Hodge-Tate weights of etale cohomology - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T14:27:20Z http://mathoverflow.net/feeds/question/117721 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117721/hodge-tate-weights-of-etale-cohomology Hodge-Tate weights of etale cohomology natura 2012-12-31T12:25:49Z 2012-12-31T12:52:07Z <p>Let $K/\mathbb Q_p$ be a local field, $X/K$ a proper scheme with semi-stable reduction. </p> <p><strong>Question</strong>: 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?</p> <p>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?</p> <p>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 $H^1$.</p> http://mathoverflow.net/questions/117721/hodge-tate-weights-of-etale-cohomology/117725#117725 Answer by wccanard for Hodge-Tate weights of etale cohomology wccanard 2012-12-31T12:52:07Z 2012-12-31T12:52:07Z <p>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.</p>