2.1 Quasi-coherent sheaves on $\mathbb{A}^1/\mathbb{G}_m$

We recall the standard dictionary between the derived category of quasi-coherent sheaves on $\mathbb{A}^1/\mathbb{G}_m$ and the filtered derived category of graded W(k)-modules. See [Reference BhattBha23, § 2.2] for additional details.

Let z denote the coordinate function on $\mathbb{G}_m$ . Consider the standard action of $\mathbb{G}_m$ on $\mathbb{A}^1=\operatorname{Spec} W(k)[v_+]$ given by $\mathbb{G}_m\times \mathbb{A}^1\to \mathbb{A}^1, v_+\mapsto z\otimes v_+$ . We shall denote the p-adic formal scheme associated to $\mathbb{A}^1$ with the above $\mathbb{G}_m$ -action by $\mathbb{A}^1_+$ , i.e. $\mathbb{A}^1_+:=\operatorname{Spf} W(k)[v_+]:= (\operatorname{Spec} W(k)[v_+])^{\wedge}$ . Similarly, we define an action of $\mathbb{G}_m$ on $\mathbb{A}^1_-:=\operatorname{Spf} W(k)[v_-]$ given by $v_-\mapsto z^{-1}\otimes v_-$ . We write $\mathbb{A}^1_{\pm}/\mathbb{G}_m$ for the quotient p-adic formal stacks over $B\mathbb{G}_m$ . Thus, for a p-nilpotent W(k)-algebra R, the groupoid of R-points of $\mathbb{A}^1_{+}/\mathbb{G}_m$ (respectively, $\mathbb{A}^1_{-}/\mathbb{G}_m$ ) classifies pairs $(L, v_+)$ (respectively, $(L, v_-)$ ), where L is a line bundle over $S:=\operatorname{Spec} R$ and $v_+: \mathcal{O}_S \to L$ is a section of L (respectively, $v_-: L \to \mathcal{O}_S$ is a section of $L^*$ ). We denote by $(\mathcal{O}(1), v_+)$ (respectively, $(\mathcal{O}(1), v_-)$ ) the universal pair over $\mathbb{A}^1_{+}/\mathbb{G}_m$ (respectively, $\mathbb{A}^1_{-}/\mathbb{G}_m$ ). We have a map $\mathbb{A}^1_{\pm}/\mathbb{G}_m \to B\mathbb{G}_m$ classifying the line bundle $\mathcal{O}(1)$ .Footnote ¹³ We identify the derived category of quasi-coherent sheaves on $\mathbb{A}^1_{\pm}/\mathbb{G}_m$ with the derived category of p-complete graded modules over $W(k)[v_{\pm}]$ Footnote ¹⁴ :

(2.1) \begin{equation}\mathcal{D}_{qc}(\mathbb{A}^1_{\pm}/\mathbb{G}_m)\simeq \widehat{\mathcal{D}}_{gr}(W(k)[v_{\pm}]).\end{equation}

The functor $\mathcal{D}_{qc}(\mathbb{A}^1_-/\mathbb{G}_m)\to \widehat{\mathcal{D}}_{gr}(W(k)[v_-])$ is given by

(2.2) \begin{equation}\mathcal{M}\mapsto (\oplus F^i, F^i\xrightarrow{v_-}F^{i-1}, \deg F^i=i),\end{equation}

where $F^i:=\mathrm{R}\Gamma(\mathbb{A}^1_-/\mathbb{G}_m,\mathcal{M}\otimes\mathcal{O}(i))$ with $\mathcal{O}(i)$ the line bundle pulled back from $B\mathbb{G}_m$ . The map $v_-: F^i\to F^{i-1}$ comes from the global section $v_-\in \Gamma(\mathbb{A}^1_-/\mathbb{G}_m,\mathcal{O}(-1))$ . Similarly, the functor $\mathcal{D}_{qc}(\mathbb{A}^1_+/\mathbb{G}_m)\to \widehat{\mathcal{D}}_{gr}(W(k)[v_+])$ is given by

(2.3) \begin{equation}\mathcal{M}\mapsto (\oplus G_i, G_i\xrightarrow{v_+}G_{i+1},\deg G_i=i)\end{equation}

where $G_i:=\mathrm{R}\Gamma(\mathbb{A}^1_+/\mathbb{G}_m,\mathcal{M}\otimes\mathcal{O}(i))$ .

Remark 2.2 Consider the map $B\mathbb{G}_m\to \mathbb{A}^1_{\pm}/\mathbb{G}_m$ given by inclusion of the origin into $\mathbb{A}^1_{\pm}$ . We identify $\mathcal{D}_{qc}(B\mathbb{G}_m)$ with the derived category of p-complete graded complexes of W(k)-modules. For any $\mathcal{M}\in \mathcal{D}_{qc,[a,b]}(\mathbb{A}^1_{\pm}/\mathbb{G}_m)$ , its restriction to $B\mathbb{G}_m$ is acyclic in grading degrees outside of [a,b]. Thus we have a well-defined functor

\[\mathcal{D}_{qc,[a,b]}(\mathbb{A}^1_{\pm}/\mathbb{G}_m)\to \mathcal{D}_{qc,[a,b]}(B\mathbb{G}_m).\]

2.2 Fontaine–Laffaille modules

Denote by $F: W(k)\to W(k)$ the Frobenius endomorphism. For $a\in W(k)$ , denote by $i_a: \operatorname{Spf} W(k)\hookrightarrow \mathbb{A}^1_- \twoheadrightarrow \mathbb{A}^1_-/\mathbb{G}_m$ the point given by equation $v_{-}=a$ . Given $\mathcal{M}\in \mathcal{D}_{qc}(\mathbb{A}^1_-/\mathbb{G}_m)$ , we denote by $\mathcal{M}_{v_-=a}:=i_a^* \mathcal{M}\in \mathcal{D}_{p\text{-}comp}(W(k))$ the derived pullback; for $\mathcal{M}\in\operatorname{Coh}(\mathbb{A}^1_-/\mathbb{G}_m)$ , we write $ L_0 i_a^* \mathcal{M} \in \textrm{Mod}(W(k))$ for the non-derived fibre.

Definition 2.4 (Cf. [Reference Fontaine and LaffailleFL82, § 1.5]Fontaine-Laffaille and [Reference WintenbergerWin84, § 1.3]). A finitely generated (‘small’) Fontaine–Laffaille module over W(k) is $\mathcal{M}\in\operatorname{Coh}(\mathbb{A}^1_-/\mathbb{G}_m)$ together with an isomorphism between the fibres

\[\varphi: F^*( L_0 i_p^* \mathcal{M})\xrightarrow{\sim} L_0 i_1^* \mathcal{M}.\]

Morphisms between Fontaine–Laffaille modules are morphisms between coherent sheaves on $\mathbb{A}^1_-/\mathbb{G}_m$ compatible with the isomorphisms $\varphi$ . We denote the category of Fontaine–Laffaille modules over W(k) by $\mathscr{MF}(W(k))$ .

1. (1) each $v_-: F^i\to F^{i-1}$ is a split injection; and

2. (2) $F^{-\infty}=\sum_i\textrm{Im} (\varphi_i)$ .

These two are precisely the conditions given in [Reference Fontaine and LaffailleFL82, § 1.5] except for the adjective split in (1) which is missing in [Reference Fontaine and LaffailleFL82]. However, using that the W(k)-module $F^{-\infty}$ has finite length, one shows (see [Reference WintenbergerWin84, Proposition 1.4.1 (ii)] or [Fal89, Theorem 2.1]) that the splitting condition for $v_-: F^i\to F^{i-1}$ is redundant.

The whole category $\mathscr{MF}(W(k))$ can be described similarly: an object of $\mathscr{MF}(W(k))$ consists of a finitely generated W(k)-module $F^{-\infty}$ equipped with an exhaustive filtration by W(k)-submodules $F^{\bullet}$ together with F-linear maps $\varphi_i: F^i\to F^{-\infty}$ , $\varphi_{i-1}v_-=p\varphi_i$ , such that the conditions (1) and (2) above hold.Footnote ¹⁵ This category was introduced in [Reference WintenbergerWin84, § 1.3]. The category of strongly divisible lattices that appeared in [Reference Fontaine and LaffailleFL82, § 7.11] is equivalent to the full subcategory of p-torsion-free objects of $\mathscr{MF}(W(k))$ .

where $i_1^*$ denotes the pullback to the point $v_-=1$ and $F^*\circ i_p^*$ denotes the pullback to the point $v_-=p$ post-composed with the Frobenius.

Part (1) of the following Proposition, with $ \mathscr{MF}(W(k))$ replaced by $\mathscr{MF}^{\mathrm{tor}}(W(k))$ , is proven in [Reference Fontaine and LaffailleFL82, § 1.8], and in [Reference WintenbergerWin84, Proposition 1.4] in general. Part (2) is proven in [Fal89, Theorem 2.1].

To prove part (1) we first verify, by induction on n, that if $(\mathcal{M},\varphi)$ is a Fontaine–Laffaille module with $p^n \mathcal{M}=0$ , then $L_1 i^*_a \mathcal{M} =0$ . For $n=1$ , we check that the pullback of $\mathcal{M}$ to $\mathbb{A}^1_- \otimes k$ is a vector bundle; this would imply the assertion. Indeed, by $\mathbb{G}_m$ -equivariance it suffices to show $\dim L_0i^*_p \mathcal{M}=\dim L_0 i^*_1 \mathcal{M}$ .Footnote ¹⁷ But this follows from the existence of $\varphi$ . For the induction step, set $\mathcal{M}'= \textrm{Ker} (\mathcal{M} \to \mathcal{M}/p)$ and consider the following commutative diagram.

(2.4)

Using that $L_1i^*_p (\mathcal{M}/p)=0$ , the rows are exact. It follows that the left downward arrow is an isomorphism. Thus, $(\mathcal{M}', \varphi_{\mathcal{M}'})$ is a Fontaine–Laffaille module. Applying the induction hypothesis to $\mathcal{M}'$ and $\mathcal{M}/p$ we infer the assertion for $\mathcal{M}$ .

To complete the proof of part (1), let us check that $L_1 i^*_a \mathcal{M} =0$ , for any Fontaine–Laffaille module. Indeed, let M denote the space of global sections of $\mathcal{M}$ pulled back to $\mathbb{A}^1_-$ . Since $\mathcal{M}$ is coherent, M is a finitely generated module over $ W(k)[v_-]^{\wedge}$ . In particular, M is p-complete in the classical (non-derived) sense, $M\buildrel{\sim}\over{\longrightarrow}R^0\,{\displaystyle\lim_{\longleftarrow}} \, M/p^n\buildrel{\sim}\over{\longrightarrow}R\,{\displaystyle\lim_{\longleftarrow}} \, M/p^n$ . The derived fibre $i^*_a \mathcal{M}$ is isomorphic to $\operatorname{Cone} (M\stackrel{v_- -a}{\longrightarrow} M)$ . We have $\operatorname{Cone} (M\stackrel{v_- -a}{\longrightarrow} M)= R\,{\displaystyle\lim_{\longleftarrow}} \,\operatorname{Cone} (M/p^n \stackrel{v_- -a}{\longrightarrow} M/p^n)$ . Since $\mathcal{M} /p^n$ has a structure of Fontaine–Laffaille module, $\operatorname{Cone} (M/p^n \stackrel{v_- -a}{\longrightarrow} M/p^n)$ is supported in cohomological degree 0, as was shown in the previous paragraph. Thus, $L_n i^*_a \mathcal{M} =0$ , for $n\gt 0$ .

