma_{\mathrm{dR}}(\widehat{\mathcal X}/R)\otimes_R R/p \xrightarrow{\sim} R\Gamma_{\mathrm{dR}}({\mathcal X}_{R/p}/(R/p)). \end{align*}

Then we are done by Lemma 5.1.

We also record a statement analogous to Corollary 5.4 but which applies to more general base prisms. We refer to [Reference Kubrak and PrikhodkoKP24, Definition 2.2.1] for the definitions of prismatic cohomology $R\Gamma_{{{\Delta}}}(\mathcal X/A)\in D(\mathrm{Mod}_{A})$ , as well as its twisted version $R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/A)\in D(\mathrm{Mod}_{A})$ .

Corollary 5.10. Let (A,I) be a transversalFootnote ⁸ prism such that the underlying ring A is Noetherian. Let $\mathcal{X}$ be a qcqs smooth Artin stack over $A/I$ such that the reduction $\mathcal X_{A/(I,p)}$ is d-de Rham-proper over $A/(I,p)$ . Then

\begin{align*} \tau^{\leqslant d}R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/A)\in \mathrm{Coh}(A). \end{align*}

If the reduction $\mathcal X_{A/(I,p)}$ is d-Hodge-proper over $A/(I,p)$ , then

\begin{align*} \tau^{\leqslant d}R\Gamma_{{{{\Delta}}}}(\mathcal X/A)\in \mathrm{Coh}(A). \end{align*}

Proof. Note that we have isomorphisms

\begin{align*}R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/A)\otimes_A A/I \xrightarrow{\sim} R\Gamma_{L{\mathrm{dR}}^\wedge_p}(\mathcal X/(A/I)) \xrightarrow{\sim} R\Gamma_{\mathrm{dR}}(\widehat{\mathcal X}/(A/I))^\wedge_p.\end{align*}

Here the first arrow is given by the de Rham comparison [Reference Kubrak and PrikhodkoKP24, Proposition 2.3.20], while the second equivalence holds for any smooth Artin $\mathcal{X}$ (see [Reference Kubrak and PrikhodkoKP24, Proposition 2.3.18]). Since (A,I) is transversal we can apply Corollary 5.9(2), getting that $\tau^{\leqslant d}R\Gamma_{\mathrm{dR}}(\widehat{\mathcal X}/(A/I))$ . The first part of the proposition then follows from Lemma 5.1.

For the second part recall that the Hodge–Tate complex $R\Gamma_{{{{\Delta}}}/I}(\mathcal X/A)\colon = R\Gamma_{{{{\Delta}}}}(\mathcal X/A)\otimes_A^{\mathbb L} A/I$ has a natural conjugate filtration with the associated graded $\mathrm{gr}_i = R\Gamma(\mathcal X, \wedge^i\mathbb L_{\mathcal X/(A/I)})^\wedge_p\{-i\}[-i]$ (see [Reference Kubrak and PrikhodkoKP24, § 2.4]). Since $\mathcal{X}$ is smooth this filtration is exhaustive [Reference Kubrak and PrikhodkoKP24, Proposition 2.4.3]) and induces a convergent spectral sequence $E^{i,j}=H^{j,i}(\widehat{\mathcal X}/(A/I))\{-i\}\Rightarrow H^{i+j}_{{{{\Delta}}}/I}(\mathcal X/A)$ (see [Reference Kubrak and PrikhodkoKP24, § 4.3.3]). Recall that the twist $\{-i\}$ here is just tensoring with the invertible $A/I$ -module $((I/I^2)^\vee)^{\otimes i}$ and so does not change coherence. It is then enough to show that $H^{j,i}(\widehat{\mathcal X}/(A/I))$ is finitely generated for $i+j\leqslant d$ , which follows from Corollary 5.9(1).

5.2 Some integral p-adic Hodge theory in a d-de Rham-proper setting

In this section we adapt some results of [Reference Kubrak and PrikhodkoKP24, §§ 2 and 4] about prismatic cohomology of Hodge-proper stacks to the setting of stacks that are only d-Hodge (or rather d-de Rham) proper.

In this section we mostly assume that the base prism $(A,I)=(\mathfrak S,E(u))$ is a Breuil–Kisin prism. The goal is to establish a truncated version of integral p-adic Hodge theory in the setting of d-de Rham-proper stacks over $\mathcal O_K$ . The main reference to some relevant background material is [Reference Kubrak and PrikhodkoKP24, § 4.3].

Let $\pi\in \mathcal O_K$ be the uniformizer to which the chosen Breuil–Kisin prism $(\mathfrak S,E(u))$ corresponds (so $\pi$ is the image of u under the identification $\mathfrak S/E(u)\xrightarrow{\sim} \mathcal O_K$ ). A compatible choice of $p^n$ -roots of $\pi$ in $\mathcal O_C$ gives an element $\pi^\flat\colon = (\pi,\pi^{1/p}, \pi^{1/p^2},\ldots)\in \mathcal O_C^\flat$ in the tilt $\mathcal O_C^\flat\colon = \lim_{x\mapsto x^p}\mathcal O_C/p$ . This then defines a homomorphism $s_{\pi^\flat}\colon \mathfrak S\rightarrow {A_{\mathrm{inf}}}\colon = W(\mathcal O_{C}^\flat)$ by sending $u \mapsto [\pi^\flat]$ . We can also compose it further with the natural map ${A_{\mathrm{inf}}}\rightarrow W(C^\flat)$ induced by $\mathcal O_C^\flat\rightarrow C^\flat\colon = \mathcal O_C^\flat[{1}/{\pi^\flat}]$ . Recall that by the étale comparison [Reference Kubrak and PrikhodkoKP24, Corollary 4.3.3] and base change we have an equivalence

(5.1) \begin{equation}R\Gamma_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_{{C}}, \mathbb Z_p) \xrightarrow{\sim} (R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/ \mathfrak S)\mathbin{\widehat{\otimes}}_{\mathfrak S} W(C^\flat))^{\varphi_{{{{\Delta}}}} = 1}\end{equation}

for any smooth qcqs Artin $\mathcal O_K$ -stack $\mathcal{X}$ .

Remark 5.11. We use the twisted version of the étale comparison here for a future convenience. The difference only affects the natural map

\begin{align*} R\Gamma_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_{{C}}, \mathbb Z_p)\mathbin{\widehat{\otimes}}_{\mathbb Z_p}W(C^\flat) \longrightarrow R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/ \mathfrak S)\mathbin{\widehat{\otimes}}_{\mathfrak S} W(C^\flat), \end{align*}

namely it becomes twisted by Frobenius on $W(C^\flat)$ (see [Reference Kubrak and PrikhodkoKP24, Remark 4.3.7]; to apply the remark, note that by base change

\begin{align*}R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/ \mathfrak S)\mathbin{\widehat{\otimes}}_{\mathfrak S} W(C^\flat)\simeq R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/ {A_{\mathrm{inf}}})\mathbin{\widehat{\otimes}}_{{A_{\mathrm{inf}}}} W(C^\flat)).\end{align*}

1. (1) There is a natural equivalence

   \begin{align*}\tau^{\leqslant d}R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\otimes_{\mathfrak S} W(C^\flat)\xrightarrow{\sim} \tau^{\leqslant d}(R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\mathbin{\widehat{\otimes}}_{\mathfrak S}W(C^\flat)).\end{align*}

2. (2) The natural map

   \begin{align*} \tau^{\leqslant d}R\Gamma_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_{{C}}, \mathbb Z_p)\otimes_{\mathbb Z_p} W(C^\flat) \longrightarrow \tau^{\leqslant d}(R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/ \mathfrak S)\mathbin{\widehat{\otimes}}_{\mathfrak S} W(C^\flat)) \end{align*}
   is also an equivalence.

3. (3) Consequently, one has isomorphisms

   \begin{align*} H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Z_p)\otimes_{\mathbb Z_p}W(C^\flat) \xrightarrow{\sim} H^i_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\otimes_{\mathfrak S}W(C^\flat) \end{align*}
   for all $i\leqslant d$ .

