Proof.

1. Step 1 We first reduce to the case when S is normal. Let $S_{\mathrm {red}}$ be the reduced quotient of the noetherian ring S, and let $S_{\mathrm {nor}}$ be the normalization of $S_{\mathrm {red}}$ . We then have the diagram

   

   Denote t, $t_1$ and $t_2$ to be the image of T in S, $S_{\mathrm {red}}$ and $S_{\mathrm {nor}}$ separately. Assume there exists a monic polynomial $g(X)\in R[X]$ over R such that $g(t_2)=0$ . As the ring $S_{\mathrm {red}}$ is noetherian, the normalization $S_{\mathrm {red}} \rightarrow S_{\mathrm {nor}}$ is injective. In particular, $g(t_2)=0$ implies $g(t_1)$ is zero, which then implies that $g(t)$ is nilpotent in S. Thus by taking $h(X):=g(X)^N\in R[X]$ for some large N, we can find a monic polynomial $h(X)$ over R with $h(t)=0$ . So it suffices to show that by passing to the normalization of S, the image of $T\in K\langle T\rangle $ is finite over R.

2. Step 2 Denote by X the scheme $\operatorname {\mathrm {Spec}}(S)$ , and $X^{\mathrm {an}}$ the relative analytification of S over the affinoid ring R, as a rigid space over $\mathrm {Spa}(R)$ (c.f [Reference ConradCon06]). Here we note that $X^{\mathrm {an}}$ is also a normal rigid space, as the normality is checked around each K-valued point of the scheme X ([Reference BerkovichBerk93, Theorem 2.2.1]). For each affinoid open subset $U=\mathrm {Spa}(A)$ of $X^{\mathrm {an}}$ , the composition of rings $K\langle T\rangle \rightarrow S\rightarrow A$ is automatically continuous by [Reference Bosch, Güntzer and RemmertBGR, 6.1.3/1]. In particular, this induces a map of rigid spaces $X^{\mathrm {an}} \rightarrow \mathbb {B}^1$ , where $\mathbb {B}^1=\mathrm {Spa}(K\langle T\rangle )$ is the unit disc over K. Here we note that the ring S is contained in the global section $\mathcal {O}_{X^{\mathrm {an}}}(X^{\mathrm {an}})$ of the structure sheaf. Moreover, to give a map from the rigid space $Y=X^{\mathrm {an}}$ to $\mathbb {B}^1$ is equivalent to give a power-bounded global analytic function over Y, i.e. an element in $\mathcal {O}_Y^+(Y)$ .Footnote ² So the image $t=\phi (T)$ is a power-bounded element in $\mathcal {O}_{X^{\mathrm {an}}}(X^{\mathrm {an}})$ .

3. Step 3 As the affine scheme $X=\operatorname {\mathrm {Spec}}(S)$ is finite type over the noetherian scheme $\operatorname {\mathrm {Spec}}(R)$ , by Nagata’s compactification theorem we could find a schematic open immersion $X\rightarrow \overline X$ with dense image, where $\overline X$ is a proper $\operatorname {\mathrm {Spec}}(R)$ -scheme. In particular, by taking the relative analytification $\overline X^{\mathrm {an}}$ of $\overline X$ , the induced morphism $X^{\mathrm {an}} \rightarrow \overline X^{\mathrm {an}}$ is an open immersion of rigid spaces over $\mathrm {Spa}(R)$ whose complement is Zariski closed of codimension $\geq 1$ in $\overline X$ . Moreover, notice that since $X^{\mathrm {an}}$ is normal, it is also an open dense subspace in the normalization of $\overline X^{\mathrm {an}}$ , so we can thus assume $\overline X^{\mathrm {an}}$ is normal ([Reference ConradCon99, Theorem 2.1.2]). We apply the first Riemann extension theorem ([Reference BartenwerferBar76, $\S$ 3, Theorem]) for the open immersion $X^{\mathrm {an}} \rightarrow \overline X^{\mathrm {an}}$ into the proper normal rigid space, to get the isomorphism

   $$\begin{align*}\mathcal{O}^+_{X^{\mathrm{an}}}(X^{\mathrm{an}})= \mathcal{O}^+_{\overline X^{\mathrm{an}}}(\overline X^{\mathrm{an}}). \end{align*}$$

   But since $\overline X^{\mathrm {an}}$ is proper over $\mathrm {Spa}(R)$ , the global section of the structure sheaf $\mathcal {O}_{\overline X^{\mathrm {an}}}(\overline X^{\mathrm {an}})$ is a finite R-algebra. In this way, the element $t=\phi (T)\in \mathcal {O}^+_{X^{\mathrm {an}}}(X^{\mathrm {an}})= \mathcal {O}^+_{\overline X^{\mathrm {an}}}(\overline X^{\mathrm {an}})\subset \mathcal {O}_{\overline X^{\mathrm {an}}}(\overline X^{\mathrm {an}})$ is also finite over R, and we are done.

Proof. By Proposition 2.1.1, the image $\phi (T)$ mod $\xi $ is finite over $R/\xi $ . In particular, there exists a polynomial $g(X)\in {\mathrm {B^+_{dR,m}}}[X]$ , such that $g(\phi (T))=0$ mod $\xi $ . Thus by taking a power of $g(X)$ we get the result.

Proof. As a finite type algebra over B is noetherian, its formal completion along $\xi =0$ identifies the $\xi ^\infty $ -elements before and after the completion ([Sta, Tag 05E9]). In particular, we can switch the order of the $\xi $ -completion and the $\xi $ -torsionfree quotient, and it suffices to show the isomorphism of the map between incomplete algebras

$$\begin{align*}\left( B[x_i]/(\xi x_i-f_i)\right) /(\xi^\infty\text{-}torsion) \longrightarrow \left( B[x_i,y]/(\xi x_i-f_i; \xi y-g)\right) /(\xi^\infty\text{-}torsion). \end{align*}$$

By assumption, since $(f_i)$ generates the ideal, we can write $g=\sum a_jf_j$ . So we get the formula in the second ring above

$$\begin{align*}\xi \cdot \left(y -\sum a_j x_j\right)=0. \end{align*}$$

Thus by the $\xi $ -torsion freeness of the target ring, the map above is surjective. Moreover, we can define a section of the above map by mapping y onto $\sum a_jx_j$ , which is possible by the $\xi $ -torsion freeness of the quotient. Hence we get the isomorphism.

