# Why is Fontaine's infinitesimal period ring $A_{\text{inf}}$ complete?

Fix a perfectoid field $$K$$ in mixed characteristic with ring of integers $$\mathcal{O}$$ and pseudo-uniformizer $$\varpi$$. Its tilt is the fraction field of $$\mathcal{O}^{\flat}=\varprojlim_{x\mapsto x^{p}}\mathcal{O}/\varpi$$. Pick an element $$\pi\in\mathcal{O}^{\flat}$$ such that $$\pi^{\sharp}/p\in\mathcal{O}^{\times}$$.

I want to understand Fontaine's infinitesimal period ring $$A_{\text{inf}}:=W(\mathcal{O}^{\flat}).$$ Many references claim that it is complete with respect to the $$(p,[\pi])$$-adic topology. However, I was not able to find a reference for this statement by myself. I would be grateful if someone could provide a full proof here.

• I don't know the original reference, but it is covered in BMS1 Lemma 3.2.
– Z. M
Commented Dec 26, 2022 at 18:15
• @Z. M I am aware of this Lemma. But I am not sure how exactly it gives an answer to my question. Commented Dec 27, 2022 at 8:18
• Erratum: the above reference to BMS1 does not seem to cover this proposition.
– Z. M
Commented Dec 27, 2022 at 13:49

We first remark that $$(p,[\pi])$$ is a regular sequence in $$\newcommand\Ainf{A_{\operatorname{inf}}}\Ainf$$, since $$p$$ is a non-zero-divisor and $$\pi$$ does not vanish in the integral domain $$\Ainf/p=\mathcal O^\flat$$. Thus the ring $$\Ainf$$ is $$(p,[\pi])$$-adically complete if and only if it is derived $$(p,[\pi])$$-complete, and since the ring $$\Ainf$$ is already derived $$p$$-complete, it suffices to check that it is derived $$[\pi]$$-complete.
Let $$\theta\colon\Ainf\to\mathcal O$$ denote the Fontaine's map, whose kernel $$I$$ is principal. Since the ring $$\Ainf$$ is derived $$I$$-complete (and in fact, $$I$$-adically complete), cf. [Hesselholt–Nikolaus, Topological Cyclic Homology, Prop 1.3.4], thus we are reduced to check that the ring $$\mathcal O$$ is derived $$\theta([\pi])$$-complete, but $$\theta([\pi])=\pi^\sharp$$ by definition, and the result follows from the derived $$p$$-completeness of $$\mathcal O$$.
• Many thanks! Just one question: Why is $A_{\text{inf}}$ $(p,[\pi])$-adically complete if and only if it is derived $(p,[\pi])$-adically complete? Commented Jan 2, 2023 at 11:18