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In $p$-adic Hodge theory, there is a nice exact sequence for quotients of $B_{\text{dr}}^+$. Denote by $t$ the typical uniformizer of $B_{\text{dr}}^+$ (the cyclotomic character), then there is a $G_{\mathbb{Q}_p}$-equivariant exact sequence: $$0\longrightarrow \mathbb{C}_p(k-1)\longrightarrow B_{\text{dr}}^+/t^kB_{\text{dr}}^+\longrightarrow B_{\text{dr}}^+/t^{k-1}B_{\text{dr}}^+\longrightarrow 0,$$ identifying the quotient $B_{\text{dr}}^+/t^kB_{\text{dr}}^+$ as an extension of the Galois representation $B_{\text{dr}}^+/t^kB_{\text{dr}}^+$ by the $k-1$'th power of the cyclotomic character.

For $B_{\text{cr}}^+$, one can write a similar $G_{\mathbb{Q}_p}$-equivariant extension: $$0\longrightarrow t^{k-1}B_{\text{cr}}^+/t^kB_{\text{cr}}^+\longrightarrow B_{\text{cr}}^+/t^kB_{\text{cr}}^+\longrightarrow B_{\text{cr}}^+/t^{k-1}B_{\text{cr}}^+\longrightarrow 0,$$

What is the Galois representation $t^{k-1}B_{\text{cr}}^+/t^kB_{\text{cr}}^+$ isomorphic to? Does it have a nice description? Being a subring of $B_{dr}^+$, it seems that the result should be a subring of $\mathbb{C}_p\otimes \xi^{k-1}$, where $\xi$ is the cyclotomic character.

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    $\begingroup$ It is just $\mathbf{C}_p (k-1)$. $\endgroup$
    – Satan's Minion
    Commented Jan 31 at 14:45
  • $\begingroup$ @Satan'sMinion could you help me see it? I find this peculiar since this somehow shows that $B_{cr}^+$ is given by iterated extensions by cyclotomic characters, in the exact same way $B_{dr}^+$ is. Is this not "weird"? $\endgroup$
    – kindasorta
    Commented Jan 31 at 14:49
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    $\begingroup$ It's not weird, because $B_{dR}^+$ is complete for the $t$-adic filtration while $B_{cr}^+$ is not. But they do have the same graded pieces. $\endgroup$
    – Satan's Minion
    Commented Jan 31 at 14:55

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