We also need the following lemma which controls the (classical) base change of prismatic cohomology to ${A_{\mathrm{inf}}}$ .

Lemma 5.13. Let $\mathcal{X}$ be a stack over $\mathcal O_K$ such that its reduction $\mathcal X_k$ is d-de Rham proper. Then we have a natural isomorphism

\begin{align*} \tau^{\leqslant d}R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\otimes_{\mathfrak S}{A_{\mathrm{inf}}}\simeq \tau^{\leqslant d}R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}}). \end{align*}

Proof. By base change for prismatic cohomology [Reference Kubrak and PrikhodkoKP24, Proposition 2.2.17] we have

\begin{align*} R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}})\simeq R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\mathbin{\widehat{\otimes}}_{\mathfrak S}{A_{\mathrm{inf}}}.\end{align*}

Note that ${A_{\mathrm{inf}}}$ is a (p,u)-completely free module over $\mathfrak S$ . Indeed, picking a basis $\{x_s\}_{s\in S}$ of $\mathcal O_C^\flat/\pi^\flat$ over k and lifts $\tilde{x}_i\in {A_{\mathrm{inf}}}$ (under the projection ${A_{\mathrm{inf}}}\rightarrow \mathcal O_C^\flat/\pi^\flat$ ) we get a map

\begin{align*}\widehat{\bigoplus_{s\in S}}\mathfrak S \longrightarrow {A_{\mathrm{inf}}},\end{align*}

which is an isomorphism modulo (p,u), and thus is itself an isomorphism by Nakayama’s lemma. Moreover, by Remark 5.6 we have that $M := \tau^{\geqslant d+1}R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)[d+1]$ satisfies the condition of Proposition A.13,, and thus . Since the completed tensor product is right t-exact, we get from this that

\begin{align*}\tau^{\leqslant d} R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}})\simeq \tau^{\leqslant d} (R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\mathbin{\widehat{\otimes}}_{\mathfrak S}{A_{\mathrm{inf}}}) \simeq \tau^{\leqslant d} (R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S))\mathbin{\widehat{\otimes}}_{\mathfrak S}{A_{\mathrm{inf}}}.\end{align*}

By the assumption on $\mathcal X_k$ we have $\tau^{\leqslant d}(R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S))\in\mathrm{Coh}(\mathfrak S)$ , and so the classical tensor product $\tau^{\leqslant d} (R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S))\otimes_{\mathfrak S}{A_{\mathrm{inf}}}$ is already derived (p,u)-complete. In this way we get an isomorphism

\begin{align*}\tau^{\leqslant d} (R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S))\otimes_{\mathfrak S}{A_{\mathrm{inf}}}\xrightarrow{\sim} \tau^{\leqslant d} R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}}),\end{align*}

as desired.

Corollary 5.14. Let $\mathcal{X}$ be a stack over $\mathcal O_K$ such that its reduction $\mathcal X_k$ is d-de Rham proper. Then for any $0\leqslant i\leqslant d$ we have a $G_K$ -equivariant isomorphism

\begin{align*} H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Z_p)\otimes_{\mathbb Z_p}{A_{\mathrm{inf}}}\bigg[\frac{1}{\mu}\bigg]\xrightarrow{\sim} H^i_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}})\otimes_{A_{\mathrm{inf}}} {A_{\mathrm{inf}}}\bigg[\frac{1}{\mu}\bigg] \end{align*}

that extends to a $(G_K,\varphi)$ -equivariant isomorphism

\begin{align*} H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Z_p)\otimes_{\mathbb Z_p}B_{\mathrm{crys}} \xrightarrow{\sim} H^i_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}})\otimes_{A_{\mathrm{inf}}} B_{\mathrm{crys}}. \end{align*}

Proof. Recall that the map $\mathfrak S\rightarrow {A_{\mathrm{inf}}}$ is flat [Reference Bhatt, Morrow and ScholzeBMS18, Lemma 4.30]. Thus, from Lemma 5.13 we get that for any $0\leqslant i\leqslant d$ there is an isomorphism $H^i_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}})\simeq H^i_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\otimes_{\mathfrak S}{A_{\mathrm{inf}}}$ . Applying Proposition 5.12(3) we then get a $\varphi$ -equivariant isomorphism

(5.2) \begin{equation} H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Z_p)\otimes_{\mathbb Z_p}W(C^\flat)\xrightarrow{\sim} H^i_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}})\otimes_{A_{\mathrm{inf}}} W(C^\flat),\end{equation}

which is in fact induced by the étale comparison map

\begin{align*} R\Gamma_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Z_p) \longrightarrow R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}})\mathbin{\widehat{\otimes}}_{{A_{\mathrm{inf}}}} W(C^\flat) \end{align*}

and thus is $G_K$ -equivariant.

By Remark 5.5, $H^i_{{{{\Delta}}}}(\mathcal X/\mathfrak S)$ is a Breuil–Kisin module in the range $0\leqslant i\leqslant d$ and this way $H^i_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}})\simeq H^i_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\otimes_{\mathfrak S}{A_{\mathrm{inf}}}$ is a Breuil–Kisin–Fargues module. Then by [Reference Bhatt, Morrow and ScholzeBMS18, Lemma 4.26] the isomorphism in (5.2) restricts to an isomorphism

\begin{align*} H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Z_p)\otimes_{\mathbb Z_p}{A_{\mathrm{inf}}}\bigg[\frac{1}{\mu}\bigg]\xrightarrow{\sim} H^i_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}})\otimes_{A_{\mathrm{inf}}} {A_{\mathrm{inf}}}\bigg[\frac{1}{\mu}\bigg], \end{align*}

which is also $G_K$ -equivariant, since $G_K$ -action on $W(C^\flat)$ preservesFootnote ¹⁰ ${A_{\mathrm{inf}}}[{1}/{\mu}]\subset W(C^\flat)$ . This then automatically extends to a $(G_K,\varphi)$ -equivariant isomorphism

\begin{align*} H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Z_p)\otimes_{\mathbb Z_p}{A_{\mathrm{inf}}}\bigg[\frac{1}{\mu},\frac{1}{\varphi(\mu)}, \frac{1}{\varphi^2(\mu)},\ldots\bigg]\xrightarrow{\sim} H^i_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}})\otimes_{A_{\mathrm{inf}}} {A_{\mathrm{inf}}}\bigg[\frac{1}{\mu},\frac{1}{\varphi(\mu)}, \frac{1}{\varphi^2(\mu)},\ldots\bigg], \end{align*}

where localization ${A_{\mathrm{inf}}}[{1}/{\mu},{1}/({\varphi(\mu)}), {1}/({\varphi^2(\mu)}),\ldots]$ is the minimal $\varphi$ -invariant subring of $W(C^\flat)$ containing ${A_{\mathrm{inf}}}[{1}/{\mu}]$ . The $G_K$ -equivariant embedding ${A_{\mathrm{inf}}}[{1}/{\mu}]\rightarrow B_{\mathrm{crys}}$ extends to a $(G_K,\varphi)$ -equivariant embedding

\begin{align*} {A_{\mathrm{inf}}}\bigg[\frac{1}{\mu},\frac{1}{\varphi(\mu)}, \frac{1}{\varphi^2(\mu)},\ldots\bigg] \rightarrow B_{\mathrm{crys}}. \end{align*}

Taking base change under this morphism we arrive at $(G_K,\varphi)$ -equivariant isomorphisms

\begin{align*}H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Z_p)\otimes_{\mathbb Z_p}B_{\mathrm{crys}} \xrightarrow{\sim} H^i_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}})\otimes_{A_{\mathrm{inf}}} B_{\mathrm{crys}}.\end{align*}

We can now prove an analogue of Theorem 4.8 in the d-de Rham proper context.