Proof. Let $(B,\xi B)$ be a prism over R. We need to show that it admits a map from the prism associated to the envelope $P[\frac {I}{\xi }]^\wedge _{\mathrm {tf}} $ . We first notice the following commutative diagram

Then we show the diagram lifts to a homomorphism $P\rightarrow B$ over ${\mathrm {B_{dR}^+}}$ . By the assumption of a prism in Definition 2.2.1 (ii), the ring $B/\xi ^m$ admits a surjection from ${\mathrm {B^+_{dR,m}}}\langle a_{j_m}\rangle [b_{l_m}]$ for finite sets of $j_m$ and $l_m$ , compatible with $m\in \mathbb {N}$ . Moreover, by enlarging the set of $j_m$ and $l_m$ inductively for each m, we may assume a compatible system of surjections as below

In particular, when $m=1$ , the quotient ring $B/\xi $ is a finite type algebra over an affinoid ring $K\langle a_{j_1}\rangle $ , and it admits a map from $P/\xi =K\langle T_i\rangle $ by the diagram above. Note that by Proposition 2.1.1 the image $t_i$ of each $T_i$ in $B/\xi $ is integral over $K\langle a_{j_1}\rangle $ . Moreover, by Corollary 2.1.3 any lift $\widetilde t_i$ of $t_i$ in $B/\xi ^m$ is integral over ${\mathrm {B^+_{dR,m}}}\langle a_{j_m}\rangle $ . So by choosing compatible lifts $\widetilde t_i$ of $t_i$ in $B/\xi ^m$ for each $m\in \mathbb {N}$ , we could extend the map ${\mathrm {B^+_{dR,m}}}\langle a_{j_m}\rangle \rightarrow B/\xi $ to ${\mathrm {B^+_{dR,m}}}\langle a_{j_m}; c_i\rangle \rightarrow B/\xi ^m$ , where $c_i$ maps onto lifts $\widetilde t_i$ of $t_i\in B/\xi $ . As a consequence, we can enlarge variables $\{a_{j_m}\}$ into $\{a_{j_m}, c_i\}$ to get a surjection ${\mathrm {B^+_{dR,m}}}\langle a_{j_m, c_i}\rangle [b_{l_m}]\rightarrow B/\xi ^m\rightarrow B/\xi $ . In particular, we may assume the map $P={\mathrm {B_{dR}^+}}\langle T_i\rangle \rightarrow B/\xi $ above factors through the surjections as below:

which is compatible with $m\in \mathbb {N}$ . So by taking an inverse limit with respect to m, we get the commutative diagram

At last, since the image of the ideal $I\subset P$ in B is killed by $\xi $ and the ring B is $\xi $ -torsion free, the homomorphism $P\rightarrow B$ induces a natural commutative diagram as below

where $P[\frac {I}{\xi }]^\wedge _{\mathrm {tf}}$ is the $\xi $ -adic completion of the ${\mathrm {B_{dR}^+}}$ -algebra $P[\frac {I}{\xi }]$ . Hence the ring $P[\frac {I}{\xi }]^\wedge _{\mathrm {tf}}$ together with the R-algebra structure of its quotient forms a prism , and we get a map of prisms .

Construction 4.1.1. Let $(M,{\mathrm {Fil}}^{\bullet } M)$ be a $\mathbb {N}^{\mathrm {op}}$ -indexed descending filtered complex of ${\mathrm {B_{dR}^+}}$ -modules. Namely ${\mathrm {Fil}}^{\bullet } M$ is an object in ${\mathscr {DF}}({\mathrm {Mod}}_{{\mathrm {B_{dR}^+}}})=\textrm {Fun}(\mathbb {N}^{\mathrm {op}}, {\mathrm {Mod}}_{{\mathrm {B_{dR}^+}}})$ as in the beginning of [Reference GuoGuo24, Section 5]. Then we define the functor $\Psi (M,{\mathrm {Fil}}^{\bullet } M)$ (or $\Psi (M)$ in short if there is no confusion) as an object in $\mathscr {D}({\mathrm {B_{dR}^+}})$ by the following formula

Here $(-)^\wedge $ above is the derived $\xi $ -adic completion, and precisely the colimit is taken over the following diagram of ${\mathrm {B_{dR}^+}}$ -complexes

(4.1.1)

where all horizontal arrows are defining morphisms of the filtration.

Proof. For each $(U=\mathrm {Spa}(A), T_e=\mathrm {Spa}(B_e))$ , we construct a module over $B'=B[\frac {I}{\xi }]^\wedge _{\mathrm {tf}} $ by taking the tensor product as in the formula above. Then by the crystal condition of $\mathcal {F}\in \mathrm {Cry}_{X/{\mathrm {B_{dR}^+}}_{\mathrm {inf}}}$ , this satisfies the crystal condition with respect to prisms coming from infinitesimal thickenings. Then it suffices to notice that by Remark 4.2.5 the category of associated prisms covers the final objects of the prismatic topos.

Assumption 4.3.1. We let R be a fixed topologically finite type K-algebra, and let P be the ring ${\mathrm {B_{dR}^+}} \langle T_i\rangle $ . Assume I is a Koszul-regular ideal of P (in the sense of [Sta, Tag 07CU]), such that $P/I$ is flat over ${\mathrm {B_{dR}^+}}$ , and $P/(I,\xi )$ is isomorphic to R.

Proof. We first assume that I admits a finite set of generators $(f_i)$ such that $(\xi , f_i)$ is a Koszul-regular sequence, which is always true Zariski locally around $\operatorname {\mathrm {Spec}}(R)$ in $\operatorname {\mathrm {Spec}}(P)$ (c.f. Example 4.3.2). Then there exists a natural map from the polynomial algebra $P[x_i]$ to $\Psi (P,I^{\bullet })$ , sending $x_i$ onto $\frac {f_i}{\xi }\in \frac {I}{\xi }$ . This induces a natural homomorphism from the noetherian ring $P[\frac {f_i}{\xi }]=P[x_i]/(\xi \cdot x_i -f_i)$ to $\Psi (P,I^{\bullet })$ , and factors through the derived $\xi $ -completion of $P[\frac {f_i}{\xi }]$ . Note by [Sta, Tag 0A06] the classical $\xi $ -completion and the derived $\xi $ -completion of a noetherian ${\mathrm {B_{dR}^+}}$ -algebra are the same. Thus the above leads to a natural map from the actual noetherian algebra $P[\frac {f_i}{\xi }]^\wedge $ to $\Psi (P,I^{\bullet })$ , where the former is classically $\xi $ -completed. Moreover, as $(\xi , f_i)$ forms a regular sequence of the ring P, the map of rings ${\mathrm {B_{dR}^+}}[y_i] \rightarrow P$ , $y_i\mapsto f_i$ is flat after localizing at the ideal $(\xi )$ (c.f. [Reference HartshorneHar66, Proposition 1]). As a consequence, the ring $P[\frac {f_i}{\xi }]$ has no $\xi $ -torsion, and we get a map from $P[\frac {I}{\xi }]^\wedge _{\mathrm {tf}}$ to $\Psi (P, I^{\bullet })$ .