To prove part (2), we start with a preliminary construction. Let $\mathscr{MF}'(W(k))$ be the category formed of objects $\mathcal{M}\in \operatorname{Coh}(\mathbb{A}^1_-/\mathbb{G}_m)$ together with a morphism $\varphi:F^* L_0 i^*_p \mathcal{M} \to L_0i^*_1\mathcal{M} $ . Since the functor $L_0i^*_1\colon \operatorname{Coh}(\mathbb{A}^1_-/\mathbb{G}_m)\to \operatorname{Coh} (\operatorname{Spf} W(k))$ is exact and $L_0 i^*_p$ is right exact, the category $\mathscr{MF}'(W(k))$ is abelian, and the forgetful functor $\mathscr{MF}'(W(k)) \to \operatorname{Coh}(\mathbb{A}^1_-/\mathbb{G}_m)$ is exact and conservative. The category $\mathscr{MF}(W(k))$ is a full subcategory of the abelian category $\mathscr{MF}'(W(k))$ . Thus, to show that $\mathscr{MF}(W(k))$ is abelian, it suffices to show that, for any morphism f in $\mathscr{MF}(W(k))$ , its kernel and cokernel computed in $\mathscr{MF}'(W(k))$ lie in $\mathscr{MF}(W(k))$ . The assertion on cokernel is clear from the right exactness properties of $L_0i^*_p$ and $L_0i^*_1$ . Let us check that $\textrm{Ker} f \in \mathscr{MF}(W(k))$ . Indeed, using the vanishing of $L_1 i^*_p $ for Fontaine–Laffaille modules we conclude that $L_0i^*_p \textrm{Ker}(\mathcal{M} \stackrel{f}{\longrightarrow} \mathcal{M}') \buildrel{\sim}\over{\longrightarrow} \textrm{Ker}(L_0i^*_p \mathcal{M} \stackrel{f}{\longrightarrow} L_0i^*_p\mathcal{M}')$ which implies that $\phi: F^* L_0i^*_p \textrm{Ker}(\mathcal{M} \stackrel{f}{\longrightarrow} \mathcal{M}') \to L_0i^*_0 \textrm{Ker}(\mathcal{M} \stackrel{f}{\longrightarrow} \mathcal{M}')$ is an isomorphism.

For proof of part (3): by part (1), for any $\mathcal{M} \in \mathscr{MF}(W(k))$ , we have a canonical isomorphism $F^*i^*_p \mathcal{M}\simeq F^* i^*_p M$ , where the pullbacks are derived. This defines an exact functor $\mathscr{MF}(W(k)) \to \mathscr{DMF}^{\mathrm{big}}(W(k))$ that extends uniquely to an exact functor

(2.5) \begin{equation} \mathcal{D}^b(\mathscr{MF}(W(k))) \to \mathscr{DMF}^{\mathrm{big}}(W(k)).\end{equation}

Let us check that (2.5) is fully faithful. It suffices to show that, for any two objects $(\mathcal{M}, \varphi_\mathcal{M}), (\mathcal{N}, \varphi_\mathcal{N}) \in \mathscr{MF}(W(k))$ , the map of mapping spectra

(2.6) \begin{equation} \operatorname{Hom}_{\mathcal{D}^b(\mathscr{MF}(W(k)))} (\mathcal{M}, \mathcal{N}) \to \operatorname{Hom}_{\mathscr{DMF}^{\mathrm{big}}(W(k))} (\mathcal{M}, \mathcal{N})\end{equation}

is an equivalence. Firstly, we check this for a p-torsion-free $\mathcal{M}$ . Using part (1), this condition is equivalent to $\mathcal{M}$ being a locally free $\mathcal{O}$ -module. By definition of the equalizer category, the spectrum $ \operatorname{Hom}_{\mathscr{DMF}^{\mathrm{big}}(W(k))} (\mathcal{M}, \mathcal{N})$ is given by

(2.7) \begin{equation} \mathrm{Fib} (\operatorname{Hom}_{\mathcal{D}_{qc}(\mathbb{A}^1_-/\mathbb{G}_m)} (\mathcal{M}, \mathcal{N} ) \to \operatorname{Hom}_{\mathcal{D}_{qc}(W(k))} (F^* i^*_p\mathcal{M}, i^*_1\mathcal{N})),\end{equation}

where the map carries $f\in \operatorname{Hom}_{\mathcal{D}_{qc}(\mathbb{A}^1_-/\mathbb{G}_m)} (\mathcal{M}, \mathcal{N} )$ to $i^*_1(f) \circ \varphi_{\mathcal{M}}- \varphi_{\mathcal{N}} \circ F^* i^*_p(f)$ . In particular, if $\mathcal{M}$ is a locally free $\mathcal{O}$ -module, $ \pi_i \operatorname{Hom}_{\mathscr{DMF}^{\mathrm{big}}(W(k))} (\mathcal{M}, \mathcal{N})=0$ unless $i=0$ or $-1$ : each mapping spectrum appearing in (2.7) is isomorphic to an abelian group. Using [Reference Bloch and KatoBK90, Lemma 4.4], the same is true for $ \operatorname{Hom}_{\mathcal{D}^b(\mathscr{MF}(W(k)))} (\mathcal{M},\mathcal{N})$ . Since $\mathscr{MF}(W(k)) \to \mathscr{DMF}^{\mathrm{big}}(W(k))$ is fully faithful and its essential image is closed under extensions (that is, for any morphism $\mathcal{N} \to \mathcal{M}[1]$ in $\mathscr{DMF}^{\mathrm{big}}(W(k))$ , its fibre is isomorphic to an object of $\mathscr{MF}(W(k))$ ), it follows that (2.6) is an equivalence for torsion-free $\mathcal{M}$ . In general, for any $\mathcal{M}$ , we have a fibre sequence $\mathcal{M}_{tor} \hookrightarrow \mathcal{M} \twoheadrightarrow \mathcal{M}/\mathcal{M}_{tor}$ , where $\mathcal{M}_{tor}$ is a torsion Fontaine–Laffaille module and $\mathcal{M}/\mathcal{M}_{tor}$ is torsion-free. Thus, it remains to prove that (2.6) is an equivalence for $\mathcal{M} =\mathcal{M}_{tor} $ . By dévissage, we may assume that $p \mathcal{M}=0$ . In this case the assertion reduces to the torsion-free case using the following observation.

It remains to prove that the essential image of the functor (2.5) consists of all objects $(\mathcal{M}, \varphi)$ whose underlying complex $\mathcal{M}\in \mathcal{D}_{qc} (\mathbb{A}^1_-/\mathbb{G}_m)$ is bounded and has coherent cohomology. We induct on length of the complex. If $\mathcal{M}$ is supported in a single cohomological degree the assertion is clear. For the induction step, we assume that $\mathcal{M}$ is connective and $H^0(\mathcal{M}) \ne 0$ . Note that the isomorphism $\varphi: F^*i^*_p \mathcal{M} \xrightarrow{\sim} i^*_1 \mathcal{M}$ induces $H^0(\varphi): F^*L_0i^*_p H^0(\mathcal{M}) \xrightarrow{\sim} L_0 i^*_1 H^0(\mathcal{M})$ . Thus, $(H^0(\mathcal{M}), H^0(\varphi))\in \mathscr{MF}(W(k))$ . By part (1) of the proposition $ L_j i^*_p H^0(\mathcal{M})= L_j i^*_1 H^0(\mathcal{M}) =0$ for $j\gt 0$ . It follows that the map $\mathcal{M} \to H^0 (\mathcal{M})$ in $\mathcal{D}_{qc}(\mathbb{A}^1_-/\mathbb{G}_m)$ lifts to a morphism in $\mathscr{DMF}^{\mathrm{big}}(W(k))$ . Applying the induction assumption to its fibre we complete the proof.

Let $\mathscr{MF}_{[a,b]}(W(k))$ (respectively, $\mathscr{DMF}^{\mathrm{big}}_{[a,b]}(W(k))$ ) be a full subcategory of $\mathscr{MF}(W(k))$ (respectively, $\mathscr{DMF}^{\mathrm{big}}(W(k))$ ) formed of objects $(\mathcal{M},\varphi)\in \mathscr{MF}(W(k))$ (respectively, $\mathscr{DMF}^{\mathrm{big}}(W(k))$ ) such that $\mathcal{M}\in \mathcal{D}_{qc,[a,b]}(\mathbb{A}^1_-/\mathbb{G}_m)$ .

2.3 Syntomification via filtered Cartier–Witt divisors

Recall from [Reference DrinfeldDri24] and [Reference BhattBha23, § 5.3] definitions of the stacks $W(k)^{\mathbb{\Delta}}$ , $W(k)^{\mathcal{N}}$ , $W(k)^{syn}$ , and $X^{syn}$ .

Throughout this paper we let W be the p-typical Witt ring scheme pulled back to $\operatorname{Spec} \mathbb{Z}_{(p)}$ . For a p-nilpotent W(k)-algebra R, set $W_R := W \times \operatorname{Spec} R$ . A $W_R$ -module M is an affine group scheme over R with an action of the ring scheme $W_R$ . We say that M is invertible if, Zariski locally on $\operatorname{Spec} R$ , M is isomorphic to the free $W_R$ -module $W_R$ . Denote by $W(k)^{\mathbb{\Delta}}(R)$ the groupoid whose objects are pairs $(M, \xi)$ , where M is an invertible $W_R$ -module and $\xi:M\to W_R$ is a morphism of $W_R$ -modules such that Zariski locally on $\operatorname{Spec} R$ the pair $(M, \xi)$ is isomorphic to $(W_R, w\textrm{Id})$ , $w=(w_1, w_2, \ldots)\in W(R)$ , with $w_1\in R$ nilpotent and $w_2\in R^*$ .