Theorem 5.15. Let $\mathcal{X}$ be a $(d+1)$ -de Rham proper stack over $\mathcal O_K$ . Then the map

\begin{align*}\Upsilon_{\mathcal X,\mathbb Q_p}\colon R\Gamma_{\acute{\mathrm{e}}\mathrm{t}}(\mathcal X_{C}, \mathbb Q_p)\rightarrow R\Gamma_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_{C}, \mathbb Q_p)\end{align*}

induces an isomorphism

\begin{align*} H^i_{\acute{\mathrm{e}}\mathrm{t}}(\mathcal X_{C}, \mathbb Q_p) \simeq H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_{C}, \mathbb Q_p) \end{align*}

for $i\leqslant d$ and an embedding

\begin{align*} H^{d+1}_{\acute{\mathrm{e}}\mathrm{t}}(\mathcal X_{C}, \mathbb Q_p) \hookrightarrow H^{d+1}_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_{C}, \mathbb Q_p) \end{align*}

for $i=d+1$ .

Proof. With Proposition 4.6 in hand, it is enough to show that

\begin{align*}\dim H^i_{\acute{\mathrm{e}}\mathrm{t}}(\mathcal X_{C}, \mathbb Q_p)\geqslant \dim H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_{C}, \mathbb Q_p)\end{align*}

for $i\leqslant d$ . By Proposition 2.9(2) we have that the reduction $\mathcal X_k$ is d-de Rham proper. By Remark 5.5 we have that $H^i_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)[{1}/{p}]$ is a free $\mathfrak S[{1}/{p}]$ -module for $i\leqslant d$ . Inverting p in Proposition 5.12(3) and comparing the dimensions of both sides over $W(C^\flat)[{1}/{p}]$ we get that

\begin{align*}\dim_{\mathbb Q_p} H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_{C}, \mathbb Q_p) = \mathrm{rk}_{\mathfrak S[{1}/{p}]}H^i_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\bigg[\frac{1}{p}\bigg]\end{align*}

for $i\leqslant d$ .

Denote $H^i_{\mathrm{dR}}(\widehat{\mathcal X}_K/K)\colon = H^i_{\mathrm{dR}}(\widehat{\mathcal X}/\mathcal O_K)[{1}/{p}]$ . By de Rham comparison [Reference Kubrak and PrikhodkoKP24] we have an equivalence $R\Gamma_{\mathrm{dR}}(\widehat{\mathcal X}/\mathcal O_K)\simeq [R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)/E(u)]$ . This gives a short exact sequence

\begin{align*} 0\longrightarrow H^i_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\bigg[\frac{1}{p}\bigg]/E\longrightarrow H^i_{\mathrm{dR}}(\widehat{\mathcal X}_K/K)\longrightarrow \bigg(H^{i+1}_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\bigg[\frac{1}{p}\bigg]\bigg)[E]\longrightarrow 0 \end{align*}

for all i. In particular, for $i\leqslant d$ we get that

\begin{align*}\dim_K H^i_{\mathrm{dR}}(\widehat{\mathcal X}_K/K)\geqslant \mathrm{rk}_{\mathfrak S[{1}/{p}]}H^i_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\bigg[\frac{1}{p}\bigg]= \dim_{\mathbb Q_p} H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_{C}, \mathbb Q_p). \end{align*}

On the other hand, we claim that for $i\leqslant d$ we have an isomorphism

\begin{align*}H^i_{\mathrm{dR}}(\widehat{\mathcal X}_K/K)\simeq H^i_{\mathrm{dR}}({\mathcal X}_K/K).\end{align*}

Indeed, derived p-completion is t-exact up to a right shift by 1 and so

\begin{align*}\tau^{\leqslant d}R\Gamma_{\mathrm{dR}}(\widehat{\mathcal X}/\mathcal O_K)\simeq \tau^{\leqslant d}((\tau^{\leqslant d+1}R\Gamma_{\mathrm{dR}}({\mathcal X}/\mathcal O_K))^\wedge_p)\end{align*}

by Lemma 2.8. However, by our assumption $\tau^{\leqslant d+1}R\Gamma_{\mathrm{dR}}({\mathcal X}/\mathcal O_K)$ is coherent and so automatically p-complete. It follows that $\tau^{\leqslant d}R\Gamma_{\mathrm{dR}}(\widehat{\mathcal X}/\mathcal O_K)\simeq \tau^{\leqslant d}R\Gamma_{\mathrm{dR}}({\mathcal X}/\mathcal O_K)$ , and consequently

\begin{align*}H^i_{\mathrm{dR}}(\widehat{\mathcal X}_K/K)\simeq H^i_{\mathrm{dR}}({\mathcal X}_K/K)\end{align*}

for $i\leqslant d$ . Finally, from the left vertical isomorphism in Proposition 4.5 we know that

\begin{align*}\dim_{\mathbb Q_p}H^i_{\acute{\mathrm{e}}\mathrm{t}}(\mathcal X_{C}, \mathbb Q_p) =\dim_K H^i_{\mathrm{dR}}({\mathcal X}_K/K)\end{align*}

for all i. We get that

\begin{align*} \dim H^i_{\acute{\mathrm{e}}\mathrm{t}}(\mathcal X_{C}, \mathbb Q_p)\geqslant \dim H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_{C}, \mathbb Q_p) \end{align*}

for $i\leqslant d$ as desired.

Remark 5.16. Let us record a fact with a similar flavor to Remark 5.6. By de Rham comparison for prismatic cohomology we have a short exact sequence

\begin{align*}0\longrightarrow H^i_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)/E(u)\longrightarrow H^i_{\mathrm{dR}}(\widehat{\mathcal X}/\mathcal O_K)\longrightarrow H^{i+1}_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)[E(u)]\longrightarrow 0.\end{align*}

If $\mathcal{X}$ is $(d+1)$ -de Rham proper over $\mathcal O_K$ , then $\mathcal X_{\mathcal O_K/p}$ is d-de Rham proper over $\mathcal O_K/p$ and so $H^{d}_{\mathrm{dR}}(\widehat{\mathcal X}/\mathcal O_K)$ is a finitely generated $\mathcal O_K$ -module. Consequently, we also get some partial information about $H^{d+1}_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)$ : namely, its E(u)-torsion $M\colon = H^{d+1}_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)[E(u)]$ is finitely generated over $\mathcal O_K$ . However, from the proof of Theorem 5.15 one can see more: namely, M dies after we invert p, $M[{1}/{p}]=0$ . Indeed, $M[{1}/{p}]$ is the last term in the short exact sequence

\begin{align*}0\longrightarrow H^{d}_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\bigg[\frac{1}{p}\bigg]/E(u)\longrightarrow H^{d}_{\mathrm{dR}}(\widehat{\mathcal X}_K/K)\longrightarrow \bigg(H^{d+1}_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\bigg[\frac{1}{p}\bigg]\bigg)[E(u)]\longrightarrow 0\end{align*}

and we saw that $\dim_K H^{d}_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)[{1}/{p}]/E(u)\simeq \dim_{\mathbb Q_p} H^d_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_{C}, \mathbb Q_p)$ , as well as the equalities

\begin{align*}\dim_{\mathbb Q_p} H^{d}_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_{C}, \mathbb Q_p)=\dim_{\mathbb Q_p} H^{d}_{\acute{\mathrm{e}}\mathrm{t}}({\mathcal X}_{C}, \mathbb Q_p)=\dim_K H^{d}_{\mathrm{dR}}({\mathcal X}_K/K)=\dim_K H^{d}_{\mathrm{dR}}(\widehat{\mathcal X}_K/K).\end{align*}

Thus, $\dim_K M[{1}/{p}]=0$ , meaning that $M[{1}/{p}]=0$ .

Let us also record the following analogue of [Reference Kubrak and PrikhodkoKP24, Proposition 4.3.14] for d-Hodge-proper stacks. Let $K_0\colon = W(k)[{1}/{p}]$ .

