User:Dkmiller271/sandbox

Here's my take on the "right" definition of $$\mathbf{B}_{\text{dR}}$$, following Fontaine's Le corps des périodes $$p$$-adiques. One starts with a category that looks very deformation-theoretic. Suppose $$\mathcal{O}$$ is a ring, and $$k$$ is a $$\mathcal{O}$$-algebra that is complete in the $$p$$-adic topology. The category $$\mathcal{C}$$ consists of all $$p$$-adically complete $$\mathcal{O}$$-algebras $$A$$ with a surjection $$\theta:A\to k$$ such that $$A$$ is also complete in the $$(\ker\theta)$$-adic topology. The traditional example would be $$\mathcal{O}=\mathbb{Z}_p$$, $$k=\mathbb{F}_p$$. In that context, one would be interested in the representability of certain deformation functors from $$\mathcal{C}$$ to sets. Here, Fontaine is only interested in initial objects of $$\mathcal{C}$$. The key theorem is that if $$k$$ is "pseudo-perfectoid" in the sense that Frobenius is surjective on $$k/p$$, then $$\mathcal{C}$$ has an initial object, which is constructed in the following way. Start with $$ k^\flat = \varprojlim_{x\mapsto x^p} k/p $$ the notation $$k^\flat$$ follows Scholze (who has a general "tilting functor" in the context of perfectoid spaces). The general theory of perfectoid spaces gives a multiplicative (but not additive) map $$\sharp:k^\flat \to k$$, such that the composite $$k^\flat\xrightarrow\sharp k\twoheadrightarrow k/p$$ is a ring homomorphism. It is a general fact about Witt vectors that we get an induced map $$\theta:W(k^\flat) \to k$$. If one writes $$\mathbf{A}_{\text{inf}}(k/\mathcal{O})$$ for the completion of the base change $$\Lambda\otimes W(k^\flat)$$ with respect to the kernel of the induced map $$ \theta:\Lambda\otimes W(k^\flat) \to k $$ then $$\mathbf{A}_{\text{inf}}(k/\mathcal{O})$$ is an initial object in $$\mathcal{C}$$. Fontaine writes $$\mathbf{A}_{\text{inf}}$$ for $$\mathbf{A}_{\text{inf}}(\mathcal{O}_{\mathbb{C}_p}/\mathbb{Z}_p)$$. So there is a homomorphism $$\theta:\mathbf{A}_{\text{inf}}\to\mathcal{O}_{\mathbb{C}_p}$$. The ring $$\mathbf{B}_{\text{dR}}^+$$ is of course the completion of $$\mathbf{A}_{\text{inf}}[\frac 1 p]$$ with respect to the induced map $$ \theta:\mathbf{A}_{\text{inf}}\left[\frac 1 p\right] \to \mathbb{C}_p $$