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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].

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