To show the isomorphism, by the derived Nakayama lemma ([Sta, Tag 0G1U]) for ${\mathrm {B_{dR}^+}}$ , it suffices to check the above is an isomorphism after a derived tensor product with K. On the one hand, by Remark 4.1.6 we have

$$\begin{align*}\Psi(P,I^{\bullet})\otimes^L_{\mathrm{B_{dR}^+}} K \cong \bigoplus_{n\in \mathbb{N}} {\mathrm{gr}}^n (P\otimes^L_{\mathrm{B_{dR}^+}} K). \end{align*}$$

By the assumption of P, each $I^n/I^{n+1}$ is Tor-independent of K over ${\mathrm {B_{dR}^+}}$ . So the above tensor product is equal to the discrete module as below

$$\begin{align*}\bigoplus_{n\in \mathbb{N}} (\overline I)^n/(\overline I)^{n+1}, \end{align*}$$

where $\overline I$ is the tensor product $I\otimes _{\mathrm {B_{dR}^+}} K$ . On the other hand, the assumption of P and $I=(f_i)$ implies the ring $P[x_1, \ldots , x_m]/(\xi x_i-f_i, 1\leq i\leq m)$ is flat over ${\mathrm {B_{dR}^+}}$ . In particular, we have

$$ \begin{align*} P[\frac{f_i}{\xi}]\otimes^L_{\mathrm{B_{dR}^+}} K & \cong P[\frac{f_i}{\xi}]\otimes_{\mathrm{B_{dR}^+}} K\\ &\cong P/(\xi ,f_i)[x_i], \end{align*} $$

which by assumption is exactly the direct sum $\bigoplus _{n\in \mathbb {N}} (\overline I)^n/(\overline I)^{n+1}$ . In this way, as the above two reductions are abstractly isomorphic to each other, to finish the proof, it suffices to notice that the map we constructed identifies the generators of each direct summand.

In general, one can still form the map from $P[\frac {I}{\xi }]^\wedge _{\mathrm {tf}} $ to $\Psi (P,I^{\bullet })$ , replacing $(f_i)$ above by any finite set of generators of I. As the ring P is noetherian, we can switch the order of the torsionfree quotient and the $\xi $ -completion to get

$$\begin{align*}P[\frac{I}{\xi}]^\wedge_{\mathrm{tf}} = \left( P[\frac{f_i}{\xi}]/(\xi^\infty\text{-torsion}) \right)^\wedge. \end{align*}$$

So to show the isomorphism, it suffices to take the derived reduction mod $\xi $ and consider the following

$$\begin{align*}\left( P[\frac{f_i}{\xi}]/(\xi^\infty\text{-torsion}) \right) \otimes^L_{\mathrm{B_{dR}^+}} K \rightarrow \Psi(P, I^{\bullet})\otimes^L_{\mathrm{B_{dR}^+}} K. \end{align*}$$

But notice that the map commutes with any Zariski localization of P, and for each $\mathfrak {p}\in \operatorname {\mathrm {Spec}}(R)$ there is a Zariski open neighborhood of $\mathfrak {p}$ in $\operatorname {\mathrm {Spec}}(P)$ such that I admits a set of Koszul-regular generators and the above is an isomorphism. Thus a Čech complex argument for an open covering finishes the proof.

Proof. By proof of Proposition 3.0.12, to describe the log connection structure on (see Definition/Lemma 4.2.6), it suffices to consider the difference map

where is the isomorphism of -modules induced from the crystal condition. Here by Definition/Lemma 4.2.6, the isomorphism is naturally isomorphic to , with

On the other hand, notice that the isomorphism $\phi _{\inf }$ is filtered under the $(I,\Delta (1))$ -adic filtration, and in particular by applying $\Psi (-, (I,\Delta (1))^{\bullet })$ and Lemma 4.3.4 we have

Thus by taking the difference between $\operatorname {\mathrm {id}}\otimes 1$ and and mod the ideal $(\frac {\Delta (1)}{\xi })^2$ , we get a natural log connection on .

Moreover, by the property of colimits, there is a natural commutative diagram sending the connection $(M,\nabla )$ to the log connection on $\pi ^*M$ , compatible with the natural structure map :

where the vertical maps are natural maps compatible with the inclusion $\Omega _{P/{\mathrm {B_{dR}^+}}}^1\rightarrow \frac {\Omega _{P/{\mathrm {B_{dR}^+}}}^1}{\xi }$ over the ring P. From the diagram, we see the log connection on $\pi ^*M$ coincides with $\nabla $ when restricted to M, and is uniquely extended by $\nabla $ using the Leibniz rule of the log connection.

At last, the integrability and the quasi-nilpotence either follow from the general result in Proposition 3.0.12, or the above commutative diagram and Leibniz rule, as $\widetilde \nabla _i(x)$ is $\xi $ -divisible for any i and $x\in M$ , and thus by Definition 3.0.6 the connection $\Psi (\nabla )$ is integral and thus quasi-nilpotent.

Proof. There is a natural map below

By the commutativity between the derived tensor product functor and the colimit, the above is an isomorphism before the derived $\xi $ -adic completion for $\Psi (-,-)$ as in Construction 4.1.1. So thanks to the derived $\xi $ -completeness of and the finiteness of M over $D_{\inf }$ , we see the left hand side above is automatically $\xi $ -complete, and hence isomorphic to the right.

Proof. By the tensor product formula of $\pi ^*M$ and the compatiblility between connections and log connections, it suffices to show that under Assumption 4.3.1 for the ring $P={\mathrm {B_{dR}^+}} \langle T_i\rangle $ , the ideal I and the ring $R=P/(I,\xi )$ , the natural map is an inclusion.