For a $W_R$ -module M, precomposing the $W_R$ -action on M with the Frobenius $F: W_R\to W_R$ we obtain a new $W_R$ -module $F_*M$ whose underlying group scheme is that of M. Applying this to the free $W_R$ -module $W_R$ , we have an exact sequence

(2.8) \begin{equation} 0\to \mathbb{G}_{a,R}^{\sharp} \stackrel{}{\longrightarrow} W_R \stackrel{F}{\longrightarrow} F_*W_R \to 0\end{equation}

of $W_R$ -modules.Footnote ¹⁸ Note that the action of $W_R$ on $\mathbb{G}_{a,R}^{\sharp}$ factors through the ‘first coordinate’ homomorphism $W_R\to \mathbb{G}_{a,R}$ . In particular, we have a natural action of the group $\mathbb{G}_{m,R}$ on the $W_R$ -module $\mathbb{G}_{a,R}^{\sharp}$ . Consequently, given an invertible R-module L we can twist $\mathbb{G}_{a,R}^{\sharp}$ by L; the resulting $W_R$ -module is denoted by $ \mathbf{V}(L)^{\sharp}$ . A $W_R$ -module M is called admissible if M fits into a sequence of $W_R$ -modules of the form

(2.9) \begin{equation} 0 \to \mathbf{V}(L)^{\sharp} \to M \to F_* M' \to 0,\end{equation}

where $L \in \textrm{Pic}(R)$ and M’ is an invertible $W_R$ -module. One checks (see [Reference DrinfeldDri24, Lemma 3.12.7]) that L, M’, and the sequence (2.9) are functorial in M. In particular, every morphism $\xi:M\to W_R$ of $W_R$ -modules extends uniquely to a morphism between short exact sequences (2.9) and (2.8). Consider the category $W(k)^{\mathcal{N}, c}(R)$ consisting of pairs $(M, \xi)$ , called filtered Cartier–Witt divisors, where M is an admissible $W_R$ -module and $\xi:M\to W_R$ is a morphism of $W_R$ -modules such that the pair $(M', \bar \xi: M' \to W_R)$ , constructed from the morphism of short exact sequences, is an object $W(k)^{\mathbb{\Delta}}(R)$ . Morphisms between $(M, \xi)$ and $(M', \xi')$ in $W(k)^{\mathcal{N}, c}(R)$ are given by morphisms $f: M \to M'$ of $W_R$ -modules such that $\xi' \circ f = \xi$ . The groupoid $W(k)^{\mathcal{N}} (R)$ is obtained from $W(k)^{\mathcal{N}, c}(R)$ by discarding all morphisms which are not isomorphisms. By definition, the groupoid of R-points of the stack $W(k)^{\mathcal{N}}$ is $W(k)^{\mathcal{N}} (R)$ . The functor $W(k)^{\mathcal{N}, c}$ from the category of p-nilpotent W(k)-algebras to the 2-category of categories that carries an algebra R to $W(k)^{\mathcal{N}, c}(R)$ is an example of what Drinfeld calls in [Reference DrinfeldDri24] a c-stack. We do not use this notion in the main body of the paper except for Remark 3.9.

Given an object $(M, \xi)$ of $W(k)^{\mathcal{N}}(R)$ , the morphism of $W_R$ -modules $\mathbf{V}(L)^{\sharp}\to \mathbb{G}_{a,R}^{\sharp}$ derived from $\xi$ lifts uniquely to a morphism of R-modules $L\to R$ (see [Reference DrinfeldDri24, Lemma 3.12.4, (ii)]). This defines a morphism of stacks called the Rees map:

(2.10) \begin{equation}t_{W(k)}: W(k)^{\mathcal{N}} \to \mathbb{A}^1_-/\mathbb{G}_m.\end{equation}

Sending $(M, \xi)\in W(k)^{\mathcal{N}}(R)$ to $(M', \bar \xi: M' \to W_R)\in W(k)^{\mathbb{\Delta}}(R)$ determines a morphism $\pi : W(k)^{\mathcal{N}} \to W(k)^{\mathbb{\Delta}}$ called the structure morphism. Every invertible $W_R$ -module is admissible and, moreover, every point $(M, \xi)\in W(k)^{\mathbb{\Delta}}(R)$ is also a point of $W(k)^{\mathcal{N}}(R)$ (see [Reference DrinfeldDri24, § 5.3]). This defines a map $j_+: W(k)^{\mathbb{\Delta}} \to W(k)^{\mathcal{N}}$ which exhibits $W(k)^{\mathbb{\Delta}}$ as an open substack of $ W(k)^{\mathcal{N}}$ (see [Reference DrinfeldDri24, Lemma 5.3.1]). As in [Reference DrinfeldDri24, § 5.6], we define another open embedding $j_-: W(k)^{\mathbb{\Delta}} \to W(k)^{\mathcal{N}}$ as follows. For $(M, \theta) \in W(k)^{\mathbb{\Delta}} (R)$ , define a filtered Cartier–Witt divisor $j_-(M, \theta):= (N, \xi)\in W(k)^{\mathcal{N}}(R)$ by the following pullback diagram.

(2.11)

One can check that open substacks defined by $j_-$ and $j_+$ do not intersect (see [Reference DrinfeldDri24, Lemma 5.6.3]). The stack $W(k)^{syn}$ is obtained from $W(k)^{\mathcal{N}}$ by gluing two copies of $W(k)^{\mathbb{\Delta}}$ using the maps $j_-$ and $j_+$ .

For every $(M, \xi)\in W(k)^{\mathcal{N}}(R)$ one considers the W(k)-algebra stack $\operatorname{Cone}(M \xrightarrow[]{\xi} W_R)$ (we refer the reader to [Reference DrinfeldDri24, § 1.3 and 1.4] and the references therein for a discussion of the notion of a ring stack). As a group stack, this is simply the quotient stack $[W_R/M]$ . The algebra structure on the quotient comes from the natural DG-algebra structure on $M \xrightarrow[]{\xi} W_R$ given by the $W_R$ -module structure on M (see [Reference DrinfeldDri24, § 1.3.3]). By ‘transmutation’ (see [Reference BhattBha23, Remark 2.3.8] and [Reference DrinfeldDri24, § 1.4.2]), this defines a Nygaardization functor $X \mapsto X^{\mathcal{N}}$ from the category of bounded p-adic formal schemes over $\operatorname{Spf} W(k)$ to stacks: for a p-nilpotent W(k)-algebra R, an R-point of the groupoid $X^{\mathcal{N}}(R)$ consists of $(M, \xi)\in W(k)^{\mathcal{N}}(R)$ together with a morphism $\operatorname{Spec} (\operatorname{Cone}(M \xrightarrow[]{\xi} W_R)(R)) \to X$ of derived schemes over W(k). For every $(M, \theta)\in W(k)^{\mathbb{\Delta}}(R)$ , setting $j_+ (M, \theta)= (N_+, \xi_+)$ , $j_- (M, \theta)= (N_-, \xi_-)$ , one constructs (see [Reference DrinfeldDri24, § 5.8]) a natural isomorphism $\operatorname{Cone}(N_- \xrightarrow[]{\xi_-} W_R) \buildrel{\sim}\over{\longrightarrow} \operatorname{Cone}(N_+ \xrightarrow[]{\xi_+} W_R)$ of W(k)-algebra stacks (the map is given by the rightward arrows in diagram (2.11)). Consequently, the fibres of $X^\mathcal{N} \to W(k)^\mathcal{N}$ over the two copies of $W(k)^{\mathbb{\Delta}}$ inside $ W(k)^{\mathcal{N}}$ are naturally identified (and denoted byFootnote ¹⁹ $j_+, j_-: X^{{\mathbb{\Delta}}} \hookrightarrow X^\mathcal{N}$ ). The stack $X^{syn}$ is obtained from $X^\mathcal{N}$ by gluing these two open substacks. This procedure defines a functor $X \mapsto X^{syn}$ from bounded p-adic formal schemes over $\operatorname{Spf} W(k)$ to stacks. In particular, if $f: X \to \operatorname{Spf} W(k)$ is such a scheme, we have by functoriality a map $f^{syn}: X^{syn} \to W(k)^{syn}$ . We refer to $\mathcal{H}_{syn}(X):= Rf_*^{syn}(\mathcal{O}_{ X^{syn}})\in \mathcal{D}_{qc}(W(k)^{syn})$ as the F-gauge associated to X.

Recall from [Reference DrinfeldDri24, Lemma 5.13.4 (i)] that we have a morphism

(2.12) \begin{equation}\mathfrak{p}_{\bar{\mathrm{dR}}}: \mathbb{A}^1_-/\mathbb{G}_m\to W(k)^{\mathcal{N}}\end{equation}

constructed as follows. Recall that an R-point of $\mathbb{A}^1_-/\mathbb{G}_m$ is a pair $(L ,v_-)$ , where L is an invertible R-module equipped with an R-linear morphism $v_-: L \to R$ . The map $\mathfrak{p}_{\bar{\mathrm{dR}}}$ is given by $\mathfrak{p}_{\bar{\mathrm{dR}}}(L,v_-)=(M_{\mathfrak{p}_{\bar{\mathrm{dR}}, R}},\xi )$ , where $M_{\mathfrak{p}_{\bar{\mathrm{dR}},R}}= \mathbf{V}(L)^{\sharp} \oplus F_*W_R$ , and

(2.13) \begin{equation}\xi:M_{\mathfrak{p}_{\bar{\mathrm{dR}},R}}=\mathbf{V}(L)^{\sharp} \oplus F_*W_R\to W_R\end{equation}

is given by $\xi:=(v_-^\sharp,V)$ . Here V is the Verschiebung and $v_-^\sharp: \mathbf{V}(L)^{\sharp} \to W_R$ is the composition of the map $\mathbf{V}(L)^{\sharp} \to \mathbb{G}_{a,\,R}^\sharp$ induced by $v_-$ and the embedding $\mathbb{G}_{a,\,R}^\sharp \hookrightarrow W_R$ from (2.8). One can verify (see [Reference BhattBha23, Theorem 2.5.6]) that, for a smooth p-adic formal scheme X over $\operatorname{Spf} W(k)$ , the complex $\mathfrak{p}^*_{\bar{\mathrm{dR}}}(\mathcal{H}_{syn}(X)) \in \mathcal{D}_{qc}(\mathbb{A}^1_-/\mathbb{G}_m)$ recovers the Hodge filtered de Rham cohomology of X.

