Proof. By Definition 5, we have \(I \mathbbb{\Delta}^0_{R}(I) \subset p\mathbbb{\Delta}^0_{R}(I)\). Thus, there is a natural homomorphism \(R/I \to \mathbbb{\Delta}^0_{R}(I)/p\), which uniquely extends to the \(p\)-adic completion \(\mathbbb{\Delta}_{R}(I)\). This endows \(\mathbbb{\Delta}_{R}(I)\) with the structure of a crystalline prism over \(R\).

To verify the universal property, suppose we are given a crystalline prism \((B, \delta_B)\) and a \(\delta\)-homomorphism \(f: R \to B\) such that \(f(I) \subset pB\). Because \(B\) is a \(\delta\)-ring and \(p\)-torsion free, the condition \(f(x) \in pB\) for \(x \in I\) implies that \(f\) extends uniquely to a \(\delta\)-homomorphism on the subring \(\mathbbb{\Delta}^0_{R}(I) \subset R[\frac{1}{p}]\) by sending \(\frac{x}{p}\) to \(\frac{f(x)}{p} \in B\). Since \(B\) is \(p\)-complete, this map further extends uniquely to the \(p\)-adic completion \(\mathbbb{\Delta}_{R}(I)\). This establishes the required bijection. ◻

Proof. Suppose \(B\) and \(C\) are two prisms in \((R/A)_{\mathbbb{\Delta}}\). Let \(\bar{J} \subset B/p \otimes_{A/p} C/p\) be the ideal generated by \(\{r\otimes 1 - 1\otimes r : r \in R\}\), and let \(J\) be its preimage in the tensor product \(B \otimes_A C\) (endowed with the tensor Frobenius lift). The canonical homomorphism \(B \otimes_A C / (p, J) \to \mathbbb{\Delta}_{B\otimes_A C}(J)/p\) induces a well-defined map \(R \to \mathbbb{\Delta}_{B\otimes_A C}(J)/p\). By the universal property of prismatic envelopes, \(\mathbbb{\Delta}_{B\otimes_A C}(J)\) represents the coproduct of \(B\) and \(C\) in \((R/A)_{\mathbbb{\Delta}}\). ◻

Proof. See [5]. ◻

Proof. By [5], the right-hand side satisfies the same universal property as the prismatic envelope on the left hand side, hence they are canonically isomorphic. ◻

Proof. Since \(\phi(x_j) \equiv x_j^p \pmod p\), the sequence \(\phi(x_1), \dots, \phi(x_r)\) remains Koszul-regular modulo \(p\) (cf. [6]). The first isomorphism follows directly from Proposition 8 by extending scalars along \(\phi: R \to R\) (which corresponds to changing the base map from \(\tau_1\) to \(\tau_2\)).

For the second isomorphism, see [5]. ◻

Proof. (1) We proceed by induction on \(l\). The base case \(l=1\) is trivial. Assume the claim holds for \(l-1\). By the induction hypothesis, we have: \[\begin{align} d(x^l) &= d(x \cdot x^{l-1}) = \phi(x)d(x^{l-1}) + \phi(x)^{l-1}d(x) + p^n d(x)d(x^{l-1}) \\ &= \phi(x)\left(\sum_{i=0}^{l-2}\binom{l-1}{i}p^{n(l-2-i)}\phi(x)^i d(x)^{l-1-i}\right) + \phi(x)^{l-1}d(x) \\ &\quad + p^n d(x)\sum_{j=0}^{l-2}\binom{l-1}{j}p^{n(l-2-j)}\phi(x)^j d(x)^{l-1-j} \\ &= \sum_{i=0}^{l-1}\binom{l-1}{i-1}p^{n(l-1-i)}\phi(x)^i d(x)^{l-i} + \binom{l-1}{l-1}\phi(x)^{l-1}d(x) \\ &\quad + \sum_{j=0}^{l-2}\binom{l-1}{j}p^{n(l-1-j)}\phi(x)^j d(x)^{l-j} \\ &= \sum_{i=0}^{l-1}\left[\binom{l-1}{i-1} + \binom{l-1}{i}\right]p^{n(l-1-i)}\phi(x)^i d(x)^{l-i} \\ &= \sum_{i=0}^{l-1}\binom{l}{i}p^{n(l-1-i)}\phi(x)^i d(x)^{l-i}. \end{align}\] This completes the inductive step for \(l\).

(2) We proceed by induction on \(r\), with \(r=0\) being trivial. Assume the claim holds for all \(r < r_0\) where \(r_0 > 0\). For \(r=r_0\) and any \(x \in A\): \[\begin{align} d\left(\phi^{r_0}(x)\right) &= d\left(\phi^{r_0-1}\left(\phi(x)\right)\right) = d\left(\phi^{r_0-1}\left(x^p + p\delta(x)\right)\right) \\ &= d\left(\phi^{r_0-1}(x)^p\right) + p d\left(\phi^{r_0-1}\left(\delta(x)\right)\right). \end{align}\] Since \(\delta(x) \in A\), the term \(p d\left(\phi^{r_0-1}\left(\delta(x)\right)\right) \in p^{r_0}A\) by the induction hypothesis. It remains to show \(d\left(\phi^{r_0-1}(x)^p\right) \in p^{r_0}A\). By part (1), we have: \[\label{42} d\left(\phi^{r_0-1}(x)^p\right) = \sum_{i=0}^{p-1}\binom{p}{i}p^{n(p-1-i)}\phi\left(\phi^{r_0-1}(x)\right)^i d\left(\phi^{r_0-1}(x)\right)^{p-i}. \tag{1}\] By the induction hypothesis, \(d\left(\phi^{r_0-1}(x)\right)^{p-i} \in p^{(r_0-1)(p-i)}A\). When \(i=0\), the corresponding term lies in \(p^{p(r_0-1)+n(p-1)}A \subset p^{r_0}A\). When \(0 < i < p\), we have \(p \mid \binom{p}{i}\), and hence the term lies in \(p^{(p-i)(r_0-1)+1}A \subset p^{r_0}A\). Thus, each term in the summation 1 lies in \(p^{r_0}A\), completing the proof. ◻

Proof. (1) Observe that: \[\begin{align} \lambda(r+1,l) &= \frac{1}{p^{l}}\left[\phi_1^{l}\left(\delta_1\left(\delta_1^{r-l}\left(\frac{x}{p}\right)\right)\right) - \phi_2\circ\phi_1^{l-1}\left(\delta_1\left(\delta_1^{r-l}\left(\frac{x}{p}\right)\right)\right)\right]\\ &= \frac{1}{p^{l}}\left[\phi_1^{l}\left(\frac{\phi_1\left(\delta_1^{r-l}\left(\frac{x}{p}\right)\right) - \delta_1^{r-l}\left(\frac{x}{p}\right)^p}{p}\right) - \phi_2\circ\phi_1^{l-1}\left(\frac{\phi_1\left(\delta_1^{r-l}\left(\frac{x}{p}\right)\right) - \delta_1^{r-l}\left(\frac{x}{p}\right)^p}{p}\right)\right]\\ &= \frac{1}{p^{l+1}}\bigg[\phi_1^{l+1}\left(\delta_1^{r-l}\left(\frac{x}{p}\right)\right) - \phi_1^{l}\left(\delta_1^{r-l}\left(\frac{x}{p}\right)\right)^p \\ &\quad - \phi_2\circ\phi_1^{l}\left(\delta_1^{r-l}\left(\frac{x}{p}\right)\right) + \phi_2\circ\phi_1^{l-1}\left(\delta_1^{r-l}\left(\frac{x}{p}\right)\right)^p \bigg]\\ &= \lambda(r+1,l+1) - \frac{1}{p^{l+1}}\left[\phi_1^{l}\left(\delta_1^{r-l}\left(\frac{x}{p}\right)\right)^p - \phi_2\circ\phi_1^{l-1}\left(\delta_1^{r-l}\left(\frac{x}{p}\right)\right)^p\right]. \end{align}\]

We now prove part (2) by induction on \(r\). First note that \[\varphi_1(B_1^m)\subseteq B_1^{m+1} \qquad \text{for every } m\geq 0.\] Indeed, \(\varphi_1(B)\subseteq B\), and for every \(0\leq t\leq m\), \[\varphi_1 \left( \delta_1^t\left(\frac{x}{p}\right) \right) = \delta_1^t\left(\frac{x}{p}\right)^p + p\, \delta_1^{t+1}\left(\frac{x}{p}\right) \in B_1^{m+1}.\] Consequently, \[\varphi_1^l(B_1^m)\subseteq B_1^{m+l} \qquad \text{for all } l,m\geq 0. \]