Corollary 5.17. Let $\mathcal{X}$ be a smooth $(d+1)$ -de Rham proper stack over $\mathcal O_K$ . Then for any $0\leqslant i\leqslant d$

\begin{align*} \dim_{K_0} H^i_{\mathrm{crys}}(\mathcal X_k/W(k))\bigg[\frac{1}{p}\bigg]=\dim_K H^i_{\mathrm{dR}}(\mathcal X_K/K) = \dim_{\mathbb Q_p} H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Q_p)=\dim_{\mathbb Q_p} H^i_{\acute{\mathrm{e}}\mathrm{t}}({\mathcal X}_C,\mathbb Q_p). \end{align*}

5.3 Fontaine’s $C_{\mathrm{crys}}$ -conjecture for d-de Rham proper stacks

In this subsection we prove that for $i\leqslant d$ the ith rational étale cohomology of the generic fiber of a $(d+1)$ -de Rham-proper stack over $\mathcal O_K$ is crystalline. We deduce it from a comparison between truncated prismatic and crystalline cohomology for which we use the simplified period ring $B_{\mathrm{max}}$ .

Construction 5.18 (Rings $A_{\mathrm{max}}$ and $B_{\mathrm{max}}$ ). The period ring $A_{\mathrm{max}}$ is defined as the p-completion $({A_{\mathrm{inf}}}[{\xi}/{p}])^\wedge_p$ . It can be equivalently described as the completion of ${A_{\mathrm{inf}}}[{\xi}/{p}]={A_{\mathrm{inf}}}[{[p^\flat]}/{p}]$ given by power series

\begin{align*}A_{\mathrm{max}}\colon = \bigg\{\!\sum_{i\in \mathbb Z} [x_i]\cdot p^i\mid x_i\in \mathcal O_C^\flat, \text{ s.t. }(\mathrm{val}_\flat(x_i)+ i)\geqslant 0 \text{ and } (\mathrm{val}_\flat(x_i)+i) \rightarrow +\infty \text{ as } i\rightarrow -\infty\! \bigg\},\end{align*}

where $\mathrm{val}_\flat$ is the natural valuation on $\mathcal O_C^\flat$ satisfying $\mathrm{val}_\flat([p^\flat])=1$ . We define the period ring $B_{\mathrm{max}}$ as $A_{\mathrm{max}}[{1}/{p}]$ . Since for $x_i\in \mathcal O_C^\flat$ one has $\mathrm{val}_\flat(x_i^p)=p\cdot \mathrm{val}_\flat(x_i)$ , Frobenius on $\mathcal O_C^\flat$ induces a map $\varphi\colon A_{\mathrm{max}}\rightarrow A_{\mathrm{max}}$ which is in fact an embedding and has the image

\begin{align*}\varphi(A_{\mathrm{max}})\simeq \bigg\{\!\sum_{i\in \mathbb Z}[x_i]\cdot p^i \text{ with } (\mathrm{val}_\flat(x_i)+ pi)\geqslant 0\text{ and } (\mathrm{val}_\flat(x_i)+pi) \rightarrow +\infty \text{ as } i\rightarrow -\infty \!\bigg\}.\end{align*}

We also have natural $G_K$ -actions on $A_{\mathrm{max}}$ and $B_{\mathrm{max}}$ induced by that on $\mathcal O_C^\flat$ . Note that it commutes with $\varphi$ .

Recall (see, e.g., [Reference CarusoCar19, § 3.2.2]) that any element $x\in A_{\mathrm{crys}}$ has a (unique) expression as

\begin{align*}x=\sum_{i\in \mathbb Z}[x_i]p^i \quad\text{with } x_i\in \mathcal O_C^\flat \text{ such that } \mathrm{val}_\flat(x_i)- \nu(i) \geqslant 0 \text { and } \mathrm{val}_\flat(x_i)- \nu(i)\rightarrow +\infty \text{ as } i\rightarrow -\infty,\end{align*}

where $\nu(i)=0$ if $i\geqslant 0$ and if $i\leqslant 0$ one puts $\nu(i)$ to be the minimal integer n such that $\mathrm{val}_p(n!)+i\geqslant 0$ . It is not hard to see that for $i\leqslant 0$ one has

\begin{align*}-ip \geqslant \nu(i)\geqslant -i\end{align*}

and, in this way, comparing the convergence conditions one gets natural $G_K$ -equivariant embeddings

\begin{align*}\cdots \subset\varphi(A_{\mathrm{crys}})\subset\varphi(A_{\mathrm{max}})\subset A_{\mathrm{crys}} \subset A_{\mathrm{max}} \quad \text{and} \quad \cdots \subset \varphi(B_{\mathrm{crys}})\subset\varphi(B_{\mathrm{max}})\subset B_{\mathrm{crys}} \subset B_{\mathrm{max}}.\end{align*}

Consequently, we have embeddings $\cdots \subset \varphi(B_{\mathrm{crys}})^{G_K}\subset\varphi(B_{\mathrm{max}})^{G_K}\subset B_{\mathrm{crys}}^{G_K}\subset B_{\mathrm{max}}^{G_K}$ of $G_K$ -invariants. One has $B_{\mathrm{crys}}^{G_K}\simeq K_0$ and $\varphi\colon B_{\mathrm{crys}} \rightarrow B_{\mathrm{crys}}$ maps it to itself via Frobenius on $K_0$ ; in particular, $\varphi(B_{\mathrm{crys}})^{G_K}\simeq B_{\mathrm{crys}}^{G_K}$ , and consequently $\varphi(B_{\mathrm{max}})^{G_K}\simeq K_0 \simeq B_{\mathrm{max}}^{G_K}$ . More generally, having a finite-dimensional $G_K$ -representation V over $\mathbb Q_p$ we get maps

\begin{align*}(V\otimes_{\mathbb Q_p}\varphi(B_{\mathrm{max}}))^{G_K}\subset (V\otimes_{\mathbb Q_p}B_{\mathrm{crys}})^{G_K} \subset (V\otimes_{\mathbb Q_p}B_{\mathrm{max}})^{G_K}\end{align*}

and, since $\varphi$ induces a $G_K$ -equivariant isomorphism $B_{\mathrm{max}}\xrightarrow{\sim}\varphi(B_{\mathrm{max}})$ , we get that the equality of dimensions

\begin{align*}\dim_{K_0}(V\otimes_{\mathbb Q_p}\varphi(B_{\mathrm{max}}))^{G_K}=\dim_{K_0}(V\otimes_{\mathbb Q_p}B_{\mathrm{max}})^{G_K}\end{align*}

implies that

\begin{align*}\dim_{K_0}(V\otimes_{\mathbb Q_p}B_{\mathrm{crys}})^{G_K} = \dim_{K_0}(V\otimes_{\mathbb Q_p}B_{\mathrm{max}})^{G_K}.\end{align*}

In particular, V crystalline if and only if it is $B_{\mathrm{max}}$ -admissible: namely,

\begin{align*}\dim_{K_0}(V\otimes_{\mathbb Q_p}B_{\mathrm{max}})^{G_K}=\dim_{\mathbb Q_p}V.\end{align*}

Construction 5.19 (Ring $\mathfrak S_{\mathrm{max}}$ ). Let e be the ramification index of $\mathfrak S_{\mathrm{max}}$ . We define the ring $\mathfrak S_{\mathrm{max}}$ as the p-adic completion

\begin{align*} \mathfrak S_{\mathrm{max}}\colon =\bigg(\mathfrak S\bigg[\frac{u^e}{ p}\bigg]\bigg)_p^\wedge. \end{align*}

Picking a uniformizer $\pi\in \mathcal O_K$ with minimal polynomial E(u) we have $E(u)\equiv u^e \mod p\mathfrak S$ and so one also has $\mathfrak S_{\mathrm{max}} \simeq \mathfrak S[{E(u)}/{p}]^\wedge_p$ . Via the choice of $\pi^\flat\in \mathcal O_C^\flat$ we have a map $\mathfrak S\rightarrow {A_{\mathrm{inf}}}$ sending u to $[\pi^\flat]$ . Since $\mathrm{val}_\flat([\pi^\flat]^e)=1$ , this map naturally extends to a map

\begin{align*} \mathfrak S_{\mathrm{max}} \rightarrow A_{\mathrm{max}}. \end{align*}

With this notation the key result of this section is as follows.

