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A finiteness theorem for cohomology of surfaces over $$p$$-adic fields and an application to Witt groups. (English) Zbl 0820.14013
Jacob, Bill (ed.) et al., K-theory and algebraic geometry: connections with quadratic forms and division algebras. Summer Research Institute on quadratic forms and division algebras, July 6-24, 1992, University of California, Santa Barbara, CA (USA). Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 58, Part 2, 403-415 (1995).
Let $$X$$ be a smooth, proper, geometrically integral surface over a $$p$$- adic field $$k$$. The authors study the unramified cohomology groups $$H^ 0 (X_{Zar}, {\mathcal H}^ i (\mathbb{Q}_ \ell/ \mathbb{Z}_ \ell (i - 1)))$$ for a prime $$\ell$$ and an integer $$i \geq 1$$. They show that these groups vanish for $$i \geq 5$$ and in the range $$1 \leq i \leq 4$$ they are of finite co-type provided either $$i \neq 3$$ or $$p \neq \ell$$ or the geometric genus $$p_ g (X)$$ of $$X$$ is zero and the Kodaira dimension $$\kappa_ X$$ of $$X$$ is $$\leq 1$$. Similar results are deduced for the finiteness of the unramified cohomology groups with coefficients in the sheaf $$\mathbb{Z}/\ell^ \nu \mathbb{Z}(i - 1)$$. As an application the authors prove that the Witt group of $$X$$ is finite for $$p \neq 2$$. For $$p = 2$$ they need the additional assumption that $$p_ g (X) = 0$$ and $$\kappa_ X \leq 1$$.
For the entire collection see [Zbl 0812.00023].

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 19G12 Witt groups of rings 14J25 Special surfaces 19E15 Algebraic cycles and motivic cohomology ($$K$$-theoretic aspects) 14G20 Local ground fields in algebraic geometry 14F17 Vanishing theorems in algebraic geometry