The Brauer group of a rational function field over a perfect field (Q1363101)

Let \(K(x)\) be a rational function field over a perfect field \(K\). It is known after \textit{M. Auslander} and \textit{A. Brumer} [Nederl. Akad. Wet. Proc., Ser. A 71, 286-296 (1968; Zbl 0182.07601)] and \textit{D. Faddeev} [Am. Math. Soc., Transl., II. Ser. 3, 15-38 (1956; Zbl 0075.02901)] that the Brauer group \(\text{Br}(K(x))\) satisfies \[ \text{Br}(K(x)) \cong \text{Br}(K) \oplus \Bigl( \bigoplus _p X (K _p) \Bigr), \] where \(p\) runs over all monic irreducible polynomials in \(K[x]\), \(K _p = K[x]/(p(x))\) and \(X (K _p)\) is the character group of the absolute Galois group of \(K _p\). The author calls this formula the ABF Theorem, after the names of the authors Auslander, Brumer and Faddeev. For a central division algebra \(D\) over \(K(x)\), the map \(D \rightarrow D \otimes _{K(x)} K(x) _v\), where \(K(x) _v\) is the completion of \(K(x)\) with respect to a place \(v\), defines a homomorphism \(\varphi _v : \text{Br}(K(x)) \rightarrow \text{Br}(K(x) _v)\). Then we obtain the homomorphism \(\varphi = \prod _v \varphi _v : \text{Br}(K(x)) \rightarrow \prod _v \text{Br} (K(x) _v)\). We have that \(\text{Br}(K(x) _v) \cong B _v ^{(u)} \oplus B _v ^{(t)}\) where \(B _v ^{(u)} \cong \text{Br}(K _v)\) (resp. \(B _v ^{(t)}\)) denotes the unramified part (resp. the totally ramified part) of \(\text{Br}(K(x) _v)\). Each class of \(B _v ^{(t)}\) is represented by a totally ramified cyclic division algebra \(( \chi, \pi _v)\), where \(\chi \in X(K _v)\) and \(\pi _v\) is a fixed prime element. In the paper under review, the author shows that the isomorphism in the ABF Theorem is obtained by the injective homomorphism \(\varphi = \prod _v \varphi _v\) composed with a natural projection. More precisely \[ \varphi _{\infty} ^{(u)} \times \prod _p \varphi _p ^{(t)} : \text{Br}(K(x)) \rightarrow B _{\infty} ^{(u)} \oplus \bigoplus _p B _p ^{(t)} \] is an isomorphism, where \(\varphi _v ^{(u)} = p _v ^{(u)} \circ \varphi _v\) (resp. \(\varphi _v ^{(t)} = p _v ^{(t)} \circ \varphi _v\)) and \(p _v ^{(u)}\) (resp. \(p _v ^{(t)}\)) denotes the projection to \(B _v ^{(u)}\) (resp. \(B _v ^{(t)}\)) (Theorem 1). The ABF Theorem is a consequence of Theorem 1. The inverse of the isomorphism given in Theorem 1, composed with \(\varphi _{\infty} ^{(t)}\) or \(\varphi _p ^{(u)}\) gives homomorphisms \(\tilde {\varphi} _{v' v}\) which determine the image \(\varphi (\text{Br}(K(x)))\) in \(\prod _v \text{Br}(K(x) _v)\) (Theorem 2).