On the {\'e}tale cohomology of algebraic varieties with totally degenerate reduction over $p$-adic fields
Vol. 14 (2007), No. 2, Page 261--291.
Raskind, Wayne ; Xarles, Xavier
On the {\'e}tale cohomology of algebraic varieties with totally degenerate reduction over $p$-adic fields
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Abstract:
Let $K$ be a field of characteristic zero that is complete with respect to a discrete valuation, with perfect residue field of characteristic $p>0$. We formulate the notion of {\it totally degenerate reduction} for a smooth projective variety $X$ over $K$. We show that for all prime numbers $\ell$, the $\bQl$-\'etale cohomology of such a variety is (after passing to a suitable finite unramified extension of $K$) a successive extension of direct sums of Galois modules of the form $\bQl(r)$. More precisely, this cohomology has an increasing filtration whose $r$-th graded quotient is of the form $V\otimes_{\bQ}\bQl(r)$, where $V$ is a finite dimensional $\bQ$-vector space that is independent of $\ell$, with an unramified action of the absolute Galois group of $K$.
Mathematics Subject Classification (2000): Primary 14F20, 14F30 , Secondary 14G20
Mathematical Reviews Number: MR2351367
Received: 2004-12-24