By definition and Lemma 4.3.4, we have $D_{\inf }$ is the I-adic completion of P, and is the $\xi $ -adic completion of $P[\frac {I}{\xi }]_{\mathrm {tf}}$ . So thanks to the left exactness of the inverse limit, it suffices to show the injectivity for each $n\in \mathbb {N}$ below

(∗) $$ \begin{align} P/(I^n,\xi^n) \longrightarrow P[\frac{I}{\xi}]_{\mathrm{tf}}/\xi^n, \end{align} $$

where we use the equality $\varprojlim P/(I^n,\xi ^n) =\varprojlim P/I^n$ by the $\xi $ -adic completeness of P. This can be checked Zariski locally, where we can assume I admits a finite set of regular generators $(f_1,\ldots ,f_l)$ . In this case, the natural map below is flat

Thus the injectivity of $(\ast )$ for each n follows from that of

$$\begin{align*}{\mathrm{B^+_{dR,n}}}[y_j] \rightarrow {\mathrm{B^+_{dR,n}}}[x_j]/(\xi\cdot x_j-y_j), \end{align*}$$

by base change along the flat map ${\mathrm {B_{dR}^+}}[y_j] \rightarrow P$ .

Proof. The idea is to compare Čech-Alexander complexes of two sides. Denote $D(n)_{\inf }$ to be the formal completion of the complete tensor product $P(n)=P^{\widehat \otimes _{{\mathrm {B_{dR}^+}}} n+1}:= {\mathrm {B_{dR}^+}} \langle T_i, \delta _{i,j}; 1\leq i\leq l, 1\leq j\leq n \rangle $ along the surjection onto $P/I$ , and let $\Delta (n)$ be the ideal of $D(n)_{\inf }$ defined by $(\delta _{i,j})$ . Let be the prismatic envelope of $P/I$ inside of $P^{\widehat \otimes _{{\mathrm {B_{dR}^+}}} n+1}$ (Lemma 4.3.4). Note that by Remark 2.2.7, the ring is also naturally isomorphic to $D(n)_{\inf }[\frac {I,\Delta (n)}{\xi }]^\wedge _{\mathrm {tf}}$ .

We then turn to the Čech-Alexander complexes of $\mathcal {F}$ and $\pi ^*\mathcal {F}$ . By [Reference GuoGuo24, Proposition 2.2.7] and its ${\mathrm {B_{dR}^+}}$ -linear version [Reference GuoGuo24, Theorem 7.2.3], we have a natural filtered isomorphism

(1) $$ \begin{align} R\Gamma(X/{\mathrm{B_{dR}^+}}_{\mathrm{pinf}},\mathcal{F}) \cong R\varprojlim_{[n]\in \Delta^{\mathrm{op}}} \mathcal{F}(D(n)_{\inf}), \end{align} $$

where the left side is endowed with the lifted filtration, and each term of the right side is equipped with the $(I,\Delta (n))$ -adic filtration (as $(I,\Delta (n))$ is the kernel ideal of the pro-infiniteisimal thickening $D(n)_{\inf } \rightarrow P/I$ ). On the other hand, by Corollary 3.0.14 we have the natural isomorphism

(2)

Moreover, by the construction of $\pi ^*\mathcal {F}$ in Theorem 4.3.5 and Lemma 4.3.6 the right hand side above is equal to

$$\begin{align*}R\varprojlim_{[n]\in \Delta^{\mathrm{op}}} \Psi(\mathcal{F}(D(n)_{\inf}),(I,\Delta(n))^{\bullet}). \end{align*}$$

In this way, by applying $\Psi $ at the sequence (1) using the lifted filtration, and notice that homotopy limit along $\Delta ^{\mathrm {op}}$ commutes with $\Psi $ , we get

Proof. Using Theorem 4.3.12, cohomology of $\pi ^*\mathcal {F}$ is isomorphic to $\Psi (R\Gamma (X/{\mathrm {B_{dR}^+}}_{\mathrm {pinf}}, \mathcal {F}), \widetilde {\mathrm {Fil}}^{\bullet })$ . Then by taking an inverse limit of [Reference GuoGuo24, Theorem 4.1.1] with respect to $\mathrm {Spa}(P/(I,\xi ^e))$ over ${\mathrm {B^+_{dR,e}}}$ , we have

$$\begin{align*}\Psi(R\Gamma(X/{\mathrm{B_{dR}^+}}_{\mathrm{pinf}}, \mathcal{F}), \widetilde {\mathrm{Fil}}^{\bullet}) \cong \Psi(M\underset{P}{\otimes} \Omega_{P/{\mathrm{B_{dR}^+}}}^{\bullet}, \widetilde{\mathrm{Fil}}^{\bullet}), \end{align*}$$

where $\widetilde {\mathrm {Fil}}^{\bullet }$ is the lifted infinitesimal filtration on the de Rham complex. Thus by writing out the right hand side above explicitly, we get the conclusion.

Proof. As a smooth lift $\widetilde R/\xi ^e$ itself is an infinitesimal envelope of R into a smooth rigid space over ${\mathrm {B^+_{dR,e}}}$ , we can apply Theorem 4.3.12 at the filtered isomorphism $R\Gamma (X/{\mathrm {B^+_{dR,e}}}_{\inf }, \mathcal {F})\cong (M/\xi ^e\otimes _{\widetilde R/\xi ^e} \Omega _{(\widetilde R/\xi ^e)/{\mathrm {B^+_{dR,e}}}}^{\bullet })$ in [Reference GuoGuo24, Theorem 4.1.1] to get the result.

Proof. Recall the analytic derived de Rham complex ${\widehat {\mathrm {dR}}^{\mathrm {an}}}_{(P/I)/{\mathrm {B_{dR}^+}}}$ over ${\mathrm {B_{dR}^+}}$ is defined as the filtered object

$$\begin{align*}R\varprojlim_e {\widehat{\mathrm{dR}}^{\mathrm{an}}}_{(P/(I,\xi^e))/{\mathrm{B^+_{dR,e}}}}, \end{align*}$$

where ${\widehat {\mathrm {dR}}^{\mathrm {an}}}_{(P/(I,\xi ^e))/{\mathrm {B^+_{dR,e}}}}$ is constructed as in [Reference GuoGuo24, Section 5] for affinoid algebras over ${\mathrm {B^+_{dR,e}}}$ . By [Reference GuoGuo24, Corollary 5.5.2], assuming Assumption 4.3.1, the derived de Rham complex ${\widehat {\mathrm {dR}}^{\mathrm {an}}}_{(P/(I,\xi ^e))/{\mathrm {B^+_{dR,e}}}}$ is filtered isomorphic to $R\Gamma (\mathrm {Spa}(P/(I,\xi ^e))/{\mathrm {B^+_{dR,e}}}_{\inf }, \mathcal {O}_{e \inf })$ . Thus the formula follows from Theorem 4.3.12.