The stack $k^{\mathcal{N}}$ can be described explicitly as follows. Set

(2.14) \begin{equation}A:=W(k)[v_+,v_-]/(v_+v_-=p).\end{equation}

We endow A with a grading such that $\deg v_+=1$ and $\deg v_-=-1$ , and consider the corresponding action of $\mathbb{G}_m$ on $\operatorname{Spf} A$ . Then $k^{\mathcal{N}}$ is identified with $\operatorname{Spf} A/\mathbb{G}_m$ . We shall just explain a construction of the map $\operatorname{Spf} A/\mathbb{G}_m \to k^{\mathcal{N}}$ , referring the reader to [Reference BhattBha23, § 5.4] for a proof of the isomorphism property. First, we describe the composite map $ \mathfrak{p}_{\mathrm{cris}}:\operatorname{Spf} A/\mathbb{G}_m\xrightarrow{\sim} k^{\mathcal{N}}\to W(k)^{\mathcal{N}}$ explicitly. Here the map $k^{\mathcal{N}}\to W(k)^{\mathcal{N}}$ comes from the morphism $\operatorname{Spec} k\to\operatorname{Spf} W(k)$ by functoriality. An R-point of $\operatorname{Spf} A/\mathbb{G}_m$ is given by $R \xrightarrow{v_+}L \xrightarrow{v_-} R$ , where L is an invertible R-module and $v_+v_-=p$ . The map $ \mathfrak{p}_{\mathrm{cris}}$ takes $(R\xrightarrow{v_+} L \xrightarrow{v_-}R)$ and sends it to $(M_{\mathfrak{p}_{\mathrm{cris}},R},\xi)$ , where $M_{\mathfrak{p}_{\mathrm{cris}},R}:=\textrm{coker}(\mathbb{G}_{a,R}^{\sharp}\xrightarrow{(v_+^{\sharp},-can)} \mathbf{V}(L)^{\sharp} \oplus W_R)$ , and $\xi$ is induced by the map

(2.15) \begin{equation} \mathbf{V}(L)^{\sharp} \oplus W_R\xrightarrow{(can\,\circ\,v_-^{\sharp},\cdot p)}W_R.\end{equation}

The notation ‘can’ stands for the canonical embedding $\mathbb{G}_{a,R}^{\sharp}\hookrightarrow W_R$ , and $\cdot p$ stands for multiplication by p. This defines the map $ \mathfrak{p}_{\mathrm{cris}}:\operatorname{Spf} A/\mathbb{G}_m\to W(k)^{\mathcal{N}}$ . To lift this map to $k^{\mathcal{N}}$ we need to endow $\operatorname{Cone}(M_{\mathfrak{p}_{\mathrm{cris}},R} \xrightarrow[]{\xi} W_R)$ with the structure of a k-algebra stack. In fact, we have a map of quasi-ideal pairs $ (W_R \stackrel{\cdot p}{\longrightarrow} W_R) \to (M_{\mathfrak{p}_{\mathrm{cris}},R} \xrightarrow[]{\xi} W_R)$ given by $ W_R\xrightarrow[]{(0, \textrm{Id})} \mathbf{V}(L)^{\sharp} \oplus W_R \to M_{\mathfrak{p}_{\mathrm{cris}},R}$ on the source and by $\textrm{Id}$ on the target and, for any W(k)-algebra D, the animated ring $\operatorname{Cone}(D \xrightarrow[]{\cdot p} D)$ receives a map from $k\buildrel{\sim}\over{\longrightarrow} \operatorname{Cone}(W(k) \xrightarrow[]{\cdot p} W(k))$ . Under the isomorphism $\operatorname{Spf} A/\mathbb{G}_m \buildrel{\sim}\over{\longrightarrow} k^{\mathcal{N}}$ , the open embedding $j_+: k^{{\mathbb{\Delta}}} \hookrightarrow k^\mathcal{N}$ (respectively, $j_-: k^{{\mathbb{\Delta}}} \hookrightarrow k^\mathcal{N}$ ) is identified with $\operatorname{Spf} W(k) \stackrel{F}{\longrightarrow} \operatorname{Spf} W(k) \xrightarrow[]{v_+=1, v_-=p} \operatorname{Spf} A \to \operatorname{Spf} A/\mathbb{G}_m$ (respectively, $\operatorname{Spf} W(k) \xrightarrow[]{v_+=p, v_-=1} \operatorname{Spf} A \to \operatorname{Spf} A/\mathbb{G}_m$ ). Let us just construct an isomorphism of k-algebra stacks obtained by pulling back $\mathbb{G}_{a, k}^{\mathcal{N}}$ along $j_{\pm}: \operatorname{Spf} W(k) \to \operatorname{Spf} A/\mathbb{G}_m$ . The restriction of $\mathbb{G}_{a, k}^{\mathcal{N}}$ to $\operatorname{Spf} W(k) \xrightarrow[]{v_+=1, v_-=p} \operatorname{Spf} A/\mathbb{G}_m$ is given by $\operatorname{Cone}( W \stackrel{\cdot p}{\longrightarrow} W)$ ; its restriction along $\operatorname{Spf} W(k) \xrightarrow[]{v_+=p, v_-=1} \operatorname{Spf} A/\mathbb{G}_m$ is $\operatorname{Cone}(F_* W \stackrel{\cdot p}{\longrightarrow} F_* W)$ . As a k-algebra stack the latter is obtained from the former by precomposing the action of k with the Frobenius. Equivalently, the k-algebra stack $\operatorname{Cone}(F_* W \stackrel{\cdot p}{\longrightarrow} F_* W)$ is isomorphic to the pullback of $\operatorname{Cone}( W \stackrel{\cdot p}{\longrightarrow} W)$ along the Frobenius on $\operatorname{Spf} W(k)$ . We refer the reader to [Reference BhattBha23, § 3.3] and especially diagram (3.3.2) there for details.

For a smooth p-adic formal scheme X over $\operatorname{Spf} W(k)$ , $ \mathfrak{p}_{\mathrm{cris}}^*(\mathcal{H}_{syn}(X))$ recovers the crystalline cohomology of $X\otimes \mathbb{F}_p$ equipped with the Nygaard filtration (see [Reference BhattBha23, § 3.3]).

2.4 The reduced locus of $W(k)^{\mathcal{N}}$

Let $C_2:=\operatorname{Spec} k[v_+^p,v_-]/(v_+^pv_-)$ . We endow $C_2$ with a $\mathbb{G}_m$ -action given by the grading: $\deg v_+^p=p$ and $\deg v_-=-1$ . The inclusion of $k[v_+^p,v_-]/(v_+^pv_-)\hookrightarrow k[v_+,v_-]/(v_+v_-)$ gives the morphism $k^{\mathcal{N}}\otimes\mathbb{F}_p\to C_2/\mathbb{G}_m$ . Recall from [Reference DrinfeldDri24, § 5.16.10] a factorization of $\mathfrak{p}_{\mathrm{cris}}$ :

(2.16) \begin{equation} k^{\mathcal{N}}\otimes\mathbb{F}_p\to C_2/\mathbb{G}_m\xrightarrow{\widetilde{\mathfrak{p}}_{\mathrm{cris}}} W(k)^{\mathcal{N}}_{red}\to W(k)^{\mathcal{N}}\otimes\mathbb{F}_p.\end{equation}

Let us just explain a construction of the composite

(2.17) \begin{equation}C_2 \twoheadrightarrow C_2/\mathbb{G}_m\xrightarrow{\widetilde{\mathfrak{p}}_{\mathrm{cris}}} W(k)^{\mathcal{N}},\end{equation}

given by a pair $(M_{C_2},\alpha:M_{C_2}\to W_{C_2})$ , where $M_{C_2}$ is an admissible $W_{C_2}$ -module and $\alpha$ is a $W_{C_2}$ -module homomorphism. Explicitly, $M_{C_2}$ is given by the pullback diagram

(2.18)

where the right downward arrow $[v_+^p]$ takes a Witt vector $w\in W_{C_2}$ to the product $[v_+^p]w$ . The morphism $\alpha$ is defined by $M_{C_2}=W_{C_2}\times_{F_*W_{C_2}}F_*W_{C_2}\xrightarrow{([v_-],V)}W_{C_2}$ . The isomorphism between the composition

\[C:=\operatorname{Spec} k[v_+,v_-]/(v_+v_-)\twoheadrightarrow C/\mathbb{G}_m=k^{\mathcal{N}}\otimes\mathbb{F}_p\xrightarrow{\mathfrak{p}_{\mathrm{cris}}} W(k)^{\mathcal{N}}\]

and is given by an isomorphism of $W_C$ -modules

(2.19) \begin{equation}\textrm{coker}(W_C^{(F)}\xrightarrow{[v_+],-\textrm{Id}} W_C^{(F)}\times W_C)\xrightarrow{\sim}W_C\times_{F_*W_C}F_*W_C=:C\times_{C_2}M_{C_2}\end{equation}

defined by the matrix $(\begin{smallmatrix} \textrm{Id} & [v_+]\\ 0& F\end{smallmatrix})$ . In [Reference DrinfeldDri24, Corollary 7.5.1], Drinfeld refines the morphism $\widetilde{\mathfrak{p}}_{\mathrm{cris}}:C_2/\mathbb{G}_m\to W(k)^{\mathcal{N}}_{red}$ to an isomorphism

(2.20) \begin{equation}BH\xrightarrow{\sim} W(k)^{\mathcal{N}}_{red},\end{equation}

where H is an affine group scheme over $C_2/\mathbb{G}_m$ constructed as follows. Let $\widetilde{H}:=W^{(F)}\times C_2$ . For $(w,t), (w',t)\in \widetilde{H}$ , the formula $(w,t)*(w',t)=(w+w'-[v_-^p(t)]ww',t)$ defines a map $\widetilde{H}\times\widetilde{H}\to \widetilde{H}$ , making $\widetilde{H}$ a group scheme over $C_2$ . The restriction of $\widetilde{H}$ to the open subscheme of $C_2$ given by $v_-\neq 0$ is $\mathbb{G}_m^{\sharp}$ , and its restriction to the closed subscheme given by $v_-=0$ is $\mathbb{G}_a^{\sharp}$ . Define a $\mathbb{G}_m$ -equivariant structure on $W\times C_2$ by

(2.21) \begin{equation} \mathbb{G}_m\times W\times C_2\to W\times C_2, \quad (\lambda,w,t)\mapsto ([\lambda^p]w,\lambda t).\end{equation}

This restricts to a $\mathbb{G}_m$ -equivariant structure on $\widetilde{H}\hookrightarrow W\times C_2$ compatible with the group structure. This defines the promised group scheme H over $C_2/\mathbb{G}_m$ .

We write $(C_2)_{v_+^p=0}$ , $(C_2)_{v_-=0}$ for the closed subschemes given by the respective equations. Denote by $D_{\mathrm{HT}}$ (respectively, $D_{\mathrm{dR}}$ ) the restrictions of $C_2/\mathbb{G}_m$ -stack BH to $(C_2)_{v_-=0}/\mathbb{G}_m$ (respectively, $(C_2)_{v_+^p=0}/\mathbb{G}_m$ ). Thus, via (2.20), one has $W(k)^{\mathcal{N}}_{red}=D_{\mathrm{HT}}\cup D_{\mathrm{dR}}$ . We refer to $D_{\mathrm{HT}}$ (respectively, $D_{\mathrm{dR}}$ ) as the Hodge–Tate (respectively, de Rham) component of $W(k)^{\mathcal{N}}_{red}$ . We denote by $D_{Hod}$ the fibre of BH over $(C_2)_{v_+^p=v_-=0}/\mathbb{G}_m=B\mathbb{G}_m$ . Geometrically, one pictures $W(k)^{\mathcal{N}}_{red}$ as a union of two components (i.e. $D_{\mathrm{HT}}$ and $D_{\mathrm{dR}}$ ) meeting transversally at $D_{Hod}$ . The restriction gives a functor

(2.22) \begin{equation}\mathcal{R}^*: \mathcal{D}_{qc}(W(k)^{\mathcal{N}}_{red})\to \mathcal{D}_{qc}(D_{\mathrm{HT}})\times_{\mathcal{D}_{qc}(D_{Hod})}\mathcal{D}_{qc}(D_{\mathrm{dR}}).\end{equation}

The following result is explained in [Reference BhattBha23, § 6] as a consequence of a general statement from [Reference LurieLur18b, Theorem 16.2.0.2].