Theorem 5.21. Let $\mathcal{X}$ be a smooth $(d+1)$ -de Rham-proper stack over $\mathcal O_K$ . Then there is a $(G_K,\varphi)$ -equivariant equivalence

\begin{align*}(\tau^{\leqslant d} R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}}))\otimes_{{A_{\mathrm{inf}}}} B_{\mathrm{max}} \simeq (\tau^{\leqslant d} R\Gamma_{\mathrm{crys}}(\mathcal X_k/ W(k)))\otimes_{W(k)} B_{\mathrm{max}}.\end{align*}

We prove this theorem in the remaining sections using a bit of condensed mathematics. Before introducing some necessary notation let us deduce from Theorem 5.21 the crystallineness of the $G_K$ -representation given by the étaleFootnote ¹¹ cohomology $H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_{\mathbb C},\mathbb Q_p)$ .

Recall that a finite-dimensional $\mathbb Q_p$ representation V is called crystalline if the dimension of $D_{\mathrm{crys}}(V)\colon = (V\otimes_{\mathbb Q_p}B_{\mathrm{crys}})^{G_K}$ over $K_0$ is equal to the dimension of V.

Proof. Recall that for $0\leqslant i\leqslant d$ we have $H^i_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}})\simeq H^i_{{{{\Delta}}}^{(1)}}(\mathcal X/\mathfrak S)\otimes_{\mathfrak S}{A_{\mathrm{inf}}}$ by Lemma 5.13, and so by Remark 5.5 we have that $H^i_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}})[{1}/{p}]$ are free modules over ${A_{\mathrm{inf}}}[{1}/{p}]$ . Thus applying cohomology to both sides of the isomorphism in Theorem 5.21 we get $(G_K,\varphi)$ -equivariant isomorphisms

\begin{align*} H^i_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}})\otimes_{A_{\mathrm{inf}}} B_{\mathrm{max}}\simeq H^i_{\mathrm{crys}}(\mathcal X_k/W(k))\otimes_{W(k)}B_{\mathrm{crys}}. \end{align*}

Recall that by Corollary 5.14 we have a $(G_K,\varphi)$ -equivariant isomorphisms

\begin{align*} H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Q_p)\otimes_{\mathbb Q_p}B_{\mathrm{crys}}\xrightarrow{\sim} H^i_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}})\otimes_{A_{\mathrm{inf}}} B_{\mathrm{crys}}. \end{align*}

Tensoring it further by $B_{\mathrm{max}}$ we then get $(G_K,\varphi)$ -equivariant isomorphisms

\begin{align*} H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Q_p)\otimes_{\mathbb Q_p}B_{\mathrm{max}}\xrightarrow{\sim} H^i_{{{{\Delta}}}^{(1)}}(\mathcal X_{\mathcal O_C}/{A_{\mathrm{inf}}})\otimes_{A_{\mathrm{inf}}} B_{\mathrm{max}}\xrightarrow{\sim} H^i_{\mathrm{crys}}(\mathcal X_k/W(k))\otimes_{W(k)}B_{\mathrm{max}}. \end{align*}

Put $V\colon = H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Q_p)$ . By the discussion in the end of Construction 5.18 we then have

\begin{align*} D_{\mathrm{crys}}(V)\simeq (V\otimes_{\mathbb Q_p}B_{\mathrm{max}})^{G_K}\simeq (H^i_{\mathrm{crys}}(\mathcal X_k/W(k))\otimes_{W(k)}B_{\mathrm{max}})^{G_K}\simeq H^i_{\mathrm{crys}}(\mathcal X_k/W(k)\otimes_{W(k)}K_0, \end{align*}

since the $G_K$ -action on $H^i_{\mathrm{crys}}(\mathcal X_k/W(k))$ is trivial and $B_{\mathrm{max}}^{G_K}\simeq K_0$ . Since $H^i_{\mathrm{crys}}(\mathcal X_k/W(k)\otimes_{W(k)}K_0\simeq H^i_{\mathrm{crys}}(\mathcal X_k/W(k)[{1}/{p}]$ , from Corollary 5.17 we get

\begin{align*} \dim_{\mathbb Q_p}V=\dim_{K_0}H^i_{\mathrm{crys}}(\mathcal X_k/W(k)\bigg[\frac{1}{p}\bigg]= \dim_{K_0} D_{\mathrm{crys}}(V), \end{align*}

and so V is crystalline.

Remark 5.23. By general theory of admissible rings of Fontaine, from Corollary 4.9 we get that for any $i\leqslant d$ there is a natural $(G_K,\varphi)$ -equivariant isomorphism

\begin{align*} H^i_{\acute{\mathrm{e}}\mathrm{t}}(\mathcal X_{C}, \mathbb Q_p)\otimes_{\mathbb Q_p}B_{\mathrm{crys}} \xrightarrow{\sim} H^i_{\mathrm{crys}}(\mathcal X_k/W(k))\otimes_{W(k)} B_{\mathrm{crys}}. \end{align*}

Remark 5.25. Let $\mathcal{X}$ be a smooth $(d+1)$ -de Rham-proper stack over $\mathcal O_K$ and let

\begin{align*}H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Z_p)_{\mathrm{free}}\colon = H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Z_p)/H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Z_p)_{\mathrm{tors}} \subset H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Q_p).\end{align*}

By Corollary 5.22, for d-de Rham-proper stacks, for $i\leqslant d$ this is a lattice in a crystalline $G_K$ -representation. Following the same argument as in [Reference Kubrak and PrikhodkoKP24, Remark 4.3.27] one can identify the corresponding Breuil–Kisin module ${\mathrm{BK}}(H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Z_p)_{\mathrm{free}})$ with the free Breuil–Kisin module $H^i_{{{{\Delta}}}}(\mathcal X/\mathfrak S)_{\mathrm{free}}$ associated to $H^i_{{{{\Delta}}}}(\mathcal X/\mathfrak S)$ (see [Reference Bhatt, Morrow and ScholzeBMS18, Proposition 4.3] for the definition). If $H^i_{{{{\Delta}}}}(\mathcal X/\mathfrak S)$ is free over $\mathfrak S$ , then by Proposition 5.12 $H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Z_p)$ is p-torsion free and the statement becomes

\begin{align*} {\mathrm{BK}}(H^i_{\acute{\mathrm{e}}\mathrm{t}}(\widehat{\mathcal X}_C,\mathbb Z_p))\simeq H^i_{{{{\Delta}}}}(\mathcal X/\mathfrak S). \end{align*}

The following lemma gives a sufficient condition for $H^i_{{{{\Delta}}}}(\mathcal X/\mathfrak S)$ to be a free $\mathfrak S$ -module.

– $M[1/p]$ is free over $\mathfrak S[1/p]$ ;

– $M\otimes_{\mathfrak S} W(k)$ is free over W(k).

Then M is free over $\mathfrak S$ .

But under these assumptions

\begin{align*} \dim_k M\otimes_{\mathfrak S} k & = \mathrm{rank}_{W(k)} M\otimes_{\mathfrak S} W(k) = \mathrm{rank}_{W(k)[1/p]} M\otimes_{\mathfrak S} W(k)\bigg[\frac{1}{p}\bigg] \\ & = \mathrm{rank}_{\mathfrak S[1/p]} M\otimes_{\mathfrak S} \mathfrak S\bigg[\frac 1 p\bigg] = \dim_{\operatorname{Frac} \mathfrak S} M\otimes_{\mathfrak S} \operatorname{Frac} \mathfrak S \end{align*}

and thus M is free by the semicontinuity of stalks.