For \(r=1\), we necessarily have \(l=1\). Since \(\varphi_i(x)=x^p+p\delta_i(x)\), we obtain \[\lambda(1,1) = \frac{1}{p} \left[ \varphi_1\left(\frac{x}{p}\right) - \varphi_2\left(\frac{x}{p}\right) \right] = \frac{\delta_1(x)-\delta_2(x)}{p} \in B = B_1^0.\] Thus, the assertion holds when \(r=1\). Assume now that \(r\geq 2\) and that the assertion has been proved for \(r-1\). We first show that \[\lambda(r,r)\in B.\] Since \(\delta_1\equiv\delta_2\pmod p\), the map \[d := \frac{\varphi_2-\varphi_1}{p^2} : B\longrightarrow B\] is well-defined and additive. Moreover, the multiplicativity of \(\varphi_1\) and \(\varphi_2\) gives \[d(ab) = \varphi_1(a)d(b) + \varphi_1(b)d(a) + p^2d(a)d(b).\] Applying Lemma 2 with \(n=2\), we obtain \[d\left(\varphi_1^{r-1}(x)\right)\in p^{r-1}B.\] Therefore, \[\lambda(r,r) = \frac{1}{p^r} \left[ \varphi_1^r\left(\frac{x}{p}\right) - \varphi_2\circ\varphi_1^{r-1} \left(\frac{x}{p}\right) \right] = - \frac{ d\left(\varphi_1^{r-1}(x)\right) }{ p^{r-1} } \in B \subseteq B_1^{r-1}. \] It remains to prove that \[\lambda(r,l)\in B_1^{r-1} \qquad \text{for } 1\leq l\leq r-1.\] We argue by descending induction on \(l\). Fix such an \(l\) and assume that \[\lambda(r,l+1)\in B_1^{r-1}.\] Set \[z := \delta_1^{r-1-l}\left(\frac{x}{p}\right), \qquad a:=\varphi_1^l(z), \qquad b:=\varphi_2\circ\varphi_1^{l-1}(z).\] By the induction hypothesis on \(r\), we have \[c := \lambda(r-1,l) = \frac{a-b}{p^l} \in B_1^{r-2}.\] Thus, \[a=b+p^lc. \] Since \[z\in B_1^{r-1-l},\] formula (*) implies that \[a=\varphi_1^l(z)\in B_1^{r-1}.\] Moreover, by (***), \[b=a-p^lc\in B_1^{r-1}.\] We now expand: \[\begin{align} \frac{a^p-b^p}{p^{l+1}} &= \frac{(b+p^lc)^p-b^p}{p^{l+1}} \\ &= \sum_{j=1}^{p-1} \frac{1}{p} \binom{p}{j} p^{l(j-1)} b^{p-j}c^j + p^{l(p-1)-1}c^p. \end{align}\] For \(1\leq j\leq p-1\), we have \[\frac{1}{p}\binom{p}{j}\in\mathbf{Z}.\] Furthermore, \[l(p-1)-1\geq 0\] because \(l\geq 1\) and \(p\geq 2\). Since \(b,c\in B_1^{r-1}\), it follows that \[\frac{a^p-b^p}{p^{l+1}} \in B_1^{r-1}.\] Part (1) now yields \[\lambda(r,l) = \lambda(r,l+1) - \frac{a^p-b^p}{p^{l+1}} \in B_1^{r-1}.\] Starting from (**) and descending from \(l=r-1\) to \(l=1\), we conclude that \[\lambda(r,l)\in B_1^{r-1} \qquad \text{for every } 1\leq l\leq r.\] This completes the induction. ◻

Proof of Theorem 10. We proceed by induction on \(r\). The base case \(r=0\) is trivial. Assume the claim holds for \(r-1\). For the inductive step \(r\), by symmetry, it suffices to prove \(\delta^r_1\left(\frac{x}{p}\right) \in B_2^r\). By the construction of \(B_i^r\), we have \(\delta_i(B_i^l) \subset B_i^{l+1}\) for \(i \in \{1,2\}\) and \(l \geq 0\). Thus, by the induction hypothesis, \(\delta_1^{r-1}\left(\frac{x}{p}\right) \in B_2^{r-1}\), whence \(\delta_2\left(\delta_1^{r-1}\left(\frac{x}{p}\right)\right) \in B_2^r\).

Note that for any \(t\): \[\delta_1^r(t) - \delta_2\left(\delta_1^{r-1}(t)\right) = \frac{1}{p}\left[\phi_1\left(\delta_1^{r-1}(t)\right) - \phi_2\left(\delta_1^{r-1}(t)\right)\right].\] Hence, taking \(t = x/p\), we obtain: \[\delta^r_1\left(\frac{x}{p}\right) = \delta_2\circ\delta^{r-1}_1\left(\frac{x}{p}\right) + \lambda(r,1) \in B_2^r\] by Lemma 3(2). ◻

Proof. By Definition 5, the uncompleted prismatic envelope \(\mathbbb{\Delta}^0_{B,\delta_i}(I)\) is the subring of \(B\left[\frac{1}{p}\right]\) generated by \(B\) and all elements of the form \(\delta_i^t\left(\frac{x}{p}\right)\) for \(x \in I\) and \(t \geq 0\).

Applying Theorem 10 and taking the union over all \(r \geq 0\), we see that for any fixed \(x \in I\), the subring generated by \(\{\delta_1^t\left(\frac{x}{p}\right)\}_{t \geq 0}\) coincides exactly with the subring generated by \(\{\delta_2^t\left(\frac{x}{p}\right)\}_{t \geq 0}\) inside \(B\left[\frac{1}{p}\right]\).

Since \(\mathbbb{\Delta}^0_{B,\delta_i}(I)\) is the compositum of these subrings for all generators \(x \in I\), it follows that \[\mathbbb{\Delta}^0_{B,\delta_1}(I) = \mathbbb{\Delta}^0_{B,\delta_2}(I)\] as identical subrings of \(B\left[\frac{1}{p}\right]\). Passing to the \(p\)-adic completions on both sides yields the canonical isomorphism \(\mathbbb{\Delta}_{B,\delta_1}(I) \cong \mathbbb{\Delta}_{B,\delta_2}(I)\). ◻

Proof. This follows directly from the universal property of prismatic envelopes. ◻

Proof. Let \(A_1^m\) and \(A_2^m\) be the \(A\)-subalgebras of \(A\left[\frac{1}{p}\right]\) defined by: \[A_1^m := A\left[\frac{x^{p^i}}{p^{1+p+\dots +p^i}} : x \in I,\;0 \leq i \leq m\right], \quad A_2^m := A\left[\delta^i\left(\frac{x}{p}\right) : 0 \leq i \leq m\right].\] We will prove by induction on \(m\) that \(A_1^m = A_2^m\). Furthermore, we claim that for each \(m\), there exists a sign \(\epsilon\in \{\pm 1\}\) such that \[\frac{x^{p^m}}{p^{1+p+\dots +p^m}} - \epsilon\delta^m\left(\frac{x}{p}\right) \in A_1^{m-1}.\] The base case \(m=0\) is trivial. Assume the claim holds for all integers strictly less than \(m\). It suffices to show: \[\frac{x^{p^m}}{p^{1+p+\dots +p^m}} - \epsilon\delta^m\left(\frac{x}{p}\right) \in A_1^{m-1} (= A_2^{m-1})\] for some \(\epsilon\in \{\pm 1\}\). By the induction hypothesis, there exist \(z \in A_1^{m-2} = A_2^{m-2}\) and \(\epsilon' \in \{\pm 1\}\) such that \[\delta^{m-1}\left(\frac{x}{p}\right) = \epsilon'\frac{x^{p^{m-1}}}{p^{1+p+\dots +p^{m-1}}} + z.\] Applying \(\delta\), we obtain: \[\begin{align} \delta^m\left(\frac{x}{p}\right) &= \delta\left(\epsilon'\frac{x^{p^{m-1}}}{p^{1+p+\dots +p^{m-1}}} + z\right) \\ &= \delta\left(\epsilon'\frac{x^{p^{m-1}}}{p^{1+p+\dots +p^{m-1}}}\right) + \delta(z) - \sum_{j=1}^{p-1}\frac{1}{p}\binom{p}{j}\left(\epsilon'\frac{x^{p^{m-1}}}{p^{1+p+\dots +p^{m-1}}}\right)^j z^{p-j}. \end{align}\] Since \(z \in A_2^{m-2}\), we have \(\delta(z) \in A_2^{m-1} = A_1^{m-1}\). Moreover, powers of \(\frac{x^{p^{m-1}}}{p^{1+p+\dots +p^{m-1}}}\) belong to \(A_1^{m-1}\). Thus, we only need to show: \[\delta\left(\epsilon'\frac{x^{p^{m-1}}}{p^{1+p+\dots +p^{m-1}}}\right) - \epsilon\frac{x^{p^m}}{p^{1+p+\dots +p^m}} \in A_1^{m-1} = A_2^{m-1}\] for some \(\epsilon\in \{\pm 1\}\). The assumption \(\phi(I) \subset I\) implies \(\delta(I) \subset I\), allowing us to expand: \[\begin{align} \delta\left(\epsilon'\frac{x^{p^{m-1}}}{p^{1+p+\dots +p^{m-1}}}\right) &= \frac{1}{p}\left[ \epsilon'\phi\left(\frac{x^{p^{m-1}}}{p^{1+p+\dots +p^{m-1}}}\right) - (\epsilon')^p\left(\frac{x^{p^{m-1}}}{p^{1+p+\dots +p^{m-1}}}\right)^p \right] \\ &= \epsilon'\frac{\phi(x)^{p^{m-1}}}{p^{1+1+p+\dots +p^{m-1}}} + (-(\epsilon')^p)\frac{x^{p^m}}{p^{1+p+\dots +p^m}}. \end{align}\] Recall that \(\phi(x) = x^p + p\delta(x) = p^2\left(p^{p-2}\left(\frac{x}{p}\right)^p + \frac{\delta(x)}{p}\right) \in p^2 A_1^0 = p^2 A_2^0\). Consequently: \[\frac{\phi(x)^{p^{m-1}}}{p^{1+1+p+\dots +p^{m-1}}} \in p^{2p^{m-1} - (1+1+p+\dots +p^{m-1})}A_1^0 \subset A_1^0.\] Setting \(\epsilon= -(\epsilon')^p\) completes the inductive step. ◻

Proof of Theorem 14. By Corollary 11, we may assume without loss of generality that \(\delta_1 = \delta_2\). Then \(J\) is a \(\phi\)-invariant ideal in the \(p\)-torsion free quotient \(R \widehat{\otimes}_A R\). By Lemma 5 and the definition of the prismatic envelope, we have: \[R_{1,2} = \left( R\widehat{\otimes}_A R\left[\frac{x^{p^m}}{p^{1+p+\dots +p^m}} : x \in J\right] \right)^\wedge.\]

Consider the continuous homomorphism \(R\left[Y_1, \dots, Y_d\right]_{\mathrm{pd}}^\wedge \to R_{1,2}\) defined by \(a \mapsto a \otimes 1\) for \(a \in R\) and \(Y_i \mapsto \frac{1\otimes X_i - X_i\otimes 1}{p}\). To show this is an isomorphism, it suffices by completeness to verify it modulo \(p\), i.e., that the map \[\label{con:42} R\left[Y_1, \dots, Y_d\right]_{\mathrm{pd}} \to R\widehat{\otimes}_A R\left[\frac{x^{p^m}}{p^{1+p+\dots +p^m}} : x \in J\right]\tag{2}\] induces an isomorphism modulo \(p\).

