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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 $H^1$.

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The Hodge-Tate weights are just the numbers where the filtration in the de Rham cohomology jumps, so as you suspected will be in the range $[0,i]$. To prove this I guess one needs to observe, amongst other things, that the associated graded ring of $B_{dR}$ is $B_{HT}$, and that there's a comparison theorem relating de Rham and etale cohomology, with the field of periods being $B_{dR}$. It's all downhill from there.

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If $X$ is an abelian variety, by a result of Tate there is an isomorphism:

$(\mathbb{C}_p(j) \otimes H^{i}(X_{\overline{K}}, \overline{\mathbb{Q}}_p)^{G_K}=H^{i-j}(X,\Omega^{j}_{X})$ when $j\leq i$ or $j\leq 0$ and zero in the other case.

Generally, Falting shows the following isomorphism for a proper smooth scheme $X$. $(\mathbb{C}_p \otimes H^{i}(X_{\overline{K}}, \overline{\mathbb{Q}}_p)=\oplus_{j \in \mathbb{Z}} \mathbb{C}_p(-j) \otimes H^{i-j}(X,\Omega^{j}_{X})$.

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