Proof. By the global version of Theorem 4.3.12 and Corollary 3.0.15, the regular immersion induces an isomorphism between the left hand side above and the following

$$\begin{align*}\Psi(R\Gamma(X/{\mathrm{B_{dR}^+}}_{\mathrm{pinf}}, \mathcal{F}), \widetilde {\mathrm{Fil}}^{\bullet})\otimes^L_{\mathrm{B_{dR}^+}} K, \end{align*}$$

where $\widetilde {\mathrm {Fil}}^{\bullet }$ is the lifted infinitesimal filtration. By Remark 4.1.6 it is further equal to

$$\begin{align*}\bigoplus_n {\mathrm{gr}}^n \left( R\Gamma(X/{\mathrm{B_{dR}^+}}_{\mathrm{pinf}}, \mathcal{F})\otimes^L_{\mathrm{B_{dR}^+}} K \right)(-n). \end{align*}$$

Here the filtration is the infinitesimal filtration. At last, by [Reference GuoGuo24, Theorem 7.2.3], the base change $R\Gamma (X/{\mathrm {B_{dR}^+}}_{\mathrm {pinf}}, \mathcal {F})\otimes ^L_{\mathrm {B_{dR}^+}} K$ is isomorphic to infinitesimal cohomology $R\Gamma (X/K_{\mathrm {inf}}, \overline {\mathcal {F}})$ with the canonical infinitesimal filtration.

Proof. As we have constructed a natural map from algebraic de Rham complex of $R/K$ to the cdga , it suffices to show that the map factors through the analytic de Rham complex of $R/K$ , and induces a termwise isomorphism with the target.

Let us assume R admits a surjection from $P={\mathrm {B_{dR}^+}} \langle T_i\rangle $ , together with an ideal $I\in P$ satisfying Assumption 4.3.1. By Corollary 4.3.13, prismatic cohomology is isomorphic to $\xi $ -divided de Rham cohomology

where each differential map is the continuous differential with respect to $(\xi ,p)$ -adic topology. Denote its termwise reduction by $\overline {C}_{P}$ , which by Theorem 4.3.12 and Remark 4.1.6 is isomorphic to

As R is smooth over K, each $L\wedge ^i\mathbb {L}_{R/K}$ is isomorphic to i-th analytic differential $\Omega _{R/K}^i$ . So by applying the Bockstein operator $\beta _\xi $ and its associated complex, we have

$$\begin{align*}(\mathrm{H}^{\bullet}(\overline{C}_{P})(\bullet), \beta_\xi) = (\Omega_{R/K}^{\bullet},d), \end{align*}$$

where d is the p-adic continuous differential. Thus the universal map from algebraic de Rham complex $(\Omega _{R/K}^{\bullet ,alg}, d)$ to $(\mathrm {H}^{\bullet }(\overline {C}_{P})(\bullet ), \beta _\xi )$ identifies its analytic quotient with the target. At last, notice that since the category of regular surjections $P\rightarrow R$ is sifted as in Remark 4.3.3 (c.f. [Reference GuoGuo24, Lemma 5.5.4]), the isomorphism above is independent of the choice of $P\rightarrow R$ . So we are done.

Construction 5.2.1. Let R be a commutative ring, and A be a simplicial R-algebra. Let $d\in \mathbb {N}$ be an integer, and let $\omega \in A_{d-1}$ be an element that represents a homology class $\bar {\omega }$ of degree $d-1$ .

For each $n\in \mathbb {N}$ , let $X_n$ be the finite set of abstract variables

$$\begin{align*}X_n:=\{x_t~|~t:[n]\rightarrow [d]\text{ is surjective and monotone}\}. \end{align*}$$

Then by [Reference IyengarIy07, 4.10], there is a free simplicial A-algebra B, denoted as $A[\{x\}|\partial x=\omega ]$ in loc. cit., defined such that $B_n=A_n[X_n]$ .

The lemma below describes the properties we need for the resolution $A[\{x\}|\partial x=\omega ]$ .

In particular the homology class $\bar {\omega }\in \pi _{d-1}(A)$ is killed in B.

Proof. We proceed with the construction by induction as below.

1. Step 1 We first construct a constant simplicial pair, defined by a p-complete polynomial ring of finite variables over ${\mathrm {A_{inf,e}}}$ together with an ideal generated by a subset of variables, such that it maps surjectively onto the pair $(B,I)$ .

   More specifically, by the finiteness assumption, we may pick two finite sets X and Y that correspond to generators of A (over ${\mathrm {A_{inf,e}}}$ as a topological algebra) and I (over A as a module) separately, with surjections of pairs

   $$\begin{align*}(f^{(0)},g^{(0)}):{\mathrm{A_{inf,e}}}\langle X, Y\rangle\rightarrow A, ~\text{and}~(Y)\rightarrow I. \end{align*}$$

   We use the same notation $({\mathrm {A_{inf,e}}}\langle X, Y \rangle , (Y))$ to be the associated constant simplicial algebra with the ideal.

2. Step 2 In this step, building onto the simplicial p-complete polynomial rings constructed in Step 1, we add finitely many more variables on each simplicial term, such that the zeroth homotopy group of the new simplicial pair is isomorphic to $(A,I)$ . Our strategy is to use a p-completed version of Construction 5.2.1 to kill the difference between $(A,I)$ and ${\mathrm {A_{inf,e}}}\langle X, Y\rangle , (Y))$ .

   We now give the construction of the simplicial pair $({\mathrm {A_{inf,e}}}\langle X^{(1)}_{\bullet }, Y \rangle , (Y))$ together with a map

   $$\begin{align*}f^{(1)}:({\mathrm{A_{inf,e}}}\langle X,Y\rangle, (Y))\longrightarrow ({\mathrm{A_{inf,e}}}\langle X^{(1)}_{\bullet},Y\rangle, (Y)), \end{align*}$$

   such that the image of $(\ker (f^{(0)}),\ker (g^{(0)}))$ is killed. First, recall that the ideal $\ker (f^{(0)})$ of ${\mathrm {A_{inf,e}}}\langle X,Y\rangle $ (and hence $\ker (g^{(0)})$ , which by definition is contained in $\ker (f^{(0)})$ ) is a finitely presented ${\mathrm {A_{inf,e}}}\langle X,Y \rangle $ -module ([Reference GuoGuo24, Corollary 5.1.3.(iii)]). So we can pick finite generators $v_1,\ldots ,v_s$ of $\ker (g^{(0)})$ , and additional finitely many elements $u_1,\ldots ,u_r$ to generate $\ker (f^{(0)})$ . Using Construction 5.2.1 we then form a free simplicial algebra of finite variables

