Exact sequence, de Rham representation

Question

Let $k$ be a $p$-adic field and $G_k$ its absolute Galois group. Let $B_\text{dR}$ be the de Rham period ring with the usual filtration given by powers of $t$. For $i < j$ integers we have an exact sequence $$ 0 \rightarrow t^{i+1}B^+_\text{dR}/t^{j+1}B^+_\text{dR} \rightarrow t^{i}B^+_\text{dR}/t^{j+1}B^+_\text{dR} \rightarrow t^iB^+_\text{dR}/t^{i+1}B^+_\text{dR} \rightarrow 0.$$

Taking $G_k$-invariants gives us again a short exact sequence (we can use $t^iB^+_\text{dR}/t^{i+1}B^+_\text{dR} \cong \hat{\overline{k}}(i)$ to calculate this).

Now let $V$ be a $p$-adic representation of $G_k$. Can we show that $$0 \rightarrow (t^{i+1}B^+_\text{dR}/t^{j+1}B^+_\text{dR} \otimes V)^{G_k}\rightarrow (t^{i}B^+_\text{dR}/t^{j+1}B^+_\text{dR} \otimes V)^{G_k}\rightarrow (t^iB^+_\text{dR}/t^{i+1}B^+_\text{dR} \otimes V)^{G_k} \rightarrow 0$$ is also exact?

Is it true if we assume some nice properties of $V$, like Hodge–Tate, de Rham etc.?

Answer 1

Since $B^+_{dR} / t B^+_{dR} \cong \mathbf{C}_p$, we have an injection $$\frac{(B^+_{dR} \otimes V)^{G_K}}{(t B^+_{dR} \otimes V)^{G_K}} \hookrightarrow (V \otimes \mathbf{C}_p)^{G_K}.$$

Multiplying by $t^i$ corresponds to a cyclotomic twist so we deduce $$ \operatorname{Gr^i} D_{dR}(V) \hookrightarrow (V \otimes \mathbf{C}_p(i))^{G_K}$$ for all $i$. On the other hand, $$\operatorname{dim}_K\left(\bigoplus_i (V \otimes \mathbf{C}_p(i))^{G_K}\right) \le \dim V$$ because $B_{HT} = \bigoplus_i \mathbf{C}_p(i)$ is a sensible period ring (it is $G_K$-admissible).

If $V$ is de Rham, then $\sum_i dim_K \operatorname{Gr}^i D_{dR}(V) = \dim(V)$, so all the injections (for all $i$) have to be isomorphisms; and it follows that your sequences are exact for all $i$ and $j$. Conversely, if $V$ is Hodge-Tate but not de Rham (such representations exist!), then at least one of these maps has to fail to be an isomorphism so your sequence is not exact for some $i$ and $j$.