On the ´ Etale Cohomology of Algebraic Varieties with Totally Degenerate Reduction over p-adic Fields to J. Tate

Let K be a field of characteristic zero that is com- plete with respect to a discrete valuation, with perfect residue field of characteristic p> 0. We formulate the notion of totally degenerate reduction for a smooth projective variety X over K. We show that for all prime numbers � , the Q-´ etale cohomology of such a variety is (after passing to a suitable finite unramified extension of K) a suc- cessive extension of direct sums of Galois modules of the form Q(r). More precisely, this cohomology has an increasing filtration whose r-th graded quotient is of the form V ⊗Q Q(r), where V is a finite dimen- sional Q-vector space that is independent of � , with an unramified action of the absolute Galois group of K.

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