On the arithmetic of local rings of dimension 2 and 3
Let \(A\) be a 3-dimensional excellent henselian normal local ring. Assume that the residue field \(K\) of \(A\) is algebraically closed or finite. Suppose also that the unique singular point of \(A\) is the closed point \(x\). Let \(f: X \rightarrow \text{Spec } A \) be a resolution of \(\text{Spec } A\) where \(X\) is a 3-dimensional regular scheme. Denote by \(Y\) the surface \(f^{-1} (x)\) and by \(P\) the set of height 1 prime ideals of \(A\). Denote \(Q_{l} (i) = \bigotimes_{j=1}^{i} Q_{j}\), \(Q_{j}\) being the Tate module \(Q_{l}\) for each \(j\). The paper is basically devoted to prove two results. Firstly, when \(K\) is algebraically closed and of characteristic 0, the mapping \(H^{3} (K, Q_{l} (2)) \rightarrow \bigoplus_{v \in P} H^{3} (K_{v}, Q_{l} (2))\) contains a subgroup isomorphic to \(Q_{l}^{r_{1} (A)}\), where \(l\) is a prime number, \(K_{v}\) the completion of \(K\) in \(v\) and \(r_{1} (A)\) the dimension of the set of elements in \(H^{2} (Y, Q_{l})\) with weight \(\leq 1\). Secondly, if \(K\) is a finite field of characteristic \(p \neq 2, 3, 5\), the authors consider the mapping \(H^{4} (K, Q_{l} (3)) \rightarrow \bigoplus_{v \in P} H^{4} (K_{v}, Q_{l} (3))\) and prove that its kernel contains a subgroup of type \((Q_{l})^{r_{1}'} (A) \) where \(r_{1}' (A) = \text{rank}_{Q_{l}} H^{2} ( |\Gamma |, Q_{l})\), \(|\Gamma |\) being the geometric realization of the exceptional fiber \(Y\). Moreover, the data \(r_{1}' (A) \) and \(r_{1} (A)\) are independent of the resolution \(f\) of \(X\). Similar results when \(\dim A = 2\) are also given generalizing a previous result by \textit{S. Saito} [in: Galois representations and arithmetic algebraic geometry, Proc. Symp., Kyoto 1985 and Tokyo 1986, Adv. Stud. Pure Math. 12, 343-373 (1987; Zbl 0672.12006)].
scientific article; zbMATH DE number 1270862
scientific article