Local duality for 2-dimensional local ring
The author generalizes \textit{S. Saito}'s result [Adv. Stud. Pure Math. 12, 343--373 (1987; Zbl 0672.12006), Theorem 5.3]. Let \(A\) be a \(2\)-dimensional normal, complete local domain with quotient field \(K\). Assume that the characteristic of \(A\) is positive and that the residue field is an \(n\)-dimensional local field. He gave the following exact sequence: \[ \begin{multlined} 0 \to N/lN \to H^{3+n}(K, (\mathbb{Z}/l \mathbb{Z})^{\otimes (2+n)}) \to\\ \bigoplus_{\text{\(\mathfrak p\) is a prime ideal of height \(1\)}} H^{2+n}(\kappa(\mathfrak p), (\mathbb{Z}/l \mathbb{Z})^{\otimes (1+n)}) \to \mathbb{Z}/l \mathbb{Z} \to 0\end{multlined} \] for any integer \(l\) prime to the characteristic of \(A\). Here \(N\) is a group independent of the choice of \(l\). Saito's theorem is the case of \(n = 0\). To prove this theorem, the author also generalized Grothendieck's local duality theorem [\textit{A. Grothendieck}, Seminaire de géométrie algébrique du Bois-Marie 1965-66 SGA 5 dirige par A. Grothendieck avec la collaboration de I. Bucur, C. Houzel, L. Illusie, J.-P. Jouanolou et J. -P. Serre. Cohomologie \(\ell\)-adique et fonctions L. Lecture Notes in Mathematics. 589. Berlin-Heidelberg-New York: Springer-Verlag (1977; Zbl 0345.00011), p. 49].
scientific article; zbMATH DE number 5615369