TITLE: Why does $H^i(X_{ét},\mathbb{Q}_p)$ have a Hodge-Tate structure?
QUESTION [18 upvotes]: Let $X$ be a variety over a $p$-adic field $K$.
Is there a simple or intuitive explanation of why the $G_K$ representation $H^i(X_{ét},\mathbb{Q}_p)$ is Hodge-Tate? More precisely, why do the powers of the cyclotomic character appear and not other characters?

REPLY [2 votes]: You need to assume that your variety is proper and smooth.
The properness implies that the vector space $H^{1}_{et}(X_{\overline{K}}, \mathbb{Q}_p)$ is of finite dimension. And the smoothness implies the following decomposition which is proved by Falting:
$(\mathbb{C}_p \otimes H^{i}(X_{\overline{K}}, \mathbb{Q}_p)=\oplus_{j \in \mathbb{Z}} \mathbb{C}_p(-j) \otimes H^{i-j}(X,\Omega^{j}_{X})$