Lemma 2.9. Functor (2.22) is fully faithful and admits right adjoint $\mathcal{R}_*$ given by

\[ (\mathcal{F}_{\mathrm{HT}},\mathcal{F}_{\mathrm{dR}},\alpha: \mathcal{F}_{\mathrm{HT}}|_{D_{Hod}}\simeq \mathcal{F}_{\mathrm{dR}}|_{D_{Hod}})\mapsto \mathrm{Fib}(i_{HT*}\mathcal{F}_{\mathrm{HT}}\oplus i_{\mathrm{dR}*}\mathcal{F}_{\mathrm{dR}}\to i_{Hod*}(\mathcal{F}_{\mathrm{HT}}|_{D_{Hod}})), \]

where $i_{?}$ is the closed embedding of $D_{?}$ into $W(k)^{\mathcal{N}}_{red}$ .

The following result describes the essential image of $\mathcal{R}^*$ in (2.22). Let $\mathcal{F}:=(\mathcal{F}_{\mathrm{HT}},\mathcal{F}_{\mathrm{dR}},\alpha: \mathcal{F}_{\mathrm{HT}}|_{D_{Hod}}\simeq \mathcal{F}_{\mathrm{dR}}|_{D_{Hod}})$ be an object of the right-hand side of (2.22).

Lemma 2.10. Then $\mathcal{F}$ lies in the essential image of $\mathcal{R}^*$ if and only if its pullback to

(2.23) \begin{equation}\mathcal{D}_{qc}((C_2)_{v_-=0}/\mathbb{G}_m){\times}_{\mathcal{D}_{qc}((C_2)_{v_+^p=v_-=0}/\mathbb{G}_m)}\mathcal{D}_{qc}((C_2)_{v_+^p=0}/\mathbb{G}_m)\end{equation}

comes from an object of $\mathcal{D}_{qc}(C_2/\mathbb{G}_m)$ .

In [Reference BhattBha23, § 6.2], Bhatt and Lurie give a convenient description of $D_{\mathrm{HT}}$ , i.e. $D_{\mathrm{HT}}\simeq (\mathbb{A}^{1,\mathrm{dR}}_+/\mathbb{G}_m)\otimes\mathbb{F}_p$ . To reconcile this isomorphism with the above description, we recall that the Frobenius morphism $\mathbb{A}^1_+\otimes\mathbb{F}_p\to \operatorname{Spec} k[v_+^p]=(C_2)_{v_-=0}$ factors through $(\mathbb{A}^1_+\otimes\mathbb{F}_p)^{\mathrm{dR}}$ . Moreover, a choice of $\delta$ -structure on $\mathbb{A}^1_+$ , such that $\delta(v_+)=0$ , gives a $\mathbb{G}_m$ -equivariant isomorphism $(\mathbb{A}^1_+\otimes\mathbb{F}_p)^{\mathrm{dR}}=B\mathbb{G}_a^{\sharp}\times (C_2)_{v_-=0}$ , where the action of $\mathbb{G}_m$ on $\mathbb{G}_a^{\sharp}$ is given by $\lambda*z=\lambda^pz$ . Using the identification of the Cartier dual to $\mathbb{G}_a^{\sharp}$ and $\widehat{\mathbb{G}}_a=\operatorname{Spf} k[[D^p]]$ , we obtain an equivalence $\mathcal{D}_{qc}(D_{\mathrm{HT}})\simeq \mathcal{D}_{gr,D^p\text{-nilp}}(k[v_+^p,D^p])$ , where the right-hand side is a full subcategory of the derived category $\mathcal{D}_{gr}(k[D^p, v_+^p])$ of graded modules over the polynomial algebra $k[D^p,v_+^p]$ with $\deg v_+^p=-\deg D^p=p$ . Objects of this full subcategory $\mathcal{D}_{gr,D^p\text{-nilp}}(k[D^p,v_+^p])$ consist of complexes $\mathcal{M}$ such that the action of $D^p$ on $\bigoplus_iH^i(\mathcal{M})$ is locally nilpotent.

2.5 Coherent sheaves and vector bundles over $X^{syn}$

Let Q be a p-complete Noetherian regular local ring. In [Reference BhattBha23, Remark 5.5.19], it was shown that the category $ \operatorname{Perf}(Q^{syn})$ of perfect complexes on $Q^{syn}$ has a unique t-structure whose category of connective objects $\operatorname{Perf}^{\leqslant 0}(Q^{syn})$ consists of all $\mathcal{F}\in \operatorname{Perf}(Q^{syn})$ such that, for any p-nilpotent ring R and a map $f:\operatorname{Spec} R \to Q^{syn}$ , the pullback $f^*\mathcal{F}$ is in $D^{\leqslant 0}(R)$ . We refer to the heart of this t-structure as the category $\operatorname{Coh}(Q^{syn})$ of coherent sheaves on $Q^{syn}$ . The latter has a more concrete description. Let $(\tilde Q, (d), \delta)$ be a p-torsion prism with $\tilde Q/(d)=Q$ . Then there exists a canonical faithfully flat map

(2.24) \begin{equation} \operatorname{Spf} \tilde{Q}[v_-, v_+]/(v_-v_+-d) \to Q^{syn},\end{equation}

where $\operatorname{Spf} \tilde{Q}[v_-, v_+]/(v_-v_+-d)$ is the (p,d)-adic formal scheme $\text{colim}_N \operatorname{Spec} \tilde{Q}[v_-, v_+]/(v_-v_+-d, d^N, p^N)$ ; see [Reference BhattBha23, Remark 5.5.19], cf. the end of § 2.3. Then $\mathcal{F} \in\mathcal{D}_{qc}(Q^{syn})$ belongs to $\operatorname{Coh}(Q^{syn})$ if and only of its pullback along (2.24) is coherent, that is, a finite module overFootnote ²⁰ $\tilde{Q}[v_-, v_+]/(v_-v_+-d)$ .

Recall that, for any stack $\mathcal{X}$ , the category $\textrm{Vect}(\mathcal{X})$ of vector bundles over $\mathcal{X}$ is defined to be $\lim_{\operatorname{Spec} R \to \mathcal{X}}\textrm{Vect}(\operatorname{Spec} R)$ , where the limit is taken over the category of all affine schemes over $\mathcal{X}$ . An object $\mathcal{F} \in\mathcal{D}_{qc}(Q^{syn})$ is a vector bundle if and only if its pullback along (2.24) is a finite projective module over $\tilde{Q}[v_-, v_+]/(v_-v_+-d)$ .

2.6 Hodge–Tate weights

Recall that $k^{\mathcal{N}}$ can be identified with $\operatorname{Spf} A/\mathbb{G}_m$ (see [Reference BhattBha23, § 5.4]). Consequently, the category $\mathcal{D}_{qc}(k^{\mathcal{N}})$ is equivalent to the subcategory $\widehat{\mathcal{D}}_{gr}(A)$ of the derived category $\mathcal{D}_{gr}(A)$ of graded A-modules spanned by p-complete objects. Given a gauge $\mathcal{F}\in \mathcal{D}_{qc}(k^{\mathcal{N}})$ , we denote by

(2.25) \begin{equation}\bigg(\mathcal{N}^{\cdot}=\bigoplus_i \mathcal{N}^i, v_-:\mathcal{N}^i\to\mathcal{N}^{i-1}, v_+:\mathcal{N}^i\to \mathcal{N}^{i+1},v_-v_+=v_+v_-=p\bigg)\end{equation}

the corresponding object in $\widehat{\mathcal{D}}_{gr}(A)$ .

are all quasi-isomorphisms.

A gauge $\mathcal{F}$ is said to be effective if it has Hodge–Tate weights $\geqslant 0$ . We say that a gauge $\mathcal{F}$ has Hodge–Tate weights $\leqslant b$ if under the identification (2.25), the maps $v_+$ in are all quasi-isomorphisms.

We say that a gauge $\mathcal{F}\in \mathcal{D}_{qc}(W(k)^{\mathcal{N}})$ has Hodge–Tate weights $\geqslant a$ (respectively, $\leqslant b$ ) if its pullback to $k^{\mathcal{N}}$ has the corresponding property. Likewise, we say that an F-gauge $\mathcal{F}\in \mathcal{D}_{qc}(W(k)^{syn})$ has Hodge–Tate weights $\geqslant a$ (respectively, $\leqslant b$ ) if its pullback to $k^{\mathcal{N}}$ has the corresponding property. We denote by $\mathcal{D}_{qc,[a,b]}(k^{\mathcal{N}})$ the full subcategory of $\mathcal{D}_{qc}(k^{\mathcal{N}})$ that consists of objects with Hodge–Tate weights in [a,b]. Similarly, we define $\mathcal{D}_{qc,[a,b]}(W(k)^{\mathcal{N}})$ .

3.1 Drinfeld lemma

Recall from (2.14) that $A:=W(k)[v_+,v_-]/(v_+v_-=p)$ with a grading such that $\deg v_+=1$ and $\deg v_-=-1$ . Let $A^i$ be the subgroup of elements of degree i. Let $B\subset A\otimes\mathrm{Frac}(W(k))$ be the W(k)-subalgebra generated by $v_-$ and ${v_+^n}/{n!}$ for $n\geqslant 0$ . The grading on A induces a grading on B and we write $B=\bigoplus_iB^i$ .

Note that $[n]=n$ for $n<p$ , and $[n+1]-[n]=1$ or 0 for every $n\in\mathbb{N}_{\gt 0}$ . For every positive integer n, the ideal $(p^{[n]})\subset \mathbb{Z}_p$ is the nth divided power of the ideal (p).

Recall the maps $\mathfrak{p}_{\bar{\mathrm{dR}}}$ and $ \mathfrak{p}_{\mathrm{cris}}$ from § 2.3. Consider the following diagram

(3.1)

where a and t are induced by homomorphisms of graded W(k)-algebras: the map t is induced by the map $W(k)[v_-]\to A$ sending $v_-\mapsto v_-$ , and the map a is induced by the obvious embedding $A\hookrightarrow B$ . Warning: $ \mathfrak{p}_{\mathrm{cris}}\not\simeq \mathfrak{p}_{\bar{\mathrm{dR}}}\circ t$ .

