A duality theorem for Tate-Shafarevich groups of curves over algebraically closed fields
In this paper, the author proves a duality theorem for Tate-Shafarevich groups over function fields. Let \(K\) be the function field of an irreducible smooth projective curve over an algebraically closed field, and let \(F\) be a finite discrete Galois module of order invertible in \(K\). Recall that the Tate-Shafarevich groups \(\Sha^i(K,F)\) are the kernel of the map from the Galois cohomology groups of \(F\) to the product of all the cohomology groups at the completions of \(K\) at the discrete valuations (Definition 3.2). The main theorem of the paper (Theorem 3.9) states that there is a perfect pairing of finite groups \[ \Sha^1(K,F)\times \Sha^1(K,Hom(F,\mathbb{Q}/\mathbb{Z}(1)))\to\mathbb{Q}/\mathbb{Z}. \] Furthermore, the group \(Sha^0(K,F)\) is finite and \(Sha^r(K,F)=0\) for \(r>1\). The proof follows the lines of [\textit{D. Harari} and \textit{T. Szamuely}, J. Algebr. Geom. 25, No. 3, 571--605 (2016; Zbl 1355.14018)] by using the duality for open subschemes of smooth proper curves (Theorem 2.2). Proposition 4.2 contains another result on the vanishing of the first Galois cohomology of certain algebraic groups.
étale and other Grothendieck topologies and cohomologies
scientific article; zbMATH DE number 7006744
scientific article