The proof of this reduction essentially follows Wang [3]. The key observation is the isomorphism modulo \(p\): \[\label{con:4242} R\left[Y_1, \dots, Y_d\right]/p \cong R\widehat{\otimes}_A R\left[\frac{x}{p} : x \in J\right]/p,\tag{3}\] where \(Y_i \mapsto \frac{1\otimes X_i - X_i\otimes 1}{p}\). This follows straightforwardly from the étaleness of \(\square\). Under the isomorphism 3 , the ideal \((Y_1, \dots, Y_d)\) on the left corresponds precisely to the ideal generated by \(\{\frac{x}{p} : x \in J\}\) on the right. Taking the PD-envelopes of these corresponding ideals yields the desired isomorphism 2 . ◻

Proof. By Corollary 11, we may again assume \(\delta_1 = \delta_2\). Then, combining Corollary 13 and Theorem 14, we obtain: \[P_{1,2} \cong \mathbbb{\Delta}_{R_{1,2}}\left(\mathfrak{a}R_{1,2}\right) \cong \mathbbb{\Delta}_{R\left[Y_1, \dots, Y_d\right]_{\mathrm{pd}}^\wedge}\left(\mathfrak{a}R\left[Y_1, \dots, Y_d\right]_{\mathrm{pd}}^\wedge\right) \cong P_1\left[Y_1, \dots, Y_d\right]_{\mathrm{pd}}^\wedge.\] Here, the final isomorphism follows from the flat base change of prismatic envelopes. ◻

Proof. Uniqueness follows immediately from the linearity of \(p\mathrm{d}\). Let \(P^0 = \Delta_R^0(\mathfrak{a})\). We first extend \(p\mathrm{d}\) to \(R\left[\frac{1}{p}\right]\) and show that \(p\mathrm{d}(P^0) \subset P^0 \otimes_R \widehat{\Omega}_{R/A}\).

We proceed by lexicographic induction on the pair \((n, j)\), where \(n = i+j\). We claim that for any \(x \in \mathfrak{a}\), \[p^{-i+1}\mathrm{d}\left(\phi^i\left(\delta^j\left(\frac{x}{p}\right)\right)\right) \in P^0 \otimes_R \widehat{\Omega}_{R/A}.\]

For \(j=0\), the claim holds by a straightforward secondary induction on \(i\).

Assume the claim holds for all pairs \((i', j')\) such that either \(i' + j' \le \lambda\), or \(i' + j' = \lambda + 1\) with \(j' < j\). Now consider a pair \((i, j)\) where \(i+j = \lambda+1\) and \(j>0\). Using the identity \(\delta(t) = \frac{\phi(t) - t^p}{p}\) and the fact that \(\phi\) is a ring homomorphism, we compute: \[\begin{align} p^{-i+1}\mathrm{d}\left(\phi^i\left(\delta^j\left(\frac{x}{p}\right)\right)\right) &= p^{-i+1}\mathrm{d}\left(\phi^i\left(\frac{\phi\left(\delta^{j-1}\left(\frac{x}{p}\right)\right) - \delta^{j-1}\left(\frac{x}{p}\right)^p}{p}\right)\right) \\ &= p^{-i}\mathrm{d}\left(\phi^{i+1}\left(\delta^{j-1}\left(\frac{x}{p}\right)\right)\right) - p^{-i+1}\phi^i\left(\delta^{j-1}\left(\frac{x}{p}\right)\right)^{p-1}\mathrm{d}\left(\phi^i\left(\delta^{j-1}\left(\frac{x}{p}\right)\right)\right). \end{align}\]

The first term belongs to \(P^0 \otimes_R \widehat{\Omega}_{R/A}\) by the inductive hypothesis for \((i+1, j-1)\), since \((i+1)+(j-1) = \lambda+1\) and \(j-1 < j\).

The second term also belongs to \(P^0 \otimes_R \widehat{\Omega}_{R/A}\) because \(\phi^i\left(\delta^{j-1}\left(\frac{x}{p}\right)\right)^{p-1} \in P^0\), and the remaining factor \(p^{-i+1}\mathrm{d}\left(\phi^i\left(\delta^{j-1}\left(\frac{x}{p}\right)\right)\right)\) satisfies the inductive hypothesis for \((i, j-1)\), where \(i + (j-1) = \lambda < \lambda+1\).

In particular, taking \(i=0\) yields \(p\mathrm{d}\left(\delta^j\left(\frac{x}{p}\right)\right) \in P^0 \otimes_R \widehat{\Omega}_{R/A}\), completing the proof. ◻

Proof. By Lemma 6, the operator \(p\mathrm{d}\) extends uniquely to a continuous derivation \(p\mathrm{d}: P \to \widehat{\Omega}_P\). We can explicitly define its extension to the \(k\)-th exterior power \(P \otimes_R \widehat{\Omega}^k_{R/A}\) via the Leibniz rule: \[p\mathrm{d}(f \otimes \omega) = p\mathrm{d}(f) \wedge \omega + f \otimes p\mathrm{d}(\omega)\] for any \(f \in P\) and \(\omega \in \widehat{\Omega}^k_{R/A}\). This map is well-defined over \(R\) precisely because \(p\mathrm{d}\) acts as a derivation on both \(P\) and \(R\). The \(p\)-adic continuity and uniqueness of this higher extension follow immediately from the continuity and uniqueness of the base derivation on \(P\). ◻

Proof. Consider the following commutative diagram (with solid arrows):

By the étaleness of \(\square\), there is a unique dashed arrow lifting the identity on \(R\) that makes the diagram commute. Since \(X_i \otimes 1 + pY_i = 1 \otimes X_i\), the continuous map \(f \mapsto \sum_{\alpha \in \mathbb{N}^d} p^{|\alpha|}\partial_{\alpha}(f)\underline{Y}^{[\alpha]}\) provides exactly such a lift. By uniqueness, the claim follows. ◻

Proof. This functor is defined by evaluating a given crystal \(\mathbb{H}\) on \(X\) to obtain the module \(M := \mathbb{H}(X)\). The stratification isomorphism \(\epsilon\) is naturally induced by the transposition morphism \(X \times X \to X \times X\) sending \((x,y)\) to \((y,x)\). ◻

Proof. By [7], the quotient \(P/p\) is isomorphic to the derived prismatic cohomology \(\mathbbb{\Delta}_{B/R} \otimes^{\mathbf{L}}_R R/p\). Consider the conjugate filtration on \(\mathbbb{\Delta}_{B/R} \otimes^{\mathbf{L}}_R R/p\). By the Hodge–Tate comparison theorem, each graded piece of the conjugate filtration is flat over \(B\). Consequently, \(P/p \cong \mathbbb{\Delta}_{B/R} \otimes^{\mathbf{L}}_R R/p\) is flat over \(B\).

Given this flatness, the remainder of the proof follows identically to the arguments found in [1], [5], or [2]. ◻

Proof. By definition, the map \(\sigma^0_0: P^1 \to P\) acts by evaluating at \(\underline{Y}_1 = 0\). Thus \(\sigma^0_0 \otimes \epsilon\) is exactly \(\theta_{\underline{0}}\), and the equivalence (1) follows immediately.

Under the identifications established above, we compute the three pullbacks of \(\epsilon\) to \(P^2\): \[\begin{align} id_{P^2} \otimes_{p^1_2, P^1} \epsilon(\underline{Y}_1^{[u]} \underline{Y}_2^{[v]} m) &= \sum_{\alpha \in \mathbb{N}^d} \theta_\alpha(m) \underline{Y}_1^{[\alpha]} \underline{Y}_1^{[u]} \underline{Y}_2^{[v]}, \\ id_{P^2} \otimes_{p^1_0, P^1} \epsilon(\underline{Y}_1^{[u]} \underline{Y}_2^{[v]} m) &= \sum_{\alpha \in \mathbb{N}^d} \theta_\alpha(m) (\underline{Y}_2 - \underline{Y}_1)^{[\alpha]} \underline{Y}_1^{[u]} \underline{Y}_2^{[v]}, \\ id_{P^2} \otimes_{p^1_1, P^1} \epsilon(\underline{Y}_1^{[u]} \underline{Y}_2^{[v]} m) &= \sum_{\alpha \in \mathbb{N}^d} \theta_\alpha(m) \underline{Y}_2^{[\alpha]} \underline{Y}_1^{[u]} \underline{Y}_2^{[v]}. \end{align}\] Therefore, the cocycle descent condition is equivalent to: \[\begin{align} &\sum_{\alpha \in \mathbb{N}^d} \sum_{\beta \in \mathbb{N}^d} \theta_\beta(\theta_\alpha(m)) (\underline{Y}_2 - \underline{Y}_1)^{[\beta]} \underline{Y}_1^{[\alpha]} = \sum_{\alpha \in \mathbb{N}^d} \theta_\alpha(m) \underline{Y}_2^{[\alpha]} \\ \Longleftrightarrow \quad & \sum_{\alpha \in \mathbb{N}^d} \theta_\alpha(m) \underline{Y}_2^{[\alpha]} = \sum_{\alpha, \beta \in \mathbb{N}^d} \theta_\beta(\theta_\alpha(m)) \sum_{k+l=\beta} (-1)^{|l|} \underline{Y}_2^{[k]} \underline{Y}_1^{[l]} \underline{Y}_1^{[\alpha]} \\ \Longleftrightarrow \quad & \theta_\gamma(m) = \sum_{\alpha, \beta \in \mathbb{N}^d} (-1)^{|\beta|} \theta_{\beta+\gamma}(\theta_\alpha(m)) \underline{Y}_1^{[\alpha]} \underline{Y}_1^{[\beta]} \quad \text{for all } \gamma \in \mathbb{N}^d. \label{con:000} \end{align}\tag{4}\]