We need the following lemma in the proof of the theorem.

Remark 3.8. The Witt vector ${[v_+^p]}/{p}\in W(B)$ determines a map $\operatorname{Spec} B \xrightarrow{d} W$ . We claim that the following diagram is commutative.

(3.2)

Here the action $\alpha$ of $\mathbb{G}_m$ on $\operatorname{Spec} B$ is given by the grading and the action of $\mathbb{G}_m$ on W is given by the formula $\beta(\lambda, w)=[\lambda^p] w$ , where $\lambda \in \mathbb{G}_m, w \in W$ , and $[\cdot]$ refers to the Teichmüller representative. Indeed, since $\mathbb{G}_m\times \operatorname{Spec} B $ is flat over $\mathbb{Z}_p$ it suffices to check that $ d \circ \alpha = \beta \circ (\textrm{Id}\times d)$ after post-composing with $W\stackrel{p}{\longrightarrow}W$ which is clear.

Using (3.2) the map d descends to a map

$$ \bar d: \operatorname{Spec} B/\mathbb{G}_m \to W/\mathbb{G}_m $$

of the quotient stacks.

Proof of Theorem 1. Fix an S-point $\mathcal{P}$ of $\operatorname{Spf} B/\mathbb{G}_m$ . Using the morphism a, we have an S-point of $\operatorname{Spf} A/\mathbb{G}_m$ , i.e. $\mathcal{O}_S\xrightarrow{v_+}L\xrightarrow{v_-}\mathcal{O}_S$ , where L is a line bundle over S and $v_+v_-=p$ . Define a morphism of $W_S$ -modules $w:W_S\to \mathbf{V}(L)^{\sharp}\hookrightarrow [L]$ , where the invertible $W_S$ -module [L] is the Teichmüller lift of L (see [Reference DrinfeldDri24, 3.11]). By Lemma 3.6, the homomorphism $[v_+^p]: W_S\to [L^{\otimes p}]$ of $W_S$ -modules is canonically divisible by p. Indeed, we first define

(3.3) \begin{equation} \frac{[v_+^p]}{p}: W_S\to [L^{\otimes p}]\end{equation}

locally on S assuming a lift $\tilde{ \mathcal{P}}: S\to \operatorname{Spf} B $ of $\mathcal{P}$ . The choice of a lift trivializes L and [L]. Composing $\tilde {\mathcal{P}}$ with $\operatorname{Spec} B \stackrel{d}{\longrightarrow} W$ from Remark 3.8 we obtain a Witt vector ${[v_+^p]}/{p} \in W(S)$ . The multiplication by ${[v_+^p]}/{p}$ defines the desired map ${[v_+^p]}/{p}: W_S\to W_S= [L^{\otimes p}]$ . Diagram (3.2) shows that (3.3) does not depend on the choice of a trivializationFootnote ²¹ and, thus, it is well-defined globally. Applying the Verschiebung, we obtain a homomorphism $V({[v_+^p]}/{p}): W_S\to [L]$ of $W_S$ -modules. Set $w:=[v_+]-V({[v_+^p]}/{p}): W_S\to [L]$ . We claim that w lands in $\mathbf{V}(L)^{\sharp}$ . Indeed, the composition $F\circ w: W_S\to [L^{\otimes p}]$ is zero.

We want to construct an isomorphism $ \mathfrak{p}_{\mathrm{cris}}\circ a(\mathcal{P})\simeq \mathfrak{p}_{\bar{\mathrm{dR}}}\circ t\circ a(\mathcal{P})$ . Define a morphism $\Psi_{\mathrm{Maz}}:M_{\mathfrak{p}_{\mathrm{cris}},S}:=\textrm{coker}(\mathbb{G}_{a,S}^{\sharp}\xrightarrow{(v_+^{\sharp},-can)} \mathbf{V}(L)^{\sharp}\oplus W_S)\longrightarrow \mathbf{V}(L)^{\sharp}\oplus F_*W_S:=M_{\mathfrak{p}_{\bar{\mathrm{dR}},S}}$ as follows: let the map $\mathbf{V}(L)^{\sharp}\oplus W_S\to \mathbf{V}(L)^{\sharp}\oplus F_*W_S$ be given by the matrix $(\begin{smallmatrix} 1& w\\ 0& F\end{smallmatrix})$ . We claim that $\Psi_{\mathrm{Maz}}\circ(v_+^{\sharp},-can)$ is zero. Indeed, $F\circ can$ is zero. We also have that $v_+^{\sharp}-w\circ can:\mathbb{G}_{a,S}^{\sharp}\to \mathbf{V}(L)^{\sharp}$ is zero because $V({[v_+^p]}/{p})\circ can =0$ .Footnote ²² This defines $\Psi_{\mathrm{Maz}}$ . The following commutative diagram shows that $\Psi_{\mathrm{Maz}}$ is an isomorphism.

Finally, one verifies that $\Psi_{\mathrm{Maz}}$ commutes with $\xi$ given by (2.13) and (2.15). This proves the existence of the isomorphism in Theorem 1.

For the uniqueness, it suffices to show that the point $\mathfrak{p}_{\bar{\mathrm{dR}}}\circ t\circ a: \operatorname{Spf} B/\mathbb{G}_m\to W(k)^{\mathcal{N}}$ has no nontrivial automorphisms. Since $\operatorname{Spf} B\to \operatorname{Spf} B/\mathbb{G}_m$ is a faithfully flat cover, it suffices to show this for the corresponding point $s:\operatorname{Spf} B\to W(k)^{\mathcal{N}}$ . Explicitly, s is given by the morphism $\mathbb{G}_a^{\sharp}\oplus F_*W_B\xrightarrow{(v_-,V)}W_B$ of $W_B$ -modules. Using [Reference BhattBha23, Proposition 5.2.1], any automorphism $\alpha$ of $W_B$ -module has the form $[\begin{smallmatrix}t & x \\0 & y\end{smallmatrix}]$ for $t \in \mathbb{G}_a(B), x \in \ker(\mathbb{G}_a^{\sharp}(B) \to \mathbb{G}_a(B)) = 0, y \in F_* W(B)$ . The group of automorphisms of s consists of $\alpha$ satisfying $(v_-, V) \circ \alpha = (v_-, V)$ . In particular, for any $(a, b) \in \mathbb{G}_a^{\sharp}(B) \oplus F_* W(B),$ we must have $v_{-} ta + V(by) = v_{-} a + V(b)$ . Substitutions $a = 0, b=1$ and $a=[v_{+}], b=0$ give $t=y=1$ i.e. $\alpha=\textrm{Id}$ .

3.2 A description of $k^{\mathcal{N}}\times_{W(k)^{\mathcal{N}}} \mathbb{A}^1_-/\mathbb{G}_m$

Consider the following Cartesian square.

(3.4)

Using Theorem 1, we have a morphism

(3.5) \begin{equation}(a,t\circ a):\operatorname{Spf} B/\mathbb{G}_m\to k^{\mathcal{N}}\times_{W(k)^{\mathcal{N}}}\mathbb{A}^1_-/\mathbb{G}_m:=\mathfrak{D}.\end{equation}

Note that the composition $\mathbb{A}^1_-/\mathbb{G}_m\stackrel{\mathfrak{p}_{\bar{\mathrm{dR}}}}{\longrightarrow}W(k)^{\mathcal{N}}\stackrel{t_{W(k)}}{\longrightarrow}\mathbb{A}^1_-/\mathbb{G}_m$ is the identity. Indeed, recall that $\mathfrak{p}_{\bar{\mathrm{dR}}}$ takes $x=(L, v_-) \in\mathbb{A}^1_{-}/\mathbb{G}_m(R)$ to an admissible module $(v_{-}^{\sharp}, V): \mathbf{V}(L)^{\sharp} \oplus F_*W_R \to W_R$ as in (2.13). This gives rise to a morphism of admissible sequences (2.9) wherethe left vertical map is $v_{-}^{\sharp}: \mathbf{V}(L)^{\sharp} \to \mathbb{G}_{a, R}^{\sharp}$ . In particular, the Rees map sends $\mathfrak{p}_{\bar{\mathrm{dR}}}(x)$ to x. Now, giving an S-point of $\mathfrak{D}$ is equivalent to giving a $y \in \mathbb{A}^1_{-}/\mathbb{G}_m(S)$ and an $x \in k^{\mathcal{N}}(S)$ together with $\mathfrak{p}_{\bar{\mathrm{dR}}}(y) \simeq \mathfrak{p}_{\mathrm{cris}}(x)$ . Applying $t_{W(k)}$ to it we get $t(x) \simeq y$ . Thus, giving an S-point of $\mathfrak{D}$ is equivalent to giving an S-point x of $k^{\mathcal{N}}$ together with an isomorphism $\mathfrak{p}_{\bar{\mathrm{dR}}}\circ t(x)\simeq \mathfrak{p}_{\mathrm{cris}}(x)$ .

Let us define a closed substack of $\mathfrak{D}$ . Note that both $\mathfrak{p}_{\bar{\mathrm{dR}}}$ and $\mathfrak{p}_{\mathrm{cris}}$ , precomposed with the structure map $\pi: W(k)^{\mathcal{N}} \to W(k)^{\mathbb{\Delta}}$ , factor through the de Rham point $\mathfrak{p}_{\mathrm{dR}}: \operatorname{Spf} W(k) \to W(k)^{\mathbb{\Delta}}$ , classifying the Cartier–Witt divisor given by $W \xrightarrow{\cdot p} W$ . This defines a map $\tilde{\pi}: \mathfrak{D} \to \operatorname{Spf} W(k) \times_{W(k)^{\mathbb{\Delta}}} \operatorname{Spf} W(k)$ . The stack on the right has a canonical point given by the diagonal map $\operatorname{Spf} W(k)\to \operatorname{Spf} W(k) \times_{W(k)^{\mathbb{\Delta}}} \operatorname{Spf} W(k)$ . Consider the fibre over this point of the morphism $\tilde{\pi}$ and denote it by $\mathfrak{D}_{0}$ .

The following remark will not be used in the remainder of the paper.