   

   over ${\mathrm {A_{inf,e}}}\langle X, Y \rangle $ , such that the images of the following induced maps vanish (Lemma 5.2.2 (iii)):

   $$ \begin{align*} \ker(f^{(0)})&\longrightarrow \pi_0(S_{\bullet}),\\ \ker(g^{(0)})&\longrightarrow \pi_0(S_{\bullet}). \end{align*} $$

   Now we take the term-wise p-adic completion at the simplicial ring $S_{\bullet }$ . Then by Lemma 5.2.2.(ii), we obtain a new simplicial pair over ${\mathrm {A_{inf,e}}}$ , denoted as

   $$\begin{align*}({\mathrm{A_{inf,e}}}\langle X^{(1)}_{\bullet}, Y \rangle, (Y)), \end{align*}$$

   where each $X^{(1)}_n$ is the finite set given by the union of X with the newly added variables from $[\partial U_i=u_i]$ and $[\partial V_j=v_j]$ , following Construction 5.2.1. Moreover, the image of the pair of ideals $(\ker (f^{(0)}),\ker (g^{(0)})$ in $\pi _0\big ({\mathrm {A_{inf,e}}}\langle X^{(1)}_{\bullet }, Y \rangle , (Y)\big )$ vanishes, thanks to the natural composition below

   

   Note that by Lemma 5.2.2 (iii) the zeroth fundamental group of $({\mathrm {A_{inf,e}}}\langle X^{(1)}_{\bullet },Y\rangle , (Y))$ is exactly isomorphic to $(A,I)$ .

3. Step 3 As homotopy groups $\pi _i$ of $(A,I)$ are trivial for $i\geq 1$ , to finish the construction, it suffices to kill the higher homotopy classes of $({\mathrm {A_{inf,e}}}\langle X^{(1)}_{\bullet },Y \rangle , (Y))$ inductively, starting from $\pi _1$ . By the long exact sequence of homotopy groups with respect to the maps

   $$\begin{align*}(Y) \longrightarrow {\mathrm{A_{inf,e}}}\langle X^{(1)}_{\bullet}, Y \rangle \longrightarrow {\mathrm{A_{inf,e}}}\langle X^{(1)}_{\bullet}, Y \rangle/(Y), \end{align*}$$

   it is equivalent to kill the higher homotopy groups of $ {\mathrm {A_{inf,e}}}\langle X^{(1)}_{\bullet }, Y \rangle $ and ${\mathrm {A_{inf,e}}}\langle X^{(1)}_{\bullet }, Y \rangle /(Y)$ . We will use the similar argument as above.

   Before the construction, we make the following observation on the finite generatedness:

   Claim 5.2.4. Let $J=J_{\bullet }$ be a simplicial finitely generated ideal of a simplicial topologically finitely presented p-torsionfree $\mathrm {A_{inf, e}}$ -algebra $A_{\bullet }$ . Then for each $n\in \mathbb {N}$ , the n-th fundamental group $\pi _n(J_{\bullet })$ is a finitely generated $A_n$ -module.

   Granting Claim 5.2.4, we can apply it at the unit ideals in the simplicial rings ${\mathrm {A_{inf,e}}}\langle X^{(n)}_{\bullet },Y \rangle /(Y)$ and ${\mathrm {A_{inf,e}}}\langle X^{(n)}_{\bullet },Y \rangle $ respectively. Then similar to Step 2, by using Construction 5.2.1 and taking the term-wise p-adic completion, we can produce the simplicial pair $({\mathrm {A_{inf,e}}}\langle X^{(n+1)}_{\bullet },Y \rangle , (Y))$ , so that $\pi _n$ of both ${\mathrm {A_{inf,e}}}\langle X^{(n)}_{\bullet },Y\rangle /(Y)$ and ${\mathrm {A_{inf,e}}}\langle X^{(n)}_{\bullet },Y\rangle $ are killed. Note that by Lemma 5.2.2.(ii), for each $i\leq n$ we have the identification of the finite sets

   $$\begin{align*}X_i^{(n+1)} = X_i^{(n)}, \end{align*}$$

   which in particular implies that $\pi _i({\mathrm {A_{inf,e}}}\langle X^{(n+1)}_{\bullet },Y\rangle , (Y)) = \pi _i({\mathrm {A_{inf,e}}}\langle X^{(n)}_{\bullet },Y\rangle , (Y))$ for each $i\leq n$ . Hence by taking the colimit with respect to n, we get a simplicial finitely presented ${\mathrm {A_{inf,e}}}$ -pair

   $$\begin{align*}({\mathrm{A_{inf,e}}}\langle X_{\bullet},Y \rangle, (Y)):=\underset{n\in \mathbb{N}}{{\mathrm{colim}}} ({\mathrm{A_{inf,e}}}\langle X^{(n)}_{\bullet},Y\rangle, (Y)). \end{align*}$$

   Note that by the construction, as $\pi _0({\mathrm {A_{inf,e}}}\langle X_{\bullet },Y \rangle , (Y))=\pi _0({\mathrm {A_{inf,e}}}\langle X_{\bullet },Y \rangle , (Y))=(A,I)$ , the natural map from $ ({\mathrm {A_{inf,e}}}\langle X_{\bullet },Y \rangle , (Y))$ to its $0$ -th fundamental group is a surjective quasi-isomorphism. This finishes the construction we need.

   Proof of the Claim.

   As in [Reference IyengarIy07, 3.4], to compute the i-th fundamental group, we can take the normalization of $J_{\bullet }$ by

   $$\begin{align*}N(J)_n=\cap_{i=1}^n \ker(d_i). \end{align*}$$

   Under the notation, the homotopy group $\pi _n(J)$ is a quotient of the kernel ideal of the $A_n$ -linear map

   $$\begin{align*}\ker(d_0|_{N(J)_n}:N(J)_n\rightarrow N(J)_{n-1}). \end{align*}$$

   We note that the group $N(J)_n$ can be written as the kernel of the $A_n$ -linear map

   $$\begin{align*}J_n\xrightarrow{\oplus_{i=1}^n d_i} \bigoplus_{i=1}^n J_{n-1}, \end{align*}$$

   where the $A_n$ -linear structure on each component of the right side is defined through the ring homomorphism $d_i:A_n\rightarrow A_{n-1}$ . But since $J_n$ is finitely generated, while $J_{n-1}$ is p-torsion free, by [Sta, Tag 0519] and [Reference GuoGuo24, Corollary 5.1.3], the kernel of the map above is finitely generated over $A_n$ . This implies that the p-torsion free $A_n$ -module $N(J)_n$ is finitely presented over $A_n$ . As a consequence, the kernel of the $A_n$ -linear map

   $$\begin{align*}d_0:N(J)_n\longrightarrow N(J)_{n-1}, \end{align*}$$

   with the source being finitely presented and the target being p-torsion free, is finitely generated over $A_n$ . Hence $\pi _n(J_{\bullet })$ is finitely generated over $A_n$ .