On the other hand, since \(\epsilon\) is \(P^1\)-linear, we have: \[\begin{align} \epsilon(fm) &= \sum_{\alpha \in \mathbb{N}^d} \theta_\alpha(fm) \underline{Y}_1^{[\alpha]} \\ &= \left( \sum_{\alpha \in \mathbb{N}^d} p^{|\alpha|} \partial_{\alpha}(f) \underline{Y}_1^{[\alpha]} \right) \left( \sum_{\alpha \in \mathbb{N}^d} \theta_\alpha(m) \underline{Y}_1^{[\alpha]} \right) \\ &= \sum_{\alpha \in \mathbb{N}^d} \underline{Y}_1^{[\alpha]} \sum_{\beta \leq \alpha} p^{|\beta|} \binom{\alpha}{\beta} \partial_{\beta}(f) \theta_{\alpha-\beta}(m). \end{align}\] Comparing coefficients, \((M,\epsilon)\) is a stratification if and only if \(\theta_{\underline{0}} = \mathrm{id}_M\), the descent condition 4 holds, and for any \(\gamma \in \mathbb{N}^d\), \(f\in P\), and \(m \in M\), we have: \[\label{con:001} \theta_{\gamma}(fm) = \sum_{\beta \leq \gamma} p^{|\beta|} \binom{\gamma}{\beta} \partial_{\beta}(f) \theta_{\gamma-\beta}(m).\tag{5}\]

Taking \(\gamma = e_i\), equation 5 simplifies to \(\theta_{e_i}(fm) = f\theta_{e_i}(m) + p\partial_i(f)m\). By comparing the coefficients of \(\underline{Y}_1^{[e_i]}\) in 4 , we obtain \(\theta_{\gamma+e_i} = \theta_{\gamma} \circ \theta_{e_i}\). Setting \(\gamma = e_j\), this yields the commutativity \(\theta_{e_i} \circ \theta_{e_j} = \theta_{e_j} \circ \theta_{e_i}\). By induction, we deduce \(\theta_{\gamma} = \prod_{i=1}^d \theta_{e_i}^{\gamma_i}\).

It remains to verify the converse: that the commutativity \(\theta_{e_i} \circ \theta_{e_j} = \theta_{e_j} \circ \theta_{e_i}\) and the product formula \(\theta_{\gamma} = \prod_{i=1}^d \theta_{e_i}^{\gamma_i}\) imply both 4 and 5 . Equation 4 follows from a straightforward combinatorial cancellation: \[\begin{align} \sum_{\alpha,\beta \in \mathbb{N}^d} (-1)^{|\beta|} \theta_{\beta+\gamma}(\theta_{\alpha}(m)) \underline{Y}_1^{[\alpha]} \underline{Y}_1^{[\beta]} &= \sum_{\alpha,\beta \in \mathbb{N}^d} (-1)^{|\beta|} \underline{\theta}^{\beta} \underline{\theta}^\gamma \underline{\theta}^\alpha \binom{\alpha+\beta}{\alpha} \underline{Y}_1^{[\alpha+\beta]} \\ &= \theta_{\gamma} \circ \sum_{\alpha,\beta \in \mathbb{N}^d} (-1)^{|\beta|} \underline{\theta}^{\beta} \underline{\theta}^\alpha \binom{\alpha+\beta}{\alpha} \underline{Y}_1^{[\alpha+\beta]} \\ &= \theta_\gamma(m). \end{align}\] Equation 5 follows by induction on \(|\gamma|\). With the convention that \(\binom{a}{b}=0\) for \(b<0\), we have: \[\begin{align} \theta_{\gamma+e_i}(fm) &= \theta_{\gamma}\left(f\theta_{e_i}(m) + p\partial_i(f)m\right) \\ &= \sum_{\beta \leq \gamma} p^{|\beta|} \binom{\gamma}{\beta} \partial_{\beta}(f) \theta_{\gamma-\beta}(\theta_{e_i}(m)) + \sum_{\beta \leq \gamma} p^{|\beta+e_i|} \binom{\gamma}{\beta} \partial_{\beta+e_i}(f) \theta_{\gamma-\beta}(m) \\ &= \sum_{\beta \leq \gamma} p^{|\beta|} \left[ \binom{\gamma+e_i}{\beta} - \binom{\gamma}{\beta-e_i} \right] \partial_{\beta}(f) \theta_{\gamma+e_i-\beta}(m) \\ &\quad + \sum_{e_i \leq \beta \leq \gamma+e_i} p^{|\beta|} \binom{\gamma}{\beta-e_i} \partial_{\beta}(f) \theta_{\gamma-\beta+e_i}(m) \\ &= \sum_{\beta \leq \gamma+e_i} p^{|\beta|} \binom{\gamma+e_i}{\beta} \partial_{\beta}(f) \theta_{\gamma+e_i-\beta}(m). \end{align}\] This completes the proof. ◻

Proof. (1) Let \(M\) be a finitely generated projective \(P\)-module. By Proposition 26, endowing \(M\) with a stratification \(\epsilon\) along \(P^\bullet\) is equivalent to specifying a family of \(A\)-linear endomorphisms \(\{\theta_\alpha\}_{\alpha \in \mathbb{N}^d}\) satisfying the given descent conditions. Specifically, the stratification is uniquely determined by the degree-one operators \(\theta_{e_i}: M \to M\). The relation \(\theta_{e_i}(fm) = f\theta_{e_i}(m) + p\partial_i(f)m\) precisely means that the operators \(\theta_{e_i}\) define a \(p\)-connection on \(M\) over \(P\). The commutativity condition \(\theta_{e_i} \circ \theta_{e_j} = \theta_{e_j} \circ \theta_{e_i}\) is equivalent to the integrability of this \(p\)-connection. Finally, the product formula \(\theta_\gamma = \prod_{i=1}^d \theta_{e_i}^{\gamma_i}\) combined with the decay condition (\(\theta_\alpha \to 0\) \(p\)-adically as \(|\alpha| \to \infty\)) is exactly the definition of topological quasi-nilpotence for the \(p\)-connection. This establishes the first equivalence.

(2) By Theorem 24, the Koszul-regularity assumption ensures that the prismatic envelope \(P\) is a cover of the final object in the prismatic site \((B/A)_{\mathbbb{\Delta}}\). Consequently, Theorem 23 guarantees that the natural evaluation functor \(\mathbb{M}_{\mathbbb{\Delta}}\) induces an equivalence of categories between prismatic crystals on \((B/A)_{\mathbbb{\Delta}}\) and stratifications along the Čech nerve \(P^\bullet\). Composing this equivalence with the one established in part (1) yields the functor \(\mathbb{D}_{\mathbbb{\Delta}}\). Since both intermediate equivalences preserve the underlying locally free modules, the composed equivalence is manifestly rank-preserving. ◻

Proof. For any \(a\in R\), the class \([a\otimes1-1\otimes a]\in J/J^{[2]}\) lies in \(pJ/J^{[2]}\). Since \(J/J^{[2]}\) is a free \(P\)-module, there is a unique element \(x\in J/J^{[2]}\) such that \(px=[a\otimes1-1\otimes a]\). The assignment \([a\otimes 1-1\otimes a]\mapsto x\) is \(R\)-linear and hence induces a homomorphism \(P\otimes_R I/I^2\to J/J^{[2]}\). The calculations on a toric chart above show that this is an isomorphism. ◻

Proof. By Proposition 26, the stratification isomorphism is given by the expansion \[\epsilon(m) = \sum_{\alpha \in \mathbb{N}^d} \theta_\alpha(m) \underline{Y}_1^{[\alpha]}.\] Modulo \(J^{[2]}\), all higher-degree terms with \(|\alpha| \ge 2\) vanish. Therefore, the projection of \(\epsilon(m) - 1\otimes_{p^0_0} m\) into \(J/J^{[2]} \otimes_P M\) is precisely the degree-one part \(\sum_{i=1}^d \theta_{e_i}(m) \otimes \underline{Y}_1^{[e_i]}\). Under the isomorphism \(J/J^{[2]} \cong \Omega_P\) from Lemma 7, the basis element \(\underline{Y}_1^{[e_i]}\) corresponds exactly to \(dX_i\). Thus, we obtain \(\theta'(m) = \sum_{i=1}^d \theta_{e_i}(m) \otimes dX_i\), which is exactly the definition of the connection \(1\)-form \(\theta(m)\) associated to the operators \(\{\theta_{e_i}\}\). ◻

Proof. We follow the explicit calculation of [3]; the case of the structure crystal is treated in [3]. Write \[e_i:=p^{-1}\mathrm{d}X_i,\qquad \eta_{j,i}:=\mathrm{d}Y_{j,i}.\] Then \(\widehat\Omega^1_{P^r}\) is freely generated over \(P^r\) by the \(e_i\) and the \(\eta_{j,i}\) for \(1\leq j\leq r\). If \(p_a^r:P^r\to P^{r+1}\) is a coface map with \(a>0\), then it preserves the zeroth vertex. Hence it does not change the chosen trivialization of the crystal, and for \(m\in \mathcal{M}\) one has \[p_a^r(m)=m,\qquad p_a^r(e_i)=e_i.\] The variables \(Y_{j,i}\) are sent to the corresponding variables described by the Čech formulas above. Therefore the equality \[p_a^r\circ \mathrm{d}_2^{r,s}=\mathrm{d}_2^{r+1,s}\circ p_a^r\] follows from the ordinary functoriality of PD-differentials and from the Leibniz rule.

