Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $H^1$.

share|cite|improve this question

1 Answer 1

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.