5.3.1 Condensed prismatic and crystalline cohomology.

First, we set up some notation. We denote by $X\mapsto \underline X$ the natural functor $ {\mathrm{Top}} \rightarrow {\mathrm{Cond}}$ . It commutes with products and so sends topological groups/rings to group/ring objects in condensed sets. We consider the full subcategory ${\mathrm{Solid}} \subset {\mathrm{Cond}}(\mathrm{Ab})$ of solid abelian groups; recall that it is closed under all limits and colimits. The restriction of the functor $A\rightarrow \underline A$ to discrete topological groups factors through ${\mathrm{Solid}}$ , inducing a fully faithful embedding $\mathrm{Ab}\rightarrow {\mathrm{Solid}}$ . It then also extends to a fully faithful embedding $D(\mathrm{Ab})\rightarrow D({\mathrm{Solid}})$ between derived categories which by slight abuse of notation we still denote by $M\mapsto \underline{M}$ . For brevity, having an $\mathbb E_{\infty}$ -ring A in $D({\mathrm{Solid}})$ we sometimes denote by $D(A):=\mathrm{Mod}_A(D({\mathrm{Solid}}))$ the category of A-modules. It is endowed with the symmetric monoidal structure given by the solid tensor product .

Next we recall some standard definitions and results about derived completions in the setting of presentable stable categories.

Definition 5.27. Let R be a commutative ring equipped with a finitely generated ideal I and let $\mathscr C$ be a presentable stable R-linear $(\infty, 1)$ -category. An object $M \in \mathscr C$ is called derived I-complete if for every $f\in I$ the limit

(5.3) \begin{equation}T(M, f) := \lim_{\leftarrow}{({\cdots \xrightarrow{\,\,\,\cdot f\,\,\,} M \xrightarrow{\,\,\,\cdot f\,\,\,} M\xrightarrow{\,\,\,\cdot f\,\,\,} M} )}\end{equation}

vanishes. We denote the full subcategory of $\mathscr C$ spanned by derived I-complete objects by $\mathscr C_{I\text{-}\mathrm{comp}}$ . The inclusion functor $\mathscr C_{I\text{-}\mathrm{comp}} \hookrightarrow \mathscr C$ admits a left adjoint $M \mapsto M_I^\wedge$ (see [Reference LurieLur18, Chapter 7.3]). Moreover, by [Reference LurieLur18, Corollary 7.3.3.3] it is enough to check the vanishing of (5.3) only for generators of the ideal I.

Remark 5.28. For $f\in I$ let $\alpha_f\colon \mathbb Z[x] \to R$ be a homomorphism determined by $x \mapsto f$ . Note that the limit (5.3) vanishes if and only if

\begin{align*}\lim_{\leftarrow}(\cdots \xrightarrow{\,\,\,\cdot x\,\,\,} M \xrightarrow{\,\,\,\cdot x\,\,\,} M \xrightarrow{\,\,\,\cdot x\,\,\,} M) \simeq 0,\end{align*}

where we consider M as a $\mathbb Z[x]$ -module via $\alpha_f$ . In particular, M is derived I-complete if and only if for all $f \in I$ it is derived x-complete as $\mathbb Z[x]$ -module.

The completion functor can be explicitly described. For a set $\{f_i\}_{i\in I}$ of elements of R and an object $M \in \mathscr C$ we denote

\begin{align*}\mathrm{Kos}(M;\{f_i\}_{i\in I}):= M\otimes_{\mathbb Z[x_i]_{i\in I}} \mathbb Z,\end{align*}

where M is considered as a $\mathbb Z[x_i]_{i\in I}$ -module via the homomorphism $\mathbb Z[x_i]_{i\in I}\rightarrow R$ sending $x_i$ to $f_i$ and all $x_i$ act on $\mathbb Z$ by 0. Note that $\mathrm{Kos}(M;\{f_i\}_{i\in I})$ is naturally a module over the derived ring $\mathrm{Kos}(R;\{f_i\}_{i\in I})$ . In the case $M\in \mathrm{Mod}_R$ , the complex $\mathrm{Kos}(M;\{f_i\}_{i\in I})$ can be explicitly computed by the ‘Koszul complex’

\begin{align*}\cdots \xrightarrow{\,\quad\,} \bigoplus_{i \lt j} M \xrightarrow{\,\quad\,} \bigoplus_i M \xrightarrow{(f_i\cdot )_{i\in I}} M,\end{align*}

where we fix some auxiliary total ordering on I (see [Sta26, Tag 0621]). Then for $I=(x_1,\ldots, x_s)\subset R$ there is a natural equivalence (see [Sta26, Tag 091N])

(5.4) \begin{equation}M^\wedge_I \simeq \lim\limits_{\stackrel{\longleftarrow}{n}} \mathrm{Kos}(M;x_1^n,\ldots,x_s^n).\end{equation}

Now we specialize to the condensed setting.

Construction 5.29.

1. (1) Let R be a commutative ring with a finitely generated ideal J. For $M\in D(\mathrm{Mod}_R)$ we define $M^\blacktriangle_J$ to be the derived J-completion of $\underline M$ (in the sense of Definition 5.27) as an object of $\mathscr C = \mathrm{Mod}_R (D({\mathrm{Solid}}))$ . Note that it is naturally a module over the solid ring $R^\blacktriangle_J \in \mathrm{CAlg} (D({\mathrm{Solid}}))$ , so the assignment $M\mapsto M^\blacktriangle_J$ is a functor

   \begin{align*}(-)^\blacktriangle_J\colon D(\mathrm{Mod}_R)\longrightarrow (\mathrm{Mod}_{R^\blacktriangle_J}(D({\mathrm{Solid}})))_{J-\mathrm{comp}}\end{align*}
   to the category of derived J-complete solid $R^\blacktriangle_J$ -modules.

2. (2) Let (R,J) be as above and let S be an R-algebra. Then we define

   \begin{align*} S^\blacktriangle_J \colon = R^\blacktriangle_J\otimes_R S. \end{align*}

3. (3) When it is clear from the context what J is, we sometimes omit it to lighten the notation. We occasionally call the operation $(-)^\blacktriangle\colon = (-)^\blacktriangle_J$ solid J-completion.

Remark 5.30. In the case $(x_1,\ldots,x_s)$ is a regular sequence, $\mathrm{Kos}(\underline R;x_1^n,\ldots,x_s^n)\simeq \underline{R/(x_1^n,\ldots,x_s^n)}$ and $R^\blacktriangle_J\simeq \underline {R^\wedge_J}\in {\mathrm{Solid}}$ where $R^\wedge_J$ is considered as a topological ring via J-adic topology.

Remark 5.31. We note that since evaluation on the singleton commutes with all limits and colimits, one has isomorphisms $R^\blacktriangle_J(*)\simeq R^\wedge_J$ and $M^\blacktriangle_J(*)\simeq M^\wedge_J$ where $(-)^\wedge_J$ is the derived J-completion. Similarly, $S^\blacktriangle_J(*)\simeq R^\wedge_J\otimes_R S$ .

Remark 5.32. Note that the solid module $M^\blacktriangle_J$ only depends on the pro-system

\begin{align*}\{[M/(x_1^n,\ldots,x_s^n)]\}_n\end{align*}

and not M itself. Consequently,Footnote ¹² the functor $(-)^\blacktriangle_J\colon D(\mathrm{Mod}_{R})\rightarrow D(R^\blacktriangle_J)_{J\text{-}\mathrm{comp}}$ factors through the usual derived completion $(-)^\wedge_{J}\colon D(\mathrm{Mod}_{R}) \rightarrow D(\mathrm{Mod}_{R})_{J\text{-}\mathrm{comp}}$ .

Lemma 5.33. The functor $(-)^\blacktriangle_J\colon D(\mathrm{Mod}_R)\rightarrow D(R^\blacktriangle_J)_{J\text{-}\mathrm{comp}}$ is right t-exact. If $J=(x_1,\ldots,x_s)\subset R$ is generated by s elements it is also left t-exact up to a shift by s.

