On the Brauer group of the product of a torus and a semisimple algebraic group
The authors give an algebraic proof of and generalise a result of [\textit{A. R. Magid}, Proc. Am. Math. Soc. 71, 164--168 (1978; Zbl 0393.20029)] from characteristic zero originally proven using the Lefschetz principle and comparison of étale and singular cohomology. The main theorem (Theorem 2.11) is: ``Let \(F\) be a field and \(n \geq 2\) an integer which is not divisible by \(\mathrm{char}\,F\). Let further \(X\) be a regular and geometrically integral \(F\)-scheme of finite type, and \(T\) an \(F\)-scheme, such that either \(T_s\) is isomorphic as an \(F_s\)-scheme to a torus, or \(T\) is the ``trivial'' torus \(\mathrm{Spec}\,F\). We assume that (i) \(\mathrm{Pic}\,X_s = 0\), (ii) \(\mathrm{H}^0(X_s, \mathrm{K}^M_i/n) = 0\) for \(i = 1,2\), and (iii) \(F_s[X_s]^\times = F_s^\times\). Then the natural homomorphism \(p_T: {}_n\mathrm{Br}(T) \to {}_n\mathrm{H}^2_{\mathrm{et}}(X \times_F T,\mathbb{G}_m)\) along the projection \(p_T: X \times_F T \to T\) is an isomorphism.'' In section 3, the authors draw several corollaries from this theorem, for example Theorem 3.2: Let \(T\) and \(n\) be as in Theorem 2.11. ``Let \(F\) be a field of characteristic \(\neq 2\), and \(X_q \hookrightarrow \mathbb{P}^m_F\) be the projective quadric defined by the equation \(q = \sum_{i=0}^ma_ix_i^2\) with \(m \geq 4\) and \(a_i \in F^\times\) for all \(i = 0, 1, \ldots, m\). Let further \(X_{q,\mathrm{aff}} \subset X_q\) be the open affine quadric defined by \(x_0 \neq 0\). Then the pull-back homomorphisms \({}_n\mathrm{Br}(T) \to {}_n\mathrm{Br}(X_{q,\mathrm{aff}} \times_F T) \quad\text{and}\quad {}_n\mathrm{Br}(T) \to {}_n\mathrm{H}^2_{\mathrm{et}}(X_q \times_F T,\mathbb{G}_m)\) along the respective projections to \(T\) are isomorphisms.'' A further Corollary 3.14: ``Let \(H\) be a connected reductive group over a field \(F\), such that \([H,H]\) is of adjoint type. Assume that the characteristic of \(F\) does not divide the order of the fundamental group of the commutator subgroup of \(H\). Then \({}_n\mathrm{Br}(\mathrm{Rad}(H)) \cong {}_n\mathrm{Br}(H)\) for all integers \(n\) which are not divisible by \(\mathrm{char}\,F\) and coprime to the order of the fundamental group of the commutator subgroup \([H,H]\).'' In section 4, the authors study the morphisms \(\mathrm{Br}(R) \to \mathrm{Br}(R[T_1,\ldots,T_d])\) for rings \(R\) of positive characteristic. In particular, they prove Theorem 4.5: ``Let \(F\) be a field of characteristic \(p > 0\) and \(d \geq 1\) an integer. Then we have \(\mathrm{Br}(F[T_1,\ldots,T_d]) \cong \mathrm{Br}(F) \oplus \mathbb{Z}(p^\infty)^{(I)}\), where the set \(I\) is non-empty if (and only if) \(F\) is not perfect or \(d \geq 2\).''
Applications of methods of algebraic \(K\)-theory
scientific article; zbMATH DE number 6361340
scientific article