Remark 3.9. Drinfeld suggested to us another interpretation of $\mathfrak{D}$ as well as its substack $\mathfrak{D}_{0}$ that makes use of the c-stack enhancement of $W(k)^{\mathcal{N}}$ introduced in [Reference DrinfeldDri24] and reviewed in § 2.3. For a scheme S, consider the category $\mathfrak{D'}(S)$ consisting of pairs (t, f), where $t: S \to \mathbb{A}_{-}^1/\mathbb{G}_m$ and $f: p_+ \to \mathfrak{p}_{\bar{\mathrm{dR}}}(t)$ is a morphism in the category $W(k)^{\mathcal{N}, c}(S)$ from § 2.3. Here $p_+$ is defined as the composition $S \to k^{\mathcal{N}} \xrightarrow{\mathfrak{p}_{\mathrm{cris}}} W(k)^{\mathcal{N}}$ , where the first morphism is given by $A \to H^0 (S, \mathcal{O}_{S})$ sending $v_-$ to p and $v_+$ to 1. We claim that the underlying groupoid of this category is equivalent to $\mathfrak{D}(S)$ . This follows from the fact [Reference DrinfeldDri24, § 5.5.3] that the Rees maps $t_{W(k)}: W(k)^{\mathcal{N}, c} \to (\mathbb{A}_{-}^1/\mathbb{G}_m)^c$ , $k^{\mathcal{N}, c} \to (\mathbb{A}_{-}^1/\mathbb{G}_m)^c $ are left fibrations of c-stacks. Indeed, since $W(k)^{\mathcal{N}, c} \to (\mathbb{A}^1_{-}/\mathbb{G}_m)^c$ is a left fibration, the category $W(k)^{\mathcal{N}, c}(S)_{p_+/}$ is identified with $(\mathbb{A}^1_{-}/\mathbb{G}_m)^c(S)_{p/} \simeq k^{\mathcal{N}, c}(S)$ .Footnote ²³ Under this identification, the natural functor $W(k)^{\mathcal{N}, c}(S)_{p_+/} \to W(k)^{\mathcal{N}, c}$ is identified with $\mathfrak{p}_{\mathrm{cris}}$ . The category $\mathfrak{D}'(S)$ is, by definition, equivalent to $W(k)^{\mathcal{N}, c}(S)_{p_+/} \times_{W(k)^{\mathcal{N}, c}(S)} (\mathbb{A}^1_{-}/\mathbb{G}_m)^c(S)$ . Its underlying groupoid is $\mathfrak{D}(S)$ by the definition of the latter (3.4). To describe the substack $\mathfrak{D}_{0},$ note that any object (t, f) of this category provides us with a triangle in $W(S)^{\mathcal{N}, c}$

(3.6)

where $p_-$ is defined by $A \to H^0 (S, \mathcal{O}_{S})$ , $v_- \mapsto 1$ , $v_+ \mapsto p$ , and the solid arrows are the canonical morphisms (constructed by observing that, for any S, $p_+$ [respectively, $p_-$ ] is the initial [respectively, final] object of the category $k^{\mathcal{N}, c}(S)$ and $p_-$ is the final object of $(\mathbb{A}_{-}^1/\mathbb{G}_m)^c(S)$ ). The triangle is not commutative in general; points that correspond to commutative triangles are classified by a substack of $\mathfrak{D}$ ; this substack is equivalent to $\mathfrak{D}_{0}$ .

Recall from [Reference DrinfeldDri24, § 1.8] that every effective gauge $\mathcal{F} \in \mathcal{D}_{qc, [0, +\infty]} (W(k)^{\mathcal{N}})$ gives rise to a complex of contravariant $\mathcal{O}$ -modules on $W(k)^{\mathcal{N}, c}$ . In particular, for every S-point of $\mathfrak{D}_{0}$ the canonical morphism $\mathcal{F}|_{p_-} \to \mathcal{F}|_{p_+}$ factors through $\mathcal{F}|_{\mathfrak{p}_{\bar{\mathrm{dR}}}(t)}$ . We also note that, for an effective F-gauge $\mathcal{F}$ , we have an isomorphism $F^*(\mathcal{F}|_{p_+}) \simeq \mathcal{F}|_{p_-}$ whose composition with $\mathcal{F}|_{p_-} \to \mathcal{F}|_{p_+}$ is the crystalline Frobenius.

In this subsection, we give an explicit description of $\mathfrak{D}$ , which was explained to us by Drinfeld and Lurie. The authors of this paper are responsible for any possible mistakes in the exposition of their results.

We endow W with the $\mathbb{G}_m$ -action given by the formula: $\lambda*w=[\lambda]w$ , where $[\cdot]$ refers to the Teichmüller representative. This gives an action of $\mathbb{G}_m$ on $\ker(W\xrightarrow{F}W)=:W^{(F)}\hookrightarrow W$ . We consider the fibre product $\operatorname{Spf} A\times_{\mathbb{A}^1_+}W^{(F)}= :\widetilde{\mathfrak{D}}'$ , where $\operatorname{Spf} A$ is viewed as a scheme over $\mathbb{A}^1_+$ via the natural embedding $W(k)[v_+]\hookrightarrow A$ , and $W^{(F)}\to\mathbb{A}^1_+$ is given by the first Witt vector coordinate.Footnote ²⁴ We endow $\operatorname{Spf} A\times_{\mathbb{A}^1_+} W^{(F)}$ with the diagonal $\mathbb{G}_m$ -action. Let $\widetilde{\mathfrak{D}}'_0\hookrightarrow\operatorname{Spf} A\times_{\mathbb{A}^1_+}W^{(F)}$ be the closed subscheme given byFootnote ²⁵

(3.7) \begin{equation}[v_-]w=p-V(1).\end{equation}

Recall from [Reference DrinfeldDri24, Lemma 3.2.6] the unique isomorphism $W^{(F)}\simeq \mathbb{G}_a^{\sharp}$ characterized by the commutative diagram

(3.8)

where the vertical map is induced by the projection $W \to \mathbb{G}_a$ to the first coordinate. Thus, under the isomorphism $W^{(F)}(\mathbb{Z}_p)\simeq \mathbb{G}_a^{\sharp}(\mathbb{Z}_p)=p\mathbb{Z}_p\subset \mathbb{Z}_p$ , the element $p-V(1)$ is sent to the number p. Consequently, under the identification $\operatorname{Spf} A\times_{\mathbb{A}^1_+} W^{(F)}\simeq \operatorname{Spf} A\times_{\mathbb{A}^1_+} \mathbb{G}_a^{\sharp}$ , Equation (3.7) has the form $v_-\cdot z=p$ , where z is a point of $\mathbb{G}_a^{\sharp}$ , and the product on the left-hand side refers to the $\mathbb{G}_a$ -module structure on $\mathbb{G}_a^{\sharp}$ . This yields a concrete description of $\widetilde{\mathfrak{D}}'_0$ as the formal spectrum of the W(k)-algebra generated by $v_-$ and $\gamma_{p^n}(v_+)$ , $n\geqslant 0$ , subject to the following relations:

(3.9) \begin{equation}\frac{p^n!}{(p^{n-1}!)^p} \gamma_{p^n}(v_+)= (\gamma_{p^{n-1}}(v_+))^p, \quad v_-^{p^n} \gamma_{p^n}(v_+)= \frac{p^{p^n}}{p^n!}.\end{equation}

Proof. To prove the first part, set $\widetilde A^\sharp:= W(k)\langle v_+\rangle[v_-]$ , $\widetilde A: = W(k)[v_+,v_-]$ , and let $J\subset \widetilde A^\sharp$ be the PD-ideal generated by ${v_+^n}/{n!}$ , $n\geqslant 1$ . Set $I:= (v_-v_+ -p) \subset \widetilde A^\sharp$ . ThenFootnote ²⁶ $I\cap J = I \cdot J $ . In particular, $I\cap J\subset J$ is a sub-PD-ideal. Consequently, there exists a unique PD structure on the image $\bar J$ of J under the projection $\widetilde A^\sharp \to \widetilde A^\sharp/I$ . We claim that the map $(A, (v_+)) \to (\widetilde A^\sharp/I, \bar J)$ exhibits the right-hand side as the PD-envelope $B^\flat$ of $(v_+)\subset A$ . Indeed, by the universal property we have a PD-homomorphism $B^\flat \to \widetilde A^\sharp/I$ . On the other hand, the projection $\widetilde A \to A$ gives a PD map $\widetilde A^\sharp \to B^\flat$ sending I to 0. By looking at generators, one readily checks that these are mutually inverse homomorphisms. Finally, we note that . Using the isomorphism $\mathbb{G}_a^\sharp\xrightarrow{\sim} W^{(F)}$ , this implies the claim.To prove the second assertion, observe that $B^{\dagger}$ is identified with the tensor product $B^{\dagger} \otimes_{D_{\mathbb{Z}_p}((p))} \mathbb{Z}_p,$ where $D_{\mathbb{Z}_p}((p))$ is the PD-envelope of $(p)\subset \mathbb{Z}_p$ and the homomorphism $D_{\mathbb{Z}_p}((p)) \to \mathbb{Z}_p$ takes each $\gamma_n(p) \in D_{\mathbb{Z}_p}((p))$ to ${p^n}/{n!}$ . Using (3.9), the claim follows.

Now we prove the proposition. Let R be a p-nilpotent ring. Recall from (3.4) that a point in $\mathfrak{D}(R)$ consists of a point $x \in k^{\mathcal{N}}(R)$ together with $\mathfrak{p}_{\mathrm{cris}}(x) \simeq \mathfrak{p}_{\bar{\mathrm{dR}}} \circ t(x)$ . Since $\widetilde{\mathfrak{D}} := \mathfrak{D} \times_{\mathbb{A}^1_{-}/\mathbb{G}_m} \mathbb{A}^1_{-}$ , we see that a point of $\widetilde{\mathfrak{D}}(R)$ consists of a homomorphism $g:A\to R$ together with an isomorphism $\beta$ of $W_R$ -modules

(3.10) \begin{equation} \textrm{coker}(\mathbb{G}_{a,R}^{\sharp}\xrightarrow{(g(v_+),-\mathrm{can})} \mathbb{G}_{a,R}^{\sharp}\oplus W_R):=M_{\mathfrak{p}_{\mathrm{cris},R}}\underset{\simeq}{\xrightarrow{\beta}} M_{\mathfrak{p}_{\bar{\mathrm{dR}},R}}:=\mathbb{G}_{a,R}^{\sharp}\oplus F_*W_R\end{equation}

such that two diagrams commute

(3.11)

(3.12)

for some $\nu \in \operatorname{Hom}_{W_R}(W_R,W_R)= W(R)$ . Explicitly, $\beta': \mathbb{G}_{a,R}^{\sharp}\oplus W_R\to M_{\mathfrak{p}_{\mathrm{cris}, R}} \xrightarrow{\beta} \mathbb{G}_{a,R}^{\sharp}\oplus F_*W_R$ is given by $(\begin{smallmatrix} 1 &w\\ 0&\nu \circ F\end{smallmatrix})$ , where $w=(w_1,w_2,\ldots)\in \operatorname{Hom}_{W_R}(W_R,\mathbb{G}_{a,R}^{\sharp})=W^{(F)}(R)$ is subject to the equation $w_1=g(v_+)$ and $[g(v_-)]w=p-V(\nu)$ .

Sending a point of $\widetilde{\mathfrak{D}}(R)$ to $(g\in (\operatorname{Spf} A)(R), w\in W^{(F)}(R))$ defines a functorial map

(3.13) \begin{equation}\widetilde{\mathfrak{D}}(R)\to (\operatorname{Spf} A\times_{\mathbb{A}^1_+} W^{(F)}) (R).\end{equation}

We next show that (3.13) is an isomorphism. To do this we construct its inverse.