Proof. Indeed, $(-)^\blacktriangle_J$ is the composition of fully faithful embedding $\underline{(-)}\colon D(\mathrm{Mod}_R)\rightarrow D(\underline(R))$ (where R is endowed with discrete topology) which is t-exact and the derived J-adic completion in $D(\mathrm{Mod}_R)$ which is right t-exact. Indeed, $M^\wedge_J\simeq (((M^\wedge_{x_1})^\wedge_{x_2})\ldots)^{\wedge}_{x_s}$ . Now, having $M\in \mathrm{Mod}_{\underline R}({\mathrm{Solid}})\subset D(\underline(R)$ and $x\in R$ the complex $[M/x]$ has two cohomology modules: $H^0([M/x])=M/x$ and $H^1([M/x])=M[x]$ (x-torsion). Since in the derived limit $\lim_{n\in \mathbb N}$ (applied to objects in the heart) we only have two terms $\lim^0$ and $\lim^1$ , we get the only non-zero term in $H^{\gt 0}(M^\wedge_x)$ could be given by $\lim^1_nM/x^n$ , which is zero by Mittag–Leffler since maps $M/x^n\rightarrow M/x^{n-1}$ are surjective.

For the second statement note that the functor $M\mapsto \underline{[M/(x_1^n,\ldots,x_s^n)]}$ is left t-exact up to a shift by s, while the homotopy limit $\lim\limits_{\stackrel{\longleftarrow}{n}}$ is left t-exact.

Warning 5.34. The functor $(-)^\blacktriangle_J$ need not be left t-exact even if we restrict to the subcategory of derived J-complete modules $D(\mathrm{Mod}_R)_{J\text{-}\mathrm{comp}} \subset D(\mathrm{Mod}_R)$ . Indeed, for an abelian group M the value of $\underline M$ on a pro-finite set $S = \lim\limits_{{\longleftarrow}} {S}_i$ is

\begin{align*}\underline M(S) = \mathrm{Cont}(S, M) \simeq \lim\limits_{\longrightarrow} M^{S_i} \simeq \lim\limits_{\longrightarrow}(\mathbb Z^{S_i}\otimes_{\mathbb Z} M) \simeq \mathrm{Cont}(S, \mathbb Z)\otimes_{\mathbb Z} M,\end{align*}

where in the second equivalence we have used that the topology on M is discrete. It is known that $\mathrm{Cont}(S, \mathbb Z)$ is a free abelian group, say $\mathrm{Cont}(S, \mathbb Z) \simeq \bigoplus_A \mathbb Z$ for some set A. It follows that

\begin{align*}\underline M(S) \simeq \bigoplus_A M.\end{align*}

The evaluation on S functor preserves limits, so

\begin{align*}M_J^\blacktriangle(S) \simeq (\underline M(S))_J^\wedge\simeq \bigg(\!\bigoplus_A M\!\bigg)_J^\wedge.\end{align*}

It follows that the left t-exactness of $(-)^\blacktriangle_J$ is equivalent to the left t-exactness of J-completed infinite direct sums. This holds, e.g., for principal J, see Lemma A.1.

Nevertheless, the functor $(-)^\blacktriangle_J$ still satisfies some ‘left t-exactness’ properties if we restrict to objects with special finiteness conditions (see Proposition A.22).

We now consider a variant of prismatic and crystalline cohomology in this setting.

Definition 5.35 (Solid (p, I)-completed prismatic cohomology). Let (A, I) be a bounded prism. Then we can consider the corresponding solid ring $A^\blacktriangle\colon = A^\blacktriangle_{(p,I)}$ . We define functors

\begin{align*}R\Gamma_{{{{\Delta}}}}^\blacktriangle(- / A), R\Gamma_{{{{\Delta}}}^{(1)}}^\blacktriangle(- / A) \colon {\mathscr{PS}\mathrm{tk}}_{A/I}^{\mathrm{op}} \longrightarrow \mathrm{Mod}_{A^\blacktriangle}(D({\mathrm{Solid}}))\end{align*}

as the application of constr:blacktriangle(2) to functors

\begin{align*}R\Gamma_{{{{\Delta}}}}(- / A), R\Gamma_{{{{\Delta}}}^{(1)}}(- / A) \colon {\mathscr{PS}\mathrm{tk}}_{A/I}^{\mathrm{op}} \longrightarrow \mathrm{Mod}_{A}(D(\mathrm{Ab}))\end{align*}

(see [Reference Kubrak and PrikhodkoKP24, Definition 2.2.1]). Namely,

\begin{align*}R\Gamma_{{{{\Delta}}}}^\blacktriangle(\mathcal X/ A) \colon = R\Gamma_{{{{\Delta}}}}(\mathcal X/ A)^\blacktriangle_{(p,I)}, \quad R\Gamma_{{{{\Delta}}}^{(1)}}^\blacktriangle(\mathcal X/ A) \colon = R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/ A)^\blacktriangle_{(p,I)}.\end{align*}

Remark 5.36. Since A is bounded, the derived (p,I)-completion of A is identified with the classical one. So in this case $A^\blacktriangle$ in fact lands in ${\mathrm{Solid}}\subset D({\mathrm{Solid}})$ , and is the solid ring $\underline A^{\mathrm{top}}$ represented by A with (p,I)-adic topology.

Definition 5.37 (Solid PD-completed crystalline cohomology). Let $S \hookrightarrow T$ be a PD-thickening, where $T=\operatorname{colim} T_n$ is a formal scheme with $T_n$ being the nth PD-neighborhood of S. Let $A \colon = \mathcal O(T)\colon = \lim_n \mathcal O(T_n)$ be the global functions on T and $A^\blacktriangle\colon = \lim_n^0 \underline{\mathcal O(T_n)}\in {\mathrm{Solid}}$ the corresponding completion in solid rings, where $\mathcal O(T_n)$ is considered as discrete topological group. We define the functor

\begin{align*}R\Gamma_{\mathrm{crys}}^\blacktriangle(-/T) \colon {\mathscr{PS}\mathrm{tk}}_{S}^{{\mathrm{op}}} \longrightarrow \mathrm{Mod}_{A^\blacktriangle}(D({\mathrm{Solid}}))\end{align*}

as the limit in $D({\mathrm{Solid}})$

\begin{align*}R\Gamma_{\mathrm{crys}}^\blacktriangle(-/T) \colon = \lim\limits_{\stackrel{\longleftarrow}{n}} \underline{R\Gamma_{\mathrm{crys}}(-/T_n)},\end{align*}

where $R\Gamma_{\mathrm{crys}}(-/T_n)$ is the right Kan extension of the crystalline cohomology functor from smooth S-schemes to prestacks.

Remark 5.38. In the following we only consider a special case when $T=\mathrm{Spf} A$ is a p-adic formal scheme. In this case there is a natural equivalence $R\Gamma_{\mathrm{crys}}^\blacktriangle(\mathcal X/T) \simeq (R\Gamma_{\mathrm{crys}}(\mathcal X/T))_p^\blacktriangle\in D(\mathrm{Mod}_{A^\blacktriangle_p}({\mathrm{Solid}}))$ .

With this definition one can formally deduce the base change and the comparison between condensation of the prismatic and crystalline cohomology from the ‘discrete’ version.

Proposition 5.39 (Base change). Let $(A,I) \to (B, IB)$ be a morphism of bounded prisms such that the (p,IB)-completed tensor product functor $- \mathbin{\widehat{\otimes}}_A B$ is left t-exact up to a shift. Also assume that $\mathcal{X}$ is qcqs syntomic Artin stack. Then the natural maps

are equivalences.

Proof. By Corollary A.27, the solid tensor product is naturally isomorphic to the solid (p,I)-completion $(R\Gamma_{{{{\Delta}}}}(\mathcal X/A)\mathbin{\widehat{\otimes}}_A B)^\blacktriangle_{(p,I)}$ , where the tensor product inside the brackets is also (p,I)-completed. By [Reference Kubrak and PrikhodkoKP24, Proposition 2.2.17], $R\Gamma_{{{{\Delta}}}}(\mathcal X/A)\mathbin{\widehat{\otimes}}_A B\simeq R\Gamma_{{{{\Delta}}}}(\mathcal X_{B/IB}/B)$ and we get the comparison. The proof for twisted cohomology is completely analogous, using the same reference.

