Fix a $p$-adic field $K$ with perfect residue field $k.$ Let $\mathbb{C}_K$ be the completion of the algebraic closure of $K,$ and let $$R = \varprojlim \mathbb{C}_K/p,$$ where the transition maps in the inverse limit are given by the Frobenius. One can then show that $R$ is a perfect valuation ring (of characteristic $p.$) Further,each element $x \in R$ can be represented by a sequence $x^{(i)} \in \mathbb{C}_p$ such that $(x^{(i+1)})^p = x^{(i)}.$ We can then form $W(R),$ the ring of Witt vectors of $R.$ There is a natural map $\theta:W(R)[1/p] \rightarrow \mathbb{C}_K,$ which takes an element $x$ of the form $$\sum_i p^i [x_i],$$ where $x_i \in R$ is the Teichmüller representative, to $\sum_{i=0}^\infty p^i x_i^{(0)}.$

One can prove that $\theta$ is surjective and that the kernel of $\theta$ is principal, generated by an element $\xi.$ One then defines $\mathbb{B}^+_{dr}$ as $$\lim_n W(R)/(\ker\theta)^n.$$ There are two natural topologies on $\mathbb{B}^+_{dr},$ the first being the $\xi$-adic topology, but there is also what is known as the "canonical" topology. The canonical topology is formed from first topologizing $W(R)[1/p]$ by taking as a basis of fundamental open neighborhoods sets of the form $p^nW(R)+p^mW(\mathfrak{a}),$ for $\mathfrak{a} \in \mathbb{C}_p$ an open ideal, and $m,n \in \mathbb{Z}.$ Then one gives each $W(R)[1/p]/(\ker \theta)^i$ the quotient topology. Finally, $\mathbb{B}_{dr}^+$ is topologized as the inverse limit of $W(R)[1/p]/(\ker \theta)^i.$

Now, Fontaine claims that there is no continuous section of $\theta:\mathbb{B}^+_{dr} \rightarrow \mathbb{C}_p,$ and even no Galois equivariant section of $\theta.$

I can prove that there is no continuous section through a quite elaborate argument, namely, that if there is such a section, then $\mathbb{B}^+_{dr}$ should be isomorphic, as a topological ring with the Galois action, to the Hodge-Tate ring, and this would imply that all Hodge-Tate representations are De Rham, and this is false.

However, I think there should be a considerably easier argument than this. For example, Fontaine claims in an article that even the projection $$W(R)[1/p]/(\ker \theta)^2 \rightarrow W(R)[1/p]/(\ker \theta) \cong \mathbb{C}_p$$has no continuous section. I can not see why this is the case, but maybe someone has an easy argument for this.

I have two questions:

1. Why is there no continuous section of $\theta,$ if $\mathbb{B}^+_{dr}$ has the canonical topology? Preferably, this argument should be fairly elementary, for example, by showing that $$W(R)[1/p]/(\ker \theta)^2 \rightarrow W(R)[1/p]/(\ker \theta) \cong \mathbb{C}_p$$has no continuous section.

2. Why is there no Galois-equivariant section of $\theta?$Here, we are dropping the continuity assumption.

Fix a $p$-adic field $K$ with perfect residue field $k.$ Let $\mathbb{C}_K$ be the completion of the algebraic closure of $K,$ and let $$R = \varprojlim \mathbb{C}_K/p,$$ where the transition maps in the inverse limit are given by the Frobenius. One can then show that $R$ is a perfect valuation ring (of characteristic $p.$) Further,each element $x \in R$ can be represented by a sequence $x^{(i)} \in \mathbb{C}_p$ such that $(x^{(i+1)})^p = x^{(i)}.$ We can then form $W(R),$ the ring of Witt vectors of $R.$ There is a natural map $\theta:W(R)[1/p] \rightarrow \mathbb{C}_K,$ which takes an element $x$ of the form $$\sum_i p^i [x_i],$$ where $x_i \in R$ is the Teichmüller representative, to $\sum_{i=0}^\infty p^i x_i^{(0)}.$

One can prove that $\theta$ is surjective and that the kernel of $\theta$ is principal, generated by an element $\xi.$ One then defines $\mathbb{B}^+_{dr}$ as $$\lim_n W(R)[1/p]/(\ker\theta)^n.$$ There are two natural topologies on $\mathbb{B}^+_{dr},$ the first being the $\xi$-adic topology, but there is also what is known as the "canonical" topology. The canonical topology is formed from first topologizing $W(R)[1/p]$ by taking as a basis of fundamental open neighborhoods sets of the form $p^nW(R)+p^mW(\mathfrak{a}),$ for $\mathfrak{a} \in \mathbb{C}_p$ an open ideal, and $m,n \in \mathbb{Z}.$ Then one gives each $W(R)[1/p]/(\ker \theta)^i$ the quotient topology. Finally, $\mathbb{B}_{dr}^+$ is topologized as the inverse limit of $W(R)[1/p]/(\ker \theta)^i.$

Now, Fontaine claims that there is no continuous section of $\theta:\mathbb{B}^+_{dr} \rightarrow \mathbb{C}_p,$ and even no Galois equivariant section of $\theta.$

I can prove that there is no continuous section through a quite elaborate argument, namely, that if there is such a section, then $\mathbb{B}^+_{dr}$ should be isomorphic, as a topological ring with the Galois action, to the Hodge-Tate ring, and this would imply that all Hodge-Tate representations are De Rham, and this is false.

However, I think there should be a considerably easier argument than this. For example, Fontaine claims in an article that even the projection $$W(R)[1/p]/(\ker \theta)^2 \rightarrow W(R)[1/p]/(\ker \theta) \cong \mathbb{C}_p$$has no continuous section. I can not see why this is the case, but maybe someone has an easy argument for this.

I have two questions:

1. Why is there no continuous section of $\theta,$ if $\mathbb{B}^+_{dr}$ has the canonical topology? Preferably, this argument should be fairly elementary, for example, by showing that $$W(R)[1/p]/(\ker \theta)^2 \rightarrow W(R)[1/p]/(\ker \theta) \cong \mathbb{C}_p$$has no continuous section.

2. Why is there no Galois-equivariant section of $\theta?$Here, we are dropping the continuity assumption.