Groupe de Brauer des corps de fractions rationnelles à coefficients complexes
The author presents the following description of the Brauer group of the rational function fields \(Br({\mathbb{C}}(t_ 1,...,t_ n))\cong \oplus_ f\{H^ 1(X_ g,{\mathbb{Q}}/{\mathbb{Z}})\oplus Div_ 0(X_ g)\oplus_{{\mathbb{Z}}}{\mathbb{Q}}/{\mathbb{Z}}\}\) where f runs through all unitary irreducible polynomials with coefficients in \({\mathbb{C}}(t_ 1,...,t_ i)\) for \(1\leq i<n_ g\). Here, \(X_ g\) is a geometrical model, i.e. a smooth variety having \(K_ g={\mathbb{C}}(t_ 1,...,t_ i)(\alpha)\), where \(\alpha\) is a root of f, as its function field and \(Div_ 0(X_ g)\) is the divisor subgroup consisting of those divisors with trivial first Chern class. Further, the author shows that \(Br({\mathbb{C}}(t_ 1,...,t_ n))\) is a divisible group so isomorphic to \(\oplus_{i\in I}({\mathbb{Q}}/{\mathbb{Z}})\) where I is in one-to-one correspondence with \({\mathbb{C}}\). In particular for all m,n\(\geq 2\) one has \(Br({\mathbb{C}}(x_ 1,...,x_ m))\cong Br({\mathbb{C}}(x_ 1,...,x_ n)).\)
Brauer group of the rational function fields
scientific article; zbMATH DE number 3884278