Proof. The case for ${\mathrm {B^+_{dR,e}}}$ follows from Theorem 5.2.3 by choosing an integral model of the pair $(B,I)$ over ${\mathrm {A_{inf,e}}}$ , since ${\mathrm {B^+_{dR,e}}}={\mathrm {A_{inf,e}}}[\frac {1}{p}]$ . To get the result for ${\mathrm {B_{dR}^+}}$ -pair $(B,I)$ , we need to adjust the proof of Theorem 5.2.3, changing its base from ${\mathrm {A_{inf,e}}}$ to ${\mathrm {B_{dR}^+}}$ . The Step 1 in Theorem 5.2.3 is identical. For Step 2, we need to find a map

$$\begin{align*}f^{(1)}:({\mathrm{B_{dR}^+}} \langle X, Y\rangle, (Y)) \longrightarrow ({\mathrm{B_{dR}^+}}\langle X^{(1)}_{\bullet},Y\rangle, (Y)), \end{align*}$$

such that the image of $(\ker (f^{(0)}),\ker (g^{(0)}))$ vanishes. Notice that the latter are $\xi $ -complete modules. Let us choose a finite set of generators $\{h_l\}$ of the kernel ideals so that their mod $\xi $ reduction are power bounded (i.e. inside of $\mathcal {O}_K \langle X, Y \rangle $ ), which exist by multiplying a sufficiently large power of p. Then imitating Step 2 in Theorem 5.2.3, we may choose a compatible set of maps

$$\begin{align*}({\mathrm{B^+_{dR,e}}}\langle X,Y \rangle, (Y))\longrightarrow ({\mathrm{B^+_{dR,e}}}\langle X^{(1)}_{\bullet}, Y \rangle, (Y)), \end{align*}$$

such that the image of the mod $\xi ^e$ reduction of $\{h_l\}$ vanishes. Thus by taking the inverse limit with respect to e, we get Step 2.

At last, to modify Step 3 of Theorem 5.2.3, we notice that the finiteness in Claim 5.2.4 holds automatically for ${\mathrm {B_{dR}^+}}$ -pairs the $({\mathrm {B_{dR}^+}}\langle X^{(n)}_{\bullet },Y \rangle , (Y))$ over ${\mathrm {B_{dR}^+}}$ , as each ${\mathrm {B_{dR}^+}}\langle X^{(n)}_i,Y\rangle $ is noetherian. For a given integer n, we fix finitely such many homotopy classes. Then similar to the above modification on Step 2, we can mod powers of $\xi $ and kills the reduction of a fixed finite set of generators for $\pi _n({\mathrm {B_{dR}^+}}\langle X^{(n)}_{\bullet },Y \rangle /(Y))$ and $\pi _n({\mathrm {B_{dR}^+}}\langle X^{(n)}_{\bullet },Y \rangle )$ . A further inverse limit of this compatible system of simplicial pairs finishes the proof.

Proof. We temporarily use the notation $(-)^\wedge _p$ to denote the derived p-completion. Let $R_0$ be a ring of definition of R, which is topologically finite type over ${\mathrm {A_{inf,e}}}$ . By induction, we can find a simplicial topologically finite type ${\mathrm {A_{inf,e}}}$ -algebras $A_{\bullet } \rightarrow R_0$ , such that each $A_n$ is a ring of definition of ${\mathrm {B^+_{dR,e}}} \langle X_n\rangle $ . The triple of rings ${\mathrm {A_{inf,e}}} \rightarrow A_{\bullet } \rightarrow R_0$ then induces the natural triangle of algebraic cotangent complexes

$$\begin{align*}\mathbb{L}_{A_{\bullet}/{\mathrm{A_{inf,e}}}}\otimes_{A_{\bullet}}^L R_0 \longrightarrow \mathbb{L}_{R_0/{\mathrm{A_{inf,e}}}} \longrightarrow \mathbb{L}_{R_0/A_{\bullet}}, \end{align*}$$

where the final term is equal to the homotopy colimit $\underset {[n]\in \Delta }{{\mathrm {colim}}}~ \mathbb {L}_{R_0/A_n}$ . We then apply the derived p-completion at the triangle and then invert at p. By [Sta, Tag 091V], the derived completion commutes with colimits. In particular we get the equalities

$$ \begin{align*} (\mathbb{L}_{R_0/A_{\bullet}})^\wedge_p\left[\frac{1}{p}\right] &= (\underset{[n]\in \Delta}{{\mathrm{colim}}} ~\mathbb{L}_{R_0/A_n})^\wedge_p \left[\frac{1}{p}\right] \\ &\cong \underset{[n]\in \Delta}{{\mathrm{colim}}}~ (\mathbb{L}_{R_0/A_n})^\wedge_p \left[\frac{1}{p}\right] \\ &= \underset{[n]\in \Delta}{{\mathrm{colim}}}~ \mathbb{L}^{\mathrm{an}}_{R/{\mathrm{B^+_{dR,e}}} \langle X_n\rangle}. \end{align*} $$

Note that by assumption the map ${\mathrm {B^+_{dR,e}}} \langle X_n\rangle \rightarrow R$ is a surjection. In particular, by [Reference GuoGuo24, Corollary 5.2.15] we have a natural isomorphism

$$\begin{align*}\mathbb{L}^{\mathrm{an}}_{R/{\mathrm{B^+_{dR,e}}} \langle X_n\rangle} \cong \mathbb{L}_{R/{\mathrm{B^+_{dR,e}}} \langle X_n\rangle}, \end{align*}$$

where the latter is the algebraic cotangent complex. So by taking the colimit, the equalities above together with the assumption that ${\mathrm {B^+_{dR,e}}} \langle X_{\bullet }\rangle $ resolves the ring R, imply the following

$$\begin{align*}(\mathbb{L}_{R_0/A_{\bullet}})^\wedge_p\left[\frac{1}{p}\right] \cong \underset{[n]\in \Delta}{{\mathrm{colim}}}~\mathbb{L}_{R/{\mathrm{B^+_{dR,e}}} \langle X_n\rangle} \cong 0. \end{align*}$$

Thus the triangle at the begining leads to the isomorphism

$$\begin{align*}\left(\mathbb{L}_{A_{\bullet}/{\mathrm{A_{inf,e}}}}\otimes_{A_{\bullet}}^L R_0\right)^\wedge_p\left[\frac{1}{p}\right] \cong (\mathbb{L}_{R_0/{\mathrm{A_{inf,e}}}})^\wedge_p\left[\frac{1}{p}\right], \end{align*}$$

namely the first formula in the statement.