Proposition 5.40 (Crystalline comparison). If $\mathcal{X}$ is smooth and $I = (p)$ , then there is a natural equivalence

\begin{align*}R\Gamma_{{{{\Delta}}}^{(1)}}^\blacktriangle(\mathcal X/A)\simeq R\Gamma_{\mathrm{crys}}^\blacktriangle(\mathcal X/A).\end{align*}

Proof. This follows from the crystalline comparison $R\Gamma_{{{{\Delta}}}^{(1)}}(\mathcal X/A)\simeq R\Gamma_{\mathrm{crys}}(\mathcal X/A)^\wedge_p$ (see [Reference Kubrak and PrikhodkoKP24, Proposition 2.5.7]) by applying $(-)^\blacktriangle$ .

Remark 5.41. One can also ‘topologize’ the étale cohomology of a stack and (formally) upgrade the étale comparison of prismatic and étale cohomology to condensed setting, but we will not need and so do not discuss this here.

Notation 5.42. In the following, we consider instances of Construction 5.29 in the following situations.

• – W(k), $\mathfrak S_{\mathrm{max}}$ , $A_{\mathrm{crys}}$ , $A_{\mathrm{max}}$ with $J=(p)$ , and the corresponding p-localizations $K_0$ , $\mathfrak S_{\mathrm{max}}[{1}/{p}]$ , $B_{\mathrm{crys}}$ , $B_{\mathrm{max}}$ . We denote the corresponding solid rings simply by $W(k)^\blacktriangle$ , $\mathfrak S_{\mathrm{max}}^\blacktriangle$ , $A_{\mathrm{crys}}^\blacktriangle$ , $A_{\mathrm{max}}^\blacktriangle$ and $K_0^\blacktriangle$ , $\mathfrak S_{\mathrm{max}}^\blacktriangle[{1}/{p}]$ , $B_{\mathrm{crys}}^\blacktriangle$ , $B_{\mathrm{max}}^\blacktriangle$ .

• – $\mathfrak S$ , $\mathfrak S\langle T\rangle\colon = \mathfrak S[T]^\wedge_{(p,u)}$ , ${A_{\mathrm{inf}}}$ with $J=(p,u)$ and $(p,[\pi^\flat])$ correspondingly. Similarly, we denote the corresponding solid rings by $\mathfrak S^\blacktriangle$ , $\mathfrak S\langle T\rangle^\blacktriangle$ , $A_{\mathrm{inf}}^\blacktriangle$ .

Furthermore, we also need the following.

Remark 5.43. As a slight variation on Proposition 5.39, let us record that, given a qcqs syntomic Artin stack $\mathcal{X}$ over $\mathcal O_C$ , there is also a natural (in particular, $(G_K,\varphi)$ -equivariant) equivalence

(5.5)

Note that even though the natural map ${A_{\mathrm{inf}}} \rightarrow A_{\mathrm{crys}}$ does not define a map of prism, the composition ${A_{\mathrm{inf}}}\xrightarrow{\varphi} {A_{\mathrm{inf}}} \rightarrow A_{\mathrm{crys}}$ does define a map of prismsFootnote ¹³ $({A_{\mathrm{inf}}}, \xi)\rightarrow (A_{\mathrm{crys}},p)$ . Applying Proposition 5.39 we get an equivalence

where by $\mathcal X^{(1)}_{A_{\mathrm{crys}}/p}$ we mean the base change of $\mathcal X$ to $A_{\mathrm{crys}}/p$ under the composite map ${A_{\mathrm{inf}}}\xrightarrow{\varphi} {A_{\mathrm{inf}}} \rightarrow A_{\mathrm{crys}} \twoheadrightarrow A_{\mathrm{crys}}/p$ . For affine schemes, by [Reference Bhatt and ScholzeBS22, Theorem 5.2] (applied to $A\colon = A_{\mathrm{crys}}$ and $I=(\xi,p)$ , and then taking further solid p-completion) we have a natural ( $\varphi$ -equivariant) equivalence

\begin{align*} R\Gamma_{{{\Delta}}}^\blacktriangle(X^{(1)}_{A_{\mathrm{crys}}/p}/A_{\mathrm{crys}})\simeq R\Gamma_{\mathrm{crys}}^\blacktriangle(X_{A_{\mathrm{crys}}/(\xi,p)}/A_{\mathrm{crys}}), \end{align*}

where on the right $X_{\mathcal O_C/p}$ is the base change of an affine $\mathcal O_C$ -scheme X via the map $\mathcal O_C\simeq {A_{\mathrm{inf}}}/\xi \rightarrow A_{\mathrm{crys}}/(\xi,p)$ . Moreover, $A_{\mathrm{crys}}/(\xi,p)$ is the PD-envelope of $\mathcal O_{C^\flat}$ along the kernel of $\mathcal O_{C^\flat} \twoheadrightarrow \mathcal O_{C^\flat}/ \pi^\flat \simeq \mathcal O_C /p$ , thus by invariance of crystalline cohomology under PD-thickenings there is a further natural equivalence

\begin{align*}R\Gamma_{\mathrm{crys}}^\blacktriangle(X_{A_{\mathrm{crys}}/(\xi,p)}/A_{\mathrm{crys}}) \simeq R\Gamma_{\mathrm{crys}}^\blacktriangle(X_{\mathcal O_{C}/p}/A_{\mathrm{crys}}).\end{align*}

Right Kan extending the composite equivalence we can then identify

\begin{align*} R\Gamma_{{{\Delta}}}^\blacktriangle(\mathcal X^{(1)}_{A_{\mathrm{crys}}/p}/A_{\mathrm{crys}}) \simeq R\Gamma_{\mathrm{crys}}^\blacktriangle(\mathcal X_{\mathcal O_{C}/p}/A_{\mathrm{crys}}). \end{align*}

It remains to note that since $\varphi$ is an isomorphism on ${A_{\mathrm{inf}}}$ we can also identify the left-hand side as

Remark 5.44. [

as a completed direct sum] Note that, by construction, the underlying $\mathfrak S^\blacktriangle$ -module of $\mathfrak S\langle T\rangle^\blacktriangle$ is given by the (p,u)-completed direct sum $\widehat\oplus_{i\in \mathbb N} \mathfrak S^\blacktriangle\cdot T^i$ . Thus, for any derived (p,u)-complete $\mathfrak S^\blacktriangle$ -module $M\in D(\mathfrak S^\blacktriangle)_{(p,u)\text{-}\mathrm{comp}}$ one has

as $\mathfrak S^\blacktriangle$ -modules. Indeed, there is a natural map

, then by Corollary A.27 the right-hand side is (p,u)-complete and so the map factors through the completion $\widehat\bigoplus_{i\in \mathbb N} M \cdot T^i$ . Moreover, it is an isomorphism modulo (p,I) since tensor product commutes with direct sums.

Remark 5.45. In addition, one has a strict exact sequence

of topological $\mathfrak S$ -modules where the topology is (p,u)-adic on $\mathfrak S\langle T\rangle$ and $\mathfrak S$ , p-adic on $\mathfrak S_{\mathrm{max}}$ and the map on the left is given by multiplication by $pT-u^e$ . This then gives a short exact sequence

\begin{align*}0\longrightarrow \mathfrak S\langle T\rangle^\blacktriangle \xrightarrow{\cdot (pT-u^e)} \mathfrak S\langle T\rangle^\blacktriangle \longrightarrow \mathfrak S_{\mathrm{max}}^\blacktriangle \longrightarrow 0\end{align*}

of solid $\mathfrak S^\blacktriangle$ -modules. Indeed, terms in the sequence are represented by the corresponding topological groups. Then the first arrow is an embedding since the functor $V\rightarrow \underline{V}$ is left exact and to see that the second arrow is a surjection one can use [Reference AndreychevAnd21, Lemma 3.1].