Why does $H^i(X_{ét},\mathbb{Q}_p)$ have a Hodge-Tate structure?

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?

• For clarity, does "variety" assume smooth and proper as well?
– Matt
Commented Sep 8, 2013 at 20:21
• Just to be precise, what is true is that, after tensoring over $\mathbb{C}_p$, the subspaces on which the (semi-linear!) Galois action is via powers of the cyclotomic character generate the whole space. As for whether there is an 'intuitive' explanation for this phenomenon, I don't know! Commented Sep 9, 2013 at 3:07
• As Keerthi pointed out, it's after you tensor by Cp that the powers of the cyclo char appear. But a statement like that is true for any Galois repn V by Sen's theory. What is specific to the étale cohomology is that it's integer powers of the cyclo char that appear. This is where you use the geometric input. As to why this is so... Commented Sep 10, 2013 at 8:02

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: $(\mathbb{C}_p \otimes H^{i}(X_{\overline{K}}, \mathbb{Q}_p)=\oplus_{j \in \mathbb{Z}} \mathbb{C}_p(-j) \otimes H^{i-j}(X,\Omega^{j}_{X})$