Let $K$ be a discretely valued extension of $\mathbb{Q}_p$ with perfect residue field $k$, and $\mathcal{C}$ a completed algebraic closure of $K$ with the ring of integers $\mathcal{O}_{\mathcal{C}}$. Denote by $\mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p)$ the p-adic completion of the divided power envelope of $W(\mathcal{O}_{\mathcal{C}}/p) \to \mathcal{O}_{\mathcal{C}}/p$. We known that $\mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p)$ is isomorphic to Fontaine's period ring $\mathbb{A}_{\text{cry}}$.

Denote by $\varphi :\mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p) \to \mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p)$ the Frobenius endomorphism induced from the absolute Frobenius of $\mathcal{O}_{\mathcal{C}}/p$, and by $\phi$ the Frobenius endomorphism of Fontaine's period ring $\mathbb{A}_{\text{cry}}$.

(1) Is the isomorphism $\mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p)\simeq \mathbb{A}_{\text{cry}}$ compatible with Frobenius endomorphisms?

Define the decreasing Nygaard filtration on $\mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p)$ by $$ \mathcal{N}^{\geq i}\mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p) =\{x\in \mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p)~|~\varphi (x) \in p^i \mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p)\} $$ for $i\geq 0$.

(2) What is the completion of $\mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p)$ with respect to the filtration?

Let $K$ be a discretely valued extension of $\mathbb{Q}_p$ with perfect residue field $k$, and $\mathcal{C}$ a completed algebraic closure of $K$ with the ring of integers $\mathcal{O}_{\mathcal{C}}$. Denote by $\mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p)$ the p-adic completion of the divided power envelope of $W(\mathcal{O}_{\mathcal{C}}/p) \to \mathcal{O}_{\mathcal{C}}/p$. We know that $\mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p)$ is isomorphic to Fontaine's period ring $\mathbb{A}_{\text{cry}}$.

Denote by $\varphi :\mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p) \to \mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p)$ the Frobenius endomorphism induced from the absolute Frobenius of $\mathcal{O}_{\mathcal{C}}/p$, and by $\phi$ the Frobenius endomorphism of Fontaine's period ring $\mathbb{A}_{\text{cry}}$.

(1) Is the isomorphism $\mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p)\simeq \mathbb{A}_{\text{cry}}$ compatible with Frobenius endomorphisms?

Define the decreasing Nygaard filtration on $\mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p)$ by $$ \mathcal{N}^{\geq i}\mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p) =\{x\in \mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p)~|~\varphi (x) \in p^i \mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p)\} $$ for $i\geq 0$.

(2) What is the completion of $\mathbb{A}_{\text{cry}}(\mathcal{O}_{\mathcal{C}}/p)$ with respect to the filtration?