The Grothendieck-Serre conjecture for principal bundles asserts that for any regular local ring \(R\) with \(K = \mathrm{Frac}(R)\) and a reductive \(R\)-group scheme \(G\), the map \(H^1_{\mathrm{\'{e}t}}(R, G) \to H^1_{\mathrm{\'{e}t}}(K, G)\) induced by \(R \hookrightarrow K\) has trivial kernel. The main theorem of this paper under review proves such an injectivity result for \(G\) isotropic, simple and simply connected and for certain rings \(R\). Here isotropy means that \(G\) contains a \(\mathbb{G}_{m,R}\).
Specifically, it is proved that that \(H^1_{\mathrm{\'{e}t}}(\mathcal{O} \otimes_k A, G) \to H^1_{\mathrm{\'{e}t}}(K \otimes_k A, G)\) has trivial kernel in the following situation: \(k\) is an infinite field, \(\mathcal{O}\) is the semi-local ring of finitely many closed points on a smooth irreducible affine \(k\)-variety, \(K = \mathrm{Frac}(\mathcal{O})\), \(G\) is an isotropic and simply connected \(\mathcal{O}\)-group scheme and \(A\) is any Noetherian \(k\)-algebra.
As a corollary, one proves for any regular domain \(R \supset \mathbb{Q}\) and any isotropic simple simply connected \(R\)-group scheme \(G\), the map \(H^1_{\mathrm{\'{e}t}}(R[t_1, \ldots, t_n], G) \to H^1_{\mathrm{\'{e}t}}(R, G)\) induced by evaluation at \(t_1 = \cdots = t_n = 0\) has trivial kernel. This result is false if the isotropy condition is dropped.