Let $K/\mathbb Q_p$ be a local field, $X/K$ a proper scheme with semi-stable reduction.

**Question**: What is the possible range of Hodge-Tate weights of the etale cohomology $H^i(X_{\overline K}, \mathbb Q_p)$ as a $G_K$ representation?

The only "understanding" that I have is that when $i=1$, things reduce to abelian schemes. And the Hodge-Tate weights are all in {0,1}. Is that right?

What will happen with $i \geq 2$? Will the range be between 0 and $i$? It is true for abelian schemes right? Since $H^i$ in this case is the $i$-th wedge product of $H^1$.