Given a point $(g,w) \in (\operatorname{Spf} A\times_{\mathbb{A}^1_+} W^{(F)})(R)$ , observe that since $w_1=g(v_+)$ the first coordinate of $p-[g(v_-)]w$ is 0. Therefore, this element can be uniquely written as $V(\nu)$ , for some $\nu \in W(R)$ . Then the matrix $(\begin{smallmatrix} 1&w\\ 0&\nu \circ F\end{smallmatrix})$ defines a homomorphism $M_{\mathfrak{p}_{\mathrm{cris},R}} {\xrightarrow{\beta}} M_{\mathfrak{p}_{\bar{\mathrm{dR}},R}}$ making the diagrams (3.11) and (3.12) commutative. It remains to check that $\beta$ is anisomorphism. Using the diagram (3.11) it is enough to check that $\nu \in W(R)$ is invertible. Let $\nu_1$ be the first ghost coordinate of $\nu$ . If $\nu$ is not invertible we can find a maximal ideal $\mathfrak{m} \subset R$ which contains $\nu_1$ . Since $W^{(F)}(R/\mathfrak{m})=0$ and $w \in W^{(F)}(R)$ , we conclude from the equation $[g(v_{-})]w=p-V(\nu)$ that $\nu_1=1$ in $R/\mathfrak{m}$ , contradicting $\nu_1 \in \mathfrak{m}$ . Thus, $\beta$ is an isomorphism. This defines a functorial map

$$(\operatorname{Spf} A\times_{\mathbb{A}^1_+} W^{(F)}) (R) \to \widetilde{\mathfrak{D}}(R),$$

which is inverse to (3.13).

For the second part, note that $\beta = (\begin{smallmatrix} 1&w\\ 0&\nu \circ F\end{smallmatrix}) \in \widetilde{\mathfrak{D}}(R)$ lands in $\widetilde{\mathfrak{D}}_{0}$ if and only if $\nu =1$ and the result follows.

Note that the derived p-completion of $B^{\dagger}$ is the quotient of the p-completion of $W(k)\langle v_+ \rangle [v_-]$ by the ideal $(v_+v_- -p)$ . Using [Reference Bhatt, Morrow and ScholzeBMS18, Remark 6.16], it follows that this quotient is not p-adically separated and, in particular, it is not p-complete. Thus, the derived stack $k^{\mathcal{N}} \times_{W(k)^{\mathcal{N}}}^{\bf L} \mathbb{A}^1_{-}$ is not classical.

Remark 3.14. Observe that the morphism

(3.14) \begin{equation} \mathfrak{D}=k^{\mathcal{N}}\times_{W(k)^{\mathcal{N}}}\mathbb{A}^1_-/\mathbb{G}_m \to k^{syn}\times_{W(k)^{syn}}\mathbb{A}^1_-/\mathbb{G}_m\end{equation}

induced by maps $k^{\mathcal{N}} \to k^{syn}$ and $W(k)^{\mathcal{N}} \to W(k)^{syn}$ is an isomorphism. Indeed, since the maps $k^{\mathcal{N}} \to k^{syn}$ , $W(k)^{\mathcal{N}} \to W(k)^{syn}$ are étale, the morphism (3.14) is also étale. Thus, it suffices to check that (3.14) induces an equivalence on points with values in every algebraically closed field K of characteristic p. This follows from [Reference DrinfeldDri24, Propositions 5.16.5 and 8.10.4 (iv)] which assert that the maps $k^\mathcal{N} \to W(k)^\mathcal{N}$ , $k^{syn} \to W(k)^{syn}$ induce equivalences on K-points.

4.1 Corollary of Drinfeld’s lemma

The isomorphism $ \mathfrak{p}_{\mathrm{cris}}\circ a\simeq \mathfrak{p}_{\bar{\mathrm{dR}}}\circ t\circ a$ from Theorem 1 gives an isomorphism of functors $\mathcal{D}_{qc}(W(k)^{\mathcal{N}})\to \mathcal{D}_{qc}(\operatorname{Spf} B/\mathbb{G}_m)$

(4.1) \begin{equation}a^*t^*\mathfrak{p}^*_{\bar{\mathrm{dR}}}\simeq a^* \mathfrak{p}_{\mathrm{cris}}^*.\end{equation}

We show that after restriction to $\mathcal{D}_{qc, [0,p-1]}(W(k)^{\mathcal{N}})$ , a stronger statement holds.

We prove the theorem by showing that

(4.2) \begin{equation} a^*:\mathcal{D}_{qc, [0,p-1]}(k^\mathcal{N}) \to \mathcal{D}_{qc}(\operatorname{Spf} B/\mathbb{G}_m)\end{equation}

is fully faithful. We start with the following.

To compute $\textrm{Tor}_1^{W(k)}(B/A,k)=\ker(B/A\xrightarrow{p}B/A)$ : by Lemma 3.2 again, one has $\ker(B^i/A^i\xrightarrow{p}B^i/A^i)=({1}/{p})v_+^ik$ for and 0 otherwise. Thus, $\textrm{Tor}_1^{W(k)}(B/A,k)= (A/(v_-))\{p\}$ as an A-module. We have that

$$(\mathcal{N}^{\bullet}\overset{\mathbb{L}}{\otimes}_A\textrm{Tor}_1^{W(k)}(B/A,k))^i\xrightarrow{\sim} \operatorname{Cone}(\mathcal{N}^{i-p+1} \stackrel{v_-}{\longrightarrow} \mathcal{N}^{i-p}).$$

Using effectivity, the cone above is acyclic for $i<p$ .

Now we can prove that (4.2) is fully faithful. For any $M,M'\in \mathcal{D}_{qc,[0,p-1]}(k^{\mathcal{N}})$ ,

\begin{align*}\mathrm{RHom}(M,M')&=\mathrm{RHom}(M,w_{\leqslant p-1}M')\simeq \mathrm{RHom}(M,w_{\leqslant p-1}a_*a^*M')\\&\simeq \mathrm{RHom}(M,a_*a^*M')\simeq \mathrm{RHom}(a^*M,a^*M'),\end{align*}

where the second and fourth isomorphisms follow by adjunction, and the third isomorphism follows by Corollary 4.2. This proves fully faithfulness and thus completes the proof of the theorem.

Let us record a generalization of Theorem 2.

Remark 4.4. In geometric context, i.e. for the prismatic gauge $\mathcal{H}_{\mathcal{N}}(X) \in \mathcal{D}_{qc}(W(k)^\mathcal{N})$ associated to a smooth p-adic formal scheme X over W(k), Equation (4.1) gives

(4.3) \begin{equation}a^*t^*\mathfrak{p}^*_{\bar{\mathrm{dR}}} (\mathcal{H}_{\mathcal{N}}(X))\simeq a^* \mathfrak{p}_{\mathrm{cris}}^* (\mathcal{H}_{\mathcal{N}}(X)),\end{equation}

which expresses a relation between the Nygaard filtration on $R\Gamma_{\mathrm{cris}}(X_k)$ , where $X_k:=X\times\operatorname{Spec} k$ , and the Hodge filtration on $R\Gamma_{\mathrm{dR}}(X)$ . In fact, $a^*t^*\mathfrak{p}^*_{\bar{{\mathrm{dR}}}} (\mathcal{H}_{\mathcal{N}}(X))$ , viewed as a complex of filtered modules over the filtered algebra $(W(k), (p)^{[\cdot]})$ (see Remark 3.5), can be identified, using [Reference Berthelot and OgusBO78, Theorem 7.2] with the crystalline Hodge filtration on $R\Gamma_{\mathrm{cris}}(X_k)$ . Thus, (4.3) says that the crystalline Hodge filtration can be obtained from the Nygaard filtration by ‘downgrading’ using the functor $a^*$ .

4.2 Construction of the functor

Define a functor

(4.4) \begin{equation} \Phi_{\mathrm{Maz}}:\mathcal{D}_{qc,[0,p-1]}(W(k)^{syn}) \to \mathscr{DMF}^{\mathrm{big}}_{[0,p-1]}(W(k))\end{equation}

as follows. Let $\mathcal{F}\in \mathcal{D}_{qc,[0,p-1]}(W(k)^{syn})$ be a prismatic F-gauge with Hodge–Tate weights in $[0,p-1]$ . We have the following diagram (see [Reference BhattBha23, Remark 1.4.2] or § 2.3)

(4.5)

where $j_-$ (respectively, $j_+ \circ F^{-1}$ ) is an open embedding onto the substack given by $v_-\neq 0$ (respectively, $v_+\neq 0$ ), $\bar j_\mathcal{N}: k^{\mathcal{N}} \to k^{syn}$ exhibits the target as the coequalizer of

(see [Reference BhattBha23, Definition 6.1.1]), and the square is commutative. The morphism $\operatorname{Spf} W(k) \xrightarrow{v_- = 1} \mathbb{A}^1_{-}/\mathbb{G}_m$ is isomorphic to $t \circ j_-$ and $\operatorname{Spf} W(k) \xrightarrow{v_- = p} \mathbb{A}^1_{-}/\mathbb{G}_m$ is isomorphic to $t \circ j_{+} \circ F^{-1}$ ; see the end of § 2.3. By the definition of $\bar j_\mathcal{N}$

(4.6) \begin{equation}\overline{j}_{\mathcal{N}}\circ j_+ \simeq \overline{j}_{\mathcal{N}}\circ j_-.\end{equation}

Let $\mathcal{M}:=\mathfrak{p}_{\bar{\mathrm{dR}}}^*\circ j_{\mathcal{N}}^*\mathcal{F}$ . Note that $F^* (\mathcal{M}|_{v_-=p})\simeq j_+^*\circ t^*\mathcal{M}$ and $\mathcal{M}|_{v_-=1}\simeq j_-^*\circ t^*\mathcal{M}$ . Then using Theorem 2, one has that $ \mathfrak{p}_{\mathrm{cris}}^*\circ j_{\mathcal{N}}^*\mathcal{F}\simeq t^*\mathcal{M}$ . Hence, we obtain $F^*(\mathcal{M}|_{v_-=p})\simeq j_+^*\circ \mathfrak{p}_{\mathrm{cris}}^*\circ j_{\mathcal{N}}^*\mathcal{F}$ and $\mathcal{M}|_{v_-=1}\simeq j_-^*\circ \mathfrak{p}_{\mathrm{cris}}^*\circ j_{\mathcal{N}}^*\mathcal{F}$ . By (4.6), we get an isomorphism $F^*(\mathcal{M}_{v_-=p})\simeq \mathcal{M}_{v_-=1}$ , which we denote by $\varphi$ . Define the functor $\Phi_{\mathrm{Maz}}$ in (4.4) by sending $\mathcal{F}$ to $(\mathcal{M}, \varphi )$ .