The only nontrivial coface is \(p_0^r\); more generally, the same calculation treats any order-preserving map \(f:[r]\to[r']\) with \(f(0)\ne0\). Under the identification \[\mathbb{H}(P^r)\cong P^r\otimes_{q_0,P}\mathcal{M},\] the map \(p_0^r\) is obtained by changing the base vertex from \(0\) to \(1\), and therefore acts through the stratification. If \[\epsilon(m)=\sum_{\alpha\in\mathbb{N}^d}\theta_\alpha(m)\underline Y_1^{[\alpha]},\] then \[p_0^r(m)=\sum_{\alpha\in\mathbb{N}^d}\theta_\alpha(m)\underline Y_1^{[\alpha]}.\] Moreover, \[p_0^r(e_i)=e_i+\eta_{1,i},\qquad p_0^r(Y_{j,i})=Y_{j+1,i}-Y_{1,i}.\] Equivalently, for a general order-preserving map \(f\) with \(f(0)\ne0\), the stratification gives \[f(m)=\sum_{\alpha\in\mathbb{N}^d}\theta_\alpha(m)\underline Y_{f(0)}^{[\alpha]}, \qquad f(e_i)=e_i+\mathrm{d}Y_{f(0),i}.\] It suffices to check the equality on \(m\in\mathcal{M}\), because both sides satisfy the same graded Leibniz rule with respect to \(P^r\) and the exterior algebra. Since the \(p\)-connection is given by \[\nabla(m)=\sum_{i=1}^d\theta_{e_i}(m)e_i,\] Proposition 26 gives \(\theta_\alpha=\theta_{e_1}^{\alpha_1}\cdots\theta_{e_d}^{\alpha_d}\) and the endomorphisms \(\theta_{e_i}\) commute. Thus \[\begin{align} \mathrm{d}_2^{r+1,0}\bigl(p_0^r(m)\bigr) &=\mathrm{d}_2^{r+1,0}\left(\sum_{\alpha}\theta_\alpha(m)\underline Y_1^{[\alpha]}\right)\\ &=\sum_{\alpha,i}\theta_{e_i}\theta_\alpha(m)\underline Y_1^{[\alpha]}e_i +\sum_{\alpha,i}\theta_\alpha(m)\underline Y_1^{[\alpha-e_i]}\eta_{1,i}\\ &=\sum_{\alpha,i}\theta_{\alpha+e_i}(m)\underline Y_1^{[\alpha]}e_i +\sum_{\alpha,i}\theta_{\alpha+e_i}(m)\underline Y_1^{[\alpha]}\eta_{1,i}\\ &=\sum_i\left(\sum_{\alpha}\theta_\alpha(\theta_{e_i}(m))\underline Y_1^{[\alpha]}\right)(e_i+\eta_{1,i})\\ &=p_0^r\left(\sum_i\theta_{e_i}(m)e_i\right) =p_0^r\bigl(\mathrm{d}_2^{r,0}(m)\bigr). \end{align}\] Here \(\underline Y_1^{[\alpha-e_i]}\) is interpreted as \(0\) if \(\alpha_i=0\), and in the third line we reindex \(\alpha\) by \(\alpha+e_i\). This is exactly the cancellation in the Taylor expansion: the terms coming from differentiating the divided-power variables \(\underline Y_1^{[\alpha]}\) are absorbed by the extra summand \(\eta_{1,i}\) in \(p_0^r(e_i)\).

The same computation after wedging with an arbitrary form \(\omega\) proves \[p_0^r\circ \mathrm{d}_2^{r,s}=\mathrm{d}_2^{r+1,s}\circ p_0^r\] for all \(s\). Therefore \(\mathrm{d}_2^{\bullet,s}\) is a morphism of cosimplicial modules. Finally, \(\mathrm{d}_1^2=0\) is the usual cosimplicial identity, \(\mathrm{d}_2^2=0\) follows from the integrability of the \(p\)-connection and from \(\mathrm{d}_P^2=0\), and the equality just proved gives \(\mathrm{d}_1\mathrm{d}_2+\mathrm{d}_2\mathrm{d}_1=0\) with the standard total-complex sign convention. ◻

Proof. We give the Čech calculation explicitly. Put \[K^{r,s}:=\mathbb{H}(P^r)\otimes_{P^r}\widehat\Omega^s_{P^r}.\] The first assertion is equivalent to saying that the quotient bicomplex \[K^{\bullet,>0}:=\bigoplus_{s>0}K^{\bullet,s}[-s]\] is acyclic for the horizontal Čech differential. It is enough to check this after choosing the trivialization \[\mathbb{H}(P^r)\cong P^r\otimes_P\mathcal{M},\] because \(\mathcal{M}\) is finite projective and the transition maps are given by the stratification.

We first treat \(s=1\). The Čech complex of the degree-one differential modules is generated by the elements \[e_i=p^{-1}\mathrm{d}X_i,\qquad \eta_{j,i}=\mathrm{d}Y_{j,i}.\] The key identity is already visible in degree one: \[p_0^0(e_i)-p_1^0(e_i)=(e_i+\eta_{1,i})-e_i=\eta_{1,i}.\] Hence the relative differential \(\eta_{1,i}\) in degree \(1\) is a Čech boundary. More generally, \[\eta_{j,i}=q_j(e_i)-q_0(e_i),\] where \(q_j:P\to P^r\) denotes the map corresponding to the \(j\)-th vertex. This is the same computation as in the standard infinitesimal Čech complex of differentials (cf. [8]). It gives an explicit contracting homotopy on the normalized Čech complex of \(\widehat\Omega^1_{P^\bullet}\): on a normalized generator represented by \(\eta_{j,i}\), the homotopy sends it to an alternating sum of the vertex differentials \(q_\ell(e_i)\) whose Čech boundary is \(\eta_{j,i}\). Thus the row \[K^{\bullet,1}:\quad K^{0,1}\to K^{1,1}\to K^{2,1}\to \cdots\] is null-homotopic.

More explicitly, [8] applies to the cosimplicial module \[N_i^r:=\bigoplus_{0\leq j\leq r}P^r e_{j,i}, \qquad f(e_{j,i})=e_{f(j),i},\] and \(\widehat\Omega^1_{P^r}\) is the quotient of the direct sum of these \(N_i^r\) by the constant summands generated by the diagonal differentials. Since Example 2.16 contracts \(N_i^\bullet\), the quotient by the constant part contracts the relative generators \(\eta_{j,i}=e_{j,i}-e_{0,i}\). This is the precise degree-one null-homotopy used above.

For higher differential degree, we use that \(\widehat\Omega^s_{P^r}\) is the derived exterior power \(\wedge^s\widehat\Omega^1_{P^r}\) and that the modules involved are finite projective over the corresponding \(P^r\). The contracting homotopy in degree one is compatible with the cosimplicial algebra structure and therefore extends, by the usual derivation rule, to the exterior algebra. This is the same passage from degree one to all degrees as in [2]. Consequently the Čech complex \(K^{\bullet,s}\) is null-homotopic for every \(s>0\). This proves that \[(\mathbb{H}(P^\bullet),\mathrm{d}_1^{\bullet,0})\longrightarrow \mathrm{Tot}_{\mathbb{H}}\] is a quasi-isomorphism.

It remains to compare the total complex with the zeroth column. We use the filtration by the horizontal Čech degree. The associated graded pieces are the de Rham complexes of the formal PD-polynomial algebras \[P^r\simeq P[\underline Y_1,\dots,\underline Y_r]_{\mathrm{pd}}^\wedge\] with coefficients in the pullback of \((\mathcal{M},p^{-1}\nabla)\). The relative part in the variables \(\underline Y_j\) is the standard PD-de Rham complex \[P[\underline Y_1,\dots,\underline Y_r]_{\mathrm{pd}}^\wedge \to \bigoplus_{j,i}P[\underline Y_1,\dots,\underline Y_r]_{\mathrm{pd}}^\wedge\mathrm{d}Y_{j,i}\to\cdots,\] which is contractible by the usual PD Poincaré homotopy. In one variable the relative PD-de Rham complex is \[P\langle Y\rangle_{\mathrm{pd}}^\wedge \xrightarrow{\mathrm{d}} P\langle Y\rangle_{\mathrm{pd}}^\wedge \mathrm{d}Y,\] and the contracting homotopy of the positive-degree part is given explicitly by \[h\left(Y^{[n]}\mathrm{d}Y\right)=Y^{[n+1]},\qquad h\left(Y^{[n]}\right)=0.\] Indeed, \(\mathrm{d}(Y^{[n+1]})=Y^{[n]}\mathrm{d}Y\). For several variables the relative PD-de Rham complex is the completed tensor product of these one-variable complexes; equivalently, one applies the above homotopy successively to the first differential variable appearing in a monomial \[\underline Y^{[\alpha]}\mathrm{d}Y_{j_1,i_1}\wedge\cdots\wedge \mathrm{d}Y_{j_t,i_t}.\] This gives a homotopy between the identity and the projection to the constant term. This is the standard calculation used in [2]. Hence the projection \(P^r\to P\) induces a quasi-isomorphism from the column in Čech degree \(r\) to the de Rham complex over \(P\).