For the equality on ${\widehat {\mathrm {dR}}^{\mathrm {an}}}/{\mathrm {Fil}}^m$ and ${\widehat {\mathrm {dR}}^{\mathrm {an}}}$ , it suffices to notice that each graded piece is functorially given by $L\wedge ^i\mathbb {L}^{\mathrm {an}}[-i]$ . Thus the last two isomorphisms follows from the first one by applying derived wedge powers.

Proof. Consider the category $\mathcal {C}$ of $\mathcal {O}_K$ -linear maps $(\mathcal {O_K} \langle E \rangle , \beta : \mathcal {O_K} \langle E \rangle \rightarrow R_0)$ for finite sets E, and equip its opposite category with the indiscrete topology. Then it suffices to show that any such resolution $\mathcal {O}_K \langle X_{\bullet }\rangle \rightarrow R_0$ is equivalent to the final object of the associated topos. This is shown as in [Sta, 08PS], with the only difference being that we use the p-completeness of $\mathcal {O}_K \langle X_{\bullet }\rangle $ to get the equality

$$\begin{align*}\mathrm{Mor}_C((\mathcal{O}_K \langle E\rangle, \beta), (\mathcal{O}_K \langle X_{\bullet}\rangle, \epsilon)) = \mathrm{Mor}_{\mathrm{Sets}}((E, \beta|_E), (\mathcal{O}_K \langle X_{\bullet}\rangle, \epsilon)). \end{align*}$$

Now we consider the presheaf of complexes over $\mathcal {C}$ sending a given map $\mathcal {O_K} \langle E \rangle \rightarrow R_0$ onto reduced prismatic cohomology together with its Hodge–Tate filtration as in Corollary 5.3.6. Then by the above argument, the global section of is isomorphic to

(∗)

In particular, it is independent of the choice of the simplicial resolution $\mathcal {O}_K \langle X_{\bullet }\rangle \rightarrow R_0$ . Moreover, by Corollary 5.3.6 the i-th graded piece of the Hodge–Tate filtration is equal to the following colimit

$$\begin{align*}\underset{\Delta}{{\mathrm{colim}}} ~\Omega^i_{K\langle X_{\bullet}\rangle/K}(-1)[-1], \end{align*}$$

which by Proposition 5.3.1 is naturally isomorphic to $L\wedge ^i\mathbb {L}^{\mathrm {an}}_{R/K}(-1)[-1]$ . In particular, when R is smooth over K, as the analytic cotangent complex coincides with its continuous differential over K, the graded pieces of the colimit of the Hodge–Tate filtrations in $(\ast )$ is isomorphic to the Hodge–Tate filtration of . So by taking colimits with respect to the exhaustive increasing filtrations of both, we finish the proof of (ii) for reduced prismatic cohomology. At last, the extension of the above to prismatic cohomology of follows from the derived Nakayama’s lemma by the reduction mod $\xi $ .

Proof. By derived Nakayama’s lemma, it suffices to show the second isomorphism. As both reduced cohomologies are étale local, we may assume R admits a regular closed immersion as in Assumption 4.3.1. Namely, we assume that there are surjections $P={\mathrm {B_{dR}^+}} \langle T_i\rangle \rightarrow P/I \rightarrow P/(I,\xi )\cong R$ , where I is a Koszul-regular ideal of P with $P/I$ being flat over ${\mathrm {B_{dR}^+}}$ . We then use Proposition 5.2.5 to find a simplicial resolution of pairs

$$\begin{align*}({\mathrm{B_{dR}^+}} \langle X_{\bullet}, Y_{\bullet}\rangle, (Y_{\bullet})) \longrightarrow (P,I), \end{align*}$$

such that each of $X_n$ and $Y_n$ is a finite set. Note that the induced simplicial K-algebra $K \langle X_{\bullet }\rangle $ resolves $P/\xi =R$ under the quotient map.

To compare two prismatic cohomology, we apply Proposition 5.3.2 at the K-resolution $K \langle X_{\bullet }\rangle \rightarrow R$ to represent derived prismatic cohomology. It then suffices to show the following isomorphism

Notice that the pairs $({\mathrm {B_{dR}^+}} \langle X_n, Y_n\rangle , (Y_n))$ and $(P,I)$ all satisfy Assumption 4.3.1. In particular, we can apply Simpson’s functor as in Theorem 4.3.12 (and in particular Corollary 4.3.16) at the pair $({\mathrm {B_{dR}^+}} \langle X_n, Y_n\rangle , (Y_n))$ to compute , which is functorial with respect to $[n]\in \Delta $ . In particular, by taking the colimit at their reduced prismatic cohomology, we get the following graded morphism

In this way, thanks to the simplicial descent for the analytic cotangent complex in Proposition 5.3.1, we get the isomorphism for the graded pieces, thus the one for reduced prismatic cohomology.

Proof. Let $R_{1,0}$ and $R_{2,0}$ be two topologically finite type $\mathcal {O}_K$ -algebras whose generic fibres are $R_1$ and $R_2$ separately. By [Reference Gabber and RameroGR, Remark 7.2.45], the analytic cotangent complex $\mathbb {L}^{\mathrm {an}}_{R/K}$ is obtained by inverting p at the derived p-completion of the algebraic cotangent complex $\mathbb {L}_{R_0/\mathcal {O}_K}$ , where R is any of $R_1, R_2$ or $R_1\widehat {\otimes }_K R_2$ (respectively for $R_0$ ). Denote by $\mathbb {L}^{\mathrm {an}}_{R_0/\mathcal {O}_K}$ the derived p-completion of the algebraic cotangent complex $\mathbb {L}_{R_0/\mathcal {O}_K}$ . Then we notice that by [Reference Gabber and RameroGR, Proposition 7.1.31], each of $\mathbb {L}^{\mathrm {an}}_{R_0/\mathcal {O}_K}$ for $R_0=R_{1,0}, R_{2,0}$ or $R_{1,0}\widehat {\otimes } R_{2,0}$ is pseudo-coherent. On the other hand, recall the natural isomorphism for the algebraic cotangent complex ([Sta, Tag 09DL]) as below