The compatibility of these homotopies with the coface maps identifies the resulting horizontal complex with the constant Čech complex attached to \(\mathrm{DR}(\mathcal{M},p^{-1}\nabla)\); this constant Čech complex is contracted by the extra degeneracy inserting the zeroth vertex. Therefore the inclusion \[\mathrm{DR}(\mathcal{M},p^{-1}\nabla)\longrightarrow \mathrm{Tot}_{\mathbb{H}}\] is also a quasi-isomorphism. Using the identification of \(\mathrm{DR}(\mathcal{M},p^{-1}\nabla)\) with \(\mathrm{DR}(\mathcal{M},\nabla)\) explained above gives the desired statement. ◻

Proof. Smoothness is stable under base change. The relative Frobenius \[\phi_{R/A}:R^{(1)}\to R,\qquad a\otimes x\mapsto ax^p,\] is faithfully flat because \(R\) is smooth over \(A\). Since \[\phi_{R/A}(f_i^{(1)})=f_i^p,\] and powers of a Koszul-regular sequence are Koszul-regular, the Koszul complex of \(f_1^{(1)},\ldots,f_r^{(1)}\) becomes acyclic in positive degrees after the faithfully flat base change \(\phi_{R/A}\). Hence, it is already acyclic in positive degrees over \(R^{(1)}\). ◻

Proof. The assertion is local on \(S\). We may therefore choose elements \(f_1,\ldots,f_r\in I_T:=\ker (S\to B)\) whose images form a Koszul-regular sequence in \(S/p\) and generate \(I_T/pS\). Let \(A\{x_1,\ldots,x_r\}\) be the \(\delta\)-polynomial algebra over \(A\), and define two \(\delta\)-\(A\)-homomorphisms \[o_1:A\{\underline{x}\}\to S^{(1)},\qquad x_i\mapsto 1\otimes f_i,\] \[o_2:A\{\underline{x}\}\to S,\qquad x_i\mapsto \phi_S(f_i).\] These are compatible with the relative Frobenius \[\phi_{S/A}:S^{(1)}\to S,\qquad 1\otimes s\mapsto \phi_S(s),\] in the sense that \(o_2=\phi_{S/A}\circ o_1\). Let \(A\{y_1,\ldots,y_r\}\) be the \(\delta\)-\(A\{\underline{x}\}\)-algebra defined by \(x_i\mapsto py_i\). By Proposition 8 and Corollary 9, we have canonical identifications \[P_T\cong S^{(1)}\widehat\otimes_{A\{\underline{x}\},o_1}A\{\underline{y}\},\] \[D_T\cong S\widehat\otimes_{A\{\underline{x}\},o_2}A\{\underline{y}\}.\] Since \(o_2=\phi_{S/A}\circ o_1\), the second ring is also \[D_T\cong S\widehat\otimes_{S^{(1)},\phi_{S/A}}P_T.\] Under this identification, \(\Phi_T\) is the completed base change of \(\phi_{S/A}\), and the square \[\xymatrix{ S^{(1)}\ar[r]\ar[d]_{\phi_{S/A}}&P_T\ar[d]^{\Phi_T}\\ S\ar[r]&D_T }\] is a pushout square of \(p\)-complete rings. The claim now follows from the fact that \(\phi_{S/A}\) is finite faithfully flat. ◻

Proof. It is enough to check compatibility with the tensor-product relation. Let \(g\in P_T\). Since \(\nabla\) is a \(p\)-connection, \[\nabla(gm)=g\nabla(m)+p\,m\otimes \mathrm{d}g.\] Applying \(\mathrm{id}_M\otimes\overline{\mathrm{d}}\Phi_T\) and using \(\mathrm{d}\Phi_T=p\,\overline{\mathrm{d}}\Phi_T\), we get \[\begin{align} \nabla_\Phi(b\otimes gm) &=b\Phi_T(g)(\mathrm{id}_M\otimes\overline{\mathrm{d}}\Phi_T)(\nabla m) +(1\otimes m)\otimes b\,\mathrm{d}(\Phi_T(g)) +(1\otimes m)\otimes \Phi_T(g)\mathrm{d}b\\ &=\nabla_\Phi(b\Phi_T(g)\otimes m). \end{align}\] The ordinary Leibniz rule over \(D_T\) follows immediately from the term \((1\otimes m)\otimes \mathrm{d}b\).

We next verify integrability directly. The connection \(\nabla_\Phi\) is the sum of the standard connection on the first factor \(D_T\) and the \(D_T\)-linear operator obtained from \(\nabla\) by the \(P_T\)-linear map \[\Omega_{P_T}\xrightarrow{\overline{\mathrm{d}}\Phi_T}\Omega_{D_T}.\] Since \(\mathrm{d}\circ\mathrm{d}\Phi_T=0\) and \(\mathrm{d}\Phi_T=p\,\overline{\mathrm{d}}\Phi_T\), the map \(\overline{\mathrm{d}}\Phi_T\) is compatible with exterior differentials. Therefore the curvature of the \(D_T\)-linear part is the image of the curvature of \(\nabla\), which is zero. The mixed terms are exactly the exterior differential of the coefficient forms \(\overline{\mathrm{d}}\Phi_T(\omega)\) and cancel by the same compatibility. Hence \(\nabla_\Phi^2=0\).

Finally we check topological quasi-nilpotence using the definition in [6]. Thus it is enough to show that, for each local section \(b\otimes m\), sufficiently high iterates of every local differential operator send \(b\otimes m\) into \(p(D_T\otimes_{P_T}M)\). Choose local coordinates \(t_1,\ldots,t_d\) on \(S\) over \(A\). Write \[\nabla(m)=\sum_i\theta_i(m)\otimes \mathrm{d}(1\otimes t_i),\qquad \overline{\mathrm{d}}\Phi_T(\mathrm{d}(1\otimes t_i))=\sum_j c_{ij}\mathrm{d}t_j,\] with \(c_{ij}\in D_T\). Then, with respect to the basis \(\mathrm{d}t_j\), the connection operators of \(\nabla_\Phi\) have the form \[\partial'_j=\partial_j^{D_T}+\sum_i c_{ij}(1\otimes\theta_i).\] Let \(b\otimes m\) be a local section. By the same definition applied to the standard connection on the \(p\)-complete PD-envelope \(D_T\), for each \(j\) there are only finitely many \(k\) such that \((\partial_j^{D_T})^k(b)\notin pD_T\). For the \(p\)-connection on \(M\), the operators \(\theta_i\) are topologically quasi-nilpotent, so for each \(m\) there are only finitely many pairs \((i,k)\) such that \(\theta_i^k(m)\notin pM\).

Expand \((\partial'_j)^N(b\otimes m)\) as a sum over words of length \(N\) in the two types of operators \[\partial_j^{D_T},\qquad c_{ij}(1\otimes\theta_i).\] Fix a bound \(N_M\) such that any word in the finitely many operators \(\theta_i\) of length at least \(N_M\) sends \(m\) into \(pM\). Such a bound exists because the \(\theta_i\) commute by integrability, and for each \(i\) some high power \(\theta_i^{N_i}\) sends \(m\) into \(pM\); one can take \(N_M> \sum_i(N_i-1)\). It remains to consider words containing fewer than \(N_M\) operators of the second type. For such a word, the coefficient of the resulting vector is obtained from \(b\) and from fewer than \(N_M\) of the elements \(c_{ij}\) by applying a total of at least \(N-N_M\) copies of \(\partial_j^{D_T}\). By the quasi-nilpotence of the standard connection on the \(p\)-complete PD-envelope \(D_T\) applied to the finite set consisting of \(b\) and the \(c_{ij}\), this coefficient lies in \(pD_T\) once \(N\) is sufficiently large. Hence, for \(N\gg 0\), every summand of \((\partial'_j)^N(b\otimes m)\) lies in \(p(D_T\otimes_{P_T}M)\). Therefore only finitely many pairs \((j,N)\) can satisfy \[(\partial'_j)^N(b\otimes m)\notin p(D_T\otimes_{P_T}M),\] which is precisely the condition in [6]. ◻

Proof. Let \((M,\nabla)\) be the \(p\)-connection corresponding to a prismatic stratification \(\epsilon\) on \(M\). The stratified construction gives a PD-stratification on \(D_T\otimes_{\Phi_T,P_T}M\) by base changing \(\epsilon\) along the functorial maps \(\Phi\) for the two-fold presentation \(S^{[1]}\to B\).

Both the connection \(\nabla_\Phi\) of Definition 33 and this base-changed stratification are integrable and topologically quasi-nilpotent, so it suffices to compare their first-order parts. Let \(J_{\mathbbb{\Delta}}\) be the diagonal ideal of the first prismatic neighborhood of \(P_T\), and let \(J_{\mathrm{crys}}\) be the diagonal PD-ideal of the first crystalline neighborhood of \(D_T\). Under the usual identifications \[J_{\mathbbb{\Delta}}/J_{\mathbbb{\Delta}}^{[2]}\cong \Omega_{P_T}, \qquad J_{\mathrm{crys}}/J_{\mathrm{crys}}^{[2]}\cong \Omega_{D_T},\] the map induced by the divided Frobenius is exactly \[\overline{\mathrm{d}}\Phi_T:\Omega_{P_T}\to\Omega_{D_T}.\] Hence the first-order part of the base-changed stratification sends \(b\otimes m\) to \[b\cdot(\mathrm{id}_M\otimes\overline{\mathrm{d}}\Phi_T)(\nabla m) +(1\otimes m)\otimes \mathrm{d}b,\] which is precisely the formula defining \(\nabla_\Phi\). This proves the compatibility. ◻

Proof. Let \((N,\eta)\) be an object of \(\mathbf{Strat}_{\mathrm{crys}}(D_T)\). Put \[D_T'{}^n:=D_T\widehat\otimes_{P_T}\cdots\widehat\otimes_{P_T}D_T \qquad (n+1\text{ factors}).\]

First, we construct a cosimplicial homomorphism \[u_T^\bullet:D_T^\bullet\longrightarrow D_T'{}^\bullet.\] Using the pushout description in Lemma 9, we identify \[D_T'{}^n\cong S\widehat\otimes_{S^{(1)}}\cdots\widehat\otimes_{S^{(1)}}S \widehat\otimes_{S^{(1)}}P_T,\] with \(n+1\) copies of \(S\). Equivalently, grouping the last copy of \(S\) with \(P_T\) gives \[D_T'{}^n\cong S\widehat\otimes_{S^{(1)}}\cdots\widehat\otimes_{S^{(1)}}S \widehat\otimes_{S^{(1)}}D_T,\] with \(n\) copies of \(S\) before the final factor \(D_T\). Let \[I_n:=\ker\left(S^{\widehat\otimes_{S^{(1)}}(n+1)}\to S\right)\] be the relative Frobenius diagonal ideal. This ideal has the standard \(p\)-adically continuous PD-structure: in an étale toric chart it is generated by the relative Frobenius diagonal differences, whose divided powers are \(p\)-adically integral, and the assertion follows by étale descent from this chart calculation. Hence \(I_nD_T'{}^n\) is a PD-ideal.

Let \[\Delta_n:D_T'{}^n\to D_T\] be the diagonal map. If \(J_T\subset D_T\) denotes the PD-ideal defining \(B\), then under the second description of \(D_T'{}^n\) one has \[\label{eq:preimage-pd-ideal-T} \Delta_n^{-1}(J_T)= I_nD_T'{}^n+ S\widehat\otimes_{S^{(1)}}\cdots\widehat\otimes_{S^{(1)}}S \widehat\otimes_{S^{(1)}}J_T,\tag{6}\] with \(n\) copies of \(S\) before the final factor \(J_T\subset D_T\). Indeed, after quotienting by \(I_nD_T'{}^n\), the diagonal identifies the quotient with \(D_T\), and the second summand maps onto \(J_T\). The first summand in 6 is a PD-ideal by the preceding paragraph, and the second is the completed base change of the PD-ideal \(J_T\subset D_T\). Their sum is a PD-ideal by the usual formula \[\gamma_m(x+y)=\sum_{a+b=m}\gamma_a(x)\gamma_b(y).\] Therefore the composite \[S^{[n]}\longrightarrow D_T'{}^n\] sends the kernel of \(S^{[n]}\to B\) into a PD-ideal. By the universal property of the \(p\)-complete PD-envelope \(D_T^n\), it extends uniquely to \[u_T^n:D_T^n\to D_T'{}^n.\] The uniqueness gives compatibility with coface and codegeneracy maps, so the \(u_T^n\) form \(u_T^\bullet\).

Next, we descend the underlying module. Pulling the PD-stratification \(\eta\) back along \(u_T^\bullet\) gives a faithfully flat descent datum on the \(D_T\)-module \(N\) relative to the finite faithfully flat map \(\Phi_T:P_T\to D_T\). Hence there is a finite projective \(P_T\)-module \(M\), unique up to unique isomorphism, and an isomorphism \[D_T\otimes_{\Phi_T,P_T}M\xrightarrow{\sim}N\] identifying the canonical descent datum with the pulled-back stratification.

We now descend the PD-stratification. For every \(n\), apply the same construction to the presentation \(S^{[n]}\to B\). The map \[\Phi_T^n:P_T^n\to D_T^n\] is finite faithfully flat by Lemma 9. The pullback of \(N\) to \(D_T^n\) descends uniquely to a finite projective \(P_T^n\)-module. By uniqueness of faithfully flat descent and compatibility with the coface maps, this descended module is the pullback of \(M\) to \(P_T^n\). The isomorphisms defining \(\eta\) therefore descend to isomorphisms between the corresponding pullbacks of \(M\) over \(P_T^1\), and the cocycle condition over \(P_T^2\) follows after the faithfully flat base change to \(D_T^2\). Thus \(M\) acquires a prismatic stratification \(\epsilon\).

By construction, \(C^{-1}_{T,\mathrm{str}}(M,\epsilon)\cong(N,\eta)\), so the functor is essentially surjective. Full faithfulness is the same descent argument applied to morphisms: a morphism between two objects after applying \(C^{-1}_{T,\mathrm{str}}\) is a \(D_T\)-linear map compatible with the pulled-back descent data along \(D_T'{}^\bullet\), hence descends uniquely to a \(P_T\)-linear map. Compatibility with stratifications descends because it can be checked after the faithfully flat base change \(\Phi_T^n\) in each cosimplicial degree. Therefore \(C^{-1}_{T,\mathrm{str}}\) is an equivalence.

The equivalence with the connection-theoretic statement follows from Lemma 11. ◻

Proof of Theorem 32. The construction above is local on \(X\). Choose an affine open \(\mathrm{Spec}(B)\subset X\) and a presentation \(T=(S\twoheadrightarrow B)\) in \(\mathcal{P}_B\). Theorem 35 gives a canonical equivalence \[\mathbf{Crys}_{(B^{(1)}/A)_{\mathbbb{\Delta}}}\xrightarrow{\sim}\mathbf{Crys}_{(B/A)_{\mathrm{crys}}}.\] This equivalence is independent of the chosen presentation. If \(T=(S\twoheadrightarrow B)\) and \(T'=(S'\twoheadrightarrow B)\) are two choices, then \[T''=(S\widehat\otimes_A S'\twoheadrightarrow B)\] dominates both. The functoriality of the maps \(\Phi_T\) and of the descent construction identifies the equivalences obtained from \(T\) and \(T'\) with the one obtained from \(T''\).

The equivalence is also compatible with restriction to principal opens. If \(f\in B\) and \(\tilde{f}\in S\) is a lift, then the \(p\)-complete localization \(S\langle \tilde{f}^{-1}\rangle\) is again a smooth \(p\)-complete \(\delta\)-\(A\)-algebra presenting \(B_f\). The prismatic envelopes, PD-envelopes, the maps \(\Phi_T\), and the descent data are compatible with this localization. Hence the local equivalences agree on overlaps.

We may therefore glue the equivalences over an affine open cover of \(X\). Since the Frobenius twist is compatible with restriction to affine opens and the local-complete-intersection condition is preserved by the lemma at the beginning of this subsection, the glued functor gives a canonical equivalence \[\mathbf{Crys}_{(X^{(1)}/A)_{\mathbbb{\Delta}}}\cong \mathbf{Crys}_{(X/A)_{\mathrm{crys}}}.\] ◻

Proof. Since \(D\) is \(p\)-torsion free, so is \(D[1/f]\). The Frobenius lift \(\phi\) on \(D\) extends to \(D[1/f]\) because \[\phi(f)=f^p+p\delta(f)=f^p\left(1+\frac{p\delta(f)}{f^p}\right)\] is invertible in the \(p\)-complete localization: the second factor is a \(p\)-adic unit. Passing to the \(p\)-adic completion gives a continuous Frobenius lift on \(\widehat D_f\). Since \(\widehat D_f\) is \(p\)-torsion free, the formula \[\delta(x)=\frac{\phi(x)-x^p}{p}\] uniquely defines a compatible \(\delta\)-structure on \(\widehat D_f\). ◻

Proof. Both sides satisfy the same universal property. Namely, for any crystalline prism \((B,\delta_B)\), a map from either side to \(B\) is equivalent to a \(\phi\)-compatible map \(R\to B\) which sends \(\mathfrak{a}\) into \(pB\) and sends \(f\) to a unit of \(B\); the latter condition is the universal property of \(p\)-complete localization. This canonically identifies the two completed prismatic envelopes. ◻

Proof. For (1), choose elements \(f_{r+1},\dots,f_s\in I\) such that \[f_1R+\cdots+f_sR=R.\] By Lemma 13, the usual sheaf property for the affine formal cover defined by \(f_1,\dots,f_s\) gives the equalizer diagram \[0\to \mathbbb{\Delta}_R(I)\to \prod_{i=1}^s\mathbbb{\Delta}_{\widehat R_{f_i}}(I\widehat R_{f_i})\rightrightarrows \prod_{i,j=1}^s\mathbbb{\Delta}_{\widehat R_{f_if_j}}(I\widehat R_{f_if_j}).\] If \(g\in I\), then \(I\widehat R_g=\widehat R_g\), so the prismatic envelope \(\mathbbb{\Delta}_{\widehat R_g}(I\widehat R_g)\) is the zero ring. Removing these zero terms gives the asserted equalizer for \(f_1,\dots,f_r\). For (2), choose principal open covers of \(U\cap\mathfrak{X}=U'\cap\mathfrak{X}\) which can be written both as \(\mathrm{Spf}(\widehat R_{f_i})\cap\mathfrak{X}\) and as \(\mathrm{Spf}(\widehat R'_{g_i})\cap\mathfrak{X}\). By Lemma 13, the corresponding localized prismatic envelopes are canonically isomorphic. Applying (1) to the two Čech diagrams shows that \(\mathbbb{\Delta}_R(I)\) and \(\mathbbb{\Delta}_{R'}(I')\) are equalizers of canonically isomorphic diagrams, hence are canonically isomorphic. ◻

Proof. Choose an affine formal open cover \[Y=\bigcup_{\lambda\in\Lambda}U_\lambda,\qquad U_\lambda=\mathrm{Spf}(R_\lambda),\] such that the given Frobenius lift modulo \(p^2\) lifts on each \(R_\lambda\) to a Frobenius lift \(\phi_\lambda:R_\lambda\to R_\lambda\). Let \(I_\lambda\subset R_\lambda\) be the ideal defining \(\mathfrak{X}\cap U_\lambda\). On each \(\mathfrak{X}\cap U_\lambda\) we first define a sheaf, still denoted \(\mathbbb{\Delta}_{R_\lambda}(I_\lambda)\). For a principal open \(V=\mathrm{Spec}((R_\lambda/I_\lambda)_{\bar f})\subset \mathfrak{X}\cap U_\lambda\), choose a lift \(f\in R_\lambda\) of \(\bar f\) and set \[\Gamma(V,\mathbbb{\Delta}_{R_\lambda}(I_\lambda)):=\mathbbb{\Delta}_{\widehat{R_{\lambda,f}}}(I_\lambda\widehat{R_{\lambda,f}}).\] Lemma 13 and Proposition 36 show that this definition is independent of the choice of \(f\) and satisfies the sheaf condition on the basis of principal opens. Thus it defines a sheaf on \(\mathfrak{X}\cap U_\lambda\) whose global sections are \(\mathbbb{\Delta}_{R_\lambda}(I_\lambda)\). It remains to glue these local sheaves. Let \(\mathrm{Spf}(C)\subset U_{\lambda_1}\cap U_{\lambda_2}\) be an affine formal open, and let \(J\subset C\) be the ideal defining \(\mathfrak{X}\cap\mathrm{Spf}(C)\). Let \(\phi_1,\phi_2:C\to C\) be the restrictions of the two local Frobenius lifts. Since both lift the same Frobenius modulo \(p^2\), we have \[\phi_1\equiv\phi_2\pmod{p^2},\qquad\text{equivalently}\qquad \delta_1\equiv\delta_2\pmod p.\] Corollary 11 gives a canonical isomorphism \[\mathbbb{\Delta}_{C,\phi_1}(J)\cong \mathbbb{\Delta}_{C,\phi_2}(J).\] By Proposition 36, these are precisely the restrictions of the two local sheaves to \(\mathfrak{X}\cap\mathrm{Spf}(C)\). Hence the displayed isomorphisms provide descent data on overlaps. The cocycle condition on triple overlaps follows from the canonicity in Corollary 11: all transition maps are induced by identifying the same \(p\)-complete subring of \(C[1/p]\) generated by the elements \(\delta^n(x/p)\). Therefore the local sheaves glue to a sheaf \(\mathbbb{\Delta}_Y(\mathfrak{X})\) on \(\mathfrak{X}\). Property (1) is built into the construction. Property (2) follows from the localization isomorphism of Lemma 13, and property (3) follows because the sheaf was defined on the basis of principal opens by \(p\)-complete rings whose reductions modulo \(p\) are compatible with the same principal localization. ◻

Proof. The injectivity of \(\tau\) and \(\tau'\) follows from the separatedness of the sheaf \(\mathbb{H}\) applied to the covers \(P_{1,2}\to P_2\) and \(P_{1,1}\to P_1\). Let \[\beta:P_{1,2}\xrightarrow{\sim}P_{1,1},\qquad \beta_*:P_2\xrightarrow{\sim}P_1,\qquad \beta':P_{1,1,2}\xrightarrow{\sim}P_{1,1,1}\] be the canonical isomorphisms coming from Corollary 11. These isomorphisms are compatible with all insertions of factors; for instance \[\nu_{*,1,1}\circ\beta=\nu_{*,1,2}\circ\beta', \qquad \nu_{1,*,1}\circ\beta=\nu_{1,*,2}\circ\beta', \qquad \beta\circ\nu_{*,2}=\nu_{*,1}\circ\beta_*.\] Thus we have a morphism between the two equalizer diagrams defining the sheaf condition: \[\xymatrix{ 0\ar[r]& P_2\ar[d]^-{\beta_*}\ar[r]&P_{1,2}\ar[d]^-{\beta}\ar@<0.5ex>[r]\ar@<-0.5ex>[r]&P_{1,1,2}\ar[d]^-{\beta'}\\ 0\ar[r]& P_1\ar[r]&P_{1,1}\ar@<0.5ex>[r]\ar@<-0.5ex>[r]&P_{1,1,1}. }\] Applying the crystal \(\mathbb{H}\) and using the crystal identifications gives the corresponding diagram for the \(M\)’s. It follows that \(\psi\circ\tau(M_2)\) and \(\tau'(M_1)\) are the same equalizer inside \(M_{1,1}\). This proves (1). The same diagram also proves \(P_2\)-linearity of \(\iota_{2,1}\) after identifying \(P_2\) with \(P_1\) through \(\beta_*\). This proves (2). For (3), consider the stratification maps obtained by pulling \(M_2\) and \(M_1\) to \(P_{2,2}\) and \(P_{1,1}\). The canonical identifications of the coproducts give an isomorphism \[M_{2,2}\xrightarrow{\sim}M_{1,1}\] obtained through the chain \[M_{2,2}\cong P_{2,2}\otimes_{P_2}M_2 \cong P_{1,2}\otimes_{P_2}M_2 \cong M_{1,2} \cong P_{1,2}\otimes_{P_1}M_1 \cong M_{1,1}.\] The compatibility of the canonical ring isomorphisms with insertion maps gives a commutative diagram \[\xymatrix{ M_2\ar[d]\ar[r]^-{\iota_{2,1}}&M_1\ar[d]\\ M_{2,2}\ar[r]^-{\sim}&M_{1,1}. }\] This is exactly the assertion that \(\iota_{2,1}\) respects the two stratifications. Finally, (4) follows from the same functoriality for triple choices of \(\delta\)-structures. Over the common triple coproduct \(P_{1,2,3}\), the three maps \(\iota_{3,2}\), \(\iota_{2,1}\), and \(\iota_{3,1}\) are all induced by the same canonical identifications of prismatic envelopes. Hence their composites agree. ◻

Proof. Choose an affine open cover of \(\mathfrak{X}\) by opens which admit toric charts and local lifts of \(\phi_Y\) to the corresponding smooth formal affine pieces of \(Y\). On each such open, Theorem 27 gives an equivalence between prismatic crystals and integrable topologically quasi-nilpotent \(p\)-connections on the local prismatic envelope. On an overlap, two choices of Frobenius lift are congruent modulo \(p^2\) because they both lift the fixed \(\phi_Y\) on \(Y/p^2\). Therefore the associated \(\delta\)-structures are congruent modulo \(p\). Theorem 40 identifies the corresponding local modules together with their stratifications, hence identifies the local \(p\)-connections. On triple overlaps, part (4) of Theorem 40 gives the cocycle condition. Thus the local equivalences glue to an equivalence over the sheaf \(\mathbbb{\Delta}_Y(\mathfrak{X})\). Rank preservation is local on \(\mathfrak{X}\), so the glued equivalence is rank preserving. ◻

Proof. The statement is local on \(\mathfrak{X}_{\mathrm{\text{\'et}}}\). On an affine chart where the Frobenius lift has been chosen, the object \(P^0\) is a cover of the final object of the local prismatic topos by Theorem 24. Therefore its Čech nerve \(P^\bullet\) computes the derived global sections of the crystal. More explicitly, this is the same Čech–Alexander calculation as [2]: the local prismatic envelope is a cover of the final object, and the higher cohomology of a crystal on each term of the Čech nerve vanishes, so the Čech totalization computes \(R\nu_*\mathbb{H}\). The local identifications are independent of the chosen lift of Frobenius by Theorem 40. Hence they agree on overlaps and glue to the asserted global quasi-isomorphism. ◻

Proof of Theorem 42. By Lemma 14, it remains to identify \(\mathrm{Tot}(\mathbb{H}^\bullet)\) with \(\mathrm{DR}(\mathcal{M},\nabla)\). This can be checked on an affine open of \(\mathfrak{X}\) admitting a toric chart and a lift of \(\phi_Y\). On such an open, Theorem 29 gives a canonical quasi-isomorphism \[\mathrm{Tot}(\mathbb{H}(P^\bullet))\simeq \mathrm{DR}(\mathcal{M},\nabla).\] The proof of Theorem 29 gives an explicit bicomplex comparison. It is modeled on Tian’s Čech–de Rham calculation and Wang’s local calculation: the horizontal Čech direction computes the crystal by [2], while the vertical direction contracts to the de Rham complex by the local calculation of Theorem 31. If two local Frobenius lifts are chosen on an overlap, Theorem 40 identifies the corresponding prismatic envelopes, the evaluated modules, and the stratifications defining the \(p\)-connections. Under these identifications the local bicomplexes used in Theorem 29 agree term by term. Consequently the local quasi-isomorphisms agree on overlaps. The cocycle condition on triple overlaps again follows from part (4) of Theorem 40. Therefore the local quasi-isomorphisms glue to a canonical global quasi-isomorphism \[R\nu_*\mathbb{H}\simeq \mathrm{DR}(\mathcal{M},\nabla).\] ◻