Affine Grassmannians of group schemes and exotic principal bundles over \(\mathbb A^1\) (Q2816467)

Let \(\mathbf{G}\) be a simply connected and simple reductive group over \(U = \mathrm{Spec}(R)\), where \(R\) is a regular local ring of finite type over an infinite field \(k\). Consider a closed subscheme \(Z \subset \mathbb{A}^1_U\) that is finite over \(U\). Assume that \(\mathcal{E}\) is a \(\mathbf{G}\)-torsor over \(\mathbb{A}^1_U\), trivialized off \(Z\). We may glue \(\mathcal{E}\) with the trivial torsor over \(\mathbb{P}^1_U \smallsetminus Z\) to obtain a \(\mathbf{G}\)-torsor \(\hat{\mathcal{E}}\) on \(\mathbb{P}^1_U\).NEWLINENEWLINEThe main Theorem 1 of the paper says that in the setting above, we have: (i) If \(\hat{\mathcal{E}}_u := \hat{\mathcal{E}}|_{\mathbb{P}^1_u}\) is trivial, where \(u \in |U|\) is the closed point, then \(\hat{\mathcal{E}}\) and thus \(\mathcal{E}\) is trivial. (ii) If \(\mathbf{G}\) is isotropic, then \(\mathcal{E}\) is trivial. (iii) If the generic fiber of \(\mathbf{G}\) is anisotropic and \(\hat{\mathcal{E}}_u\) is not trivial, then \(\mathcal{E}\) is not trivial. In fact, this is a special case of a more general Theorem 4, in which the role of \(\{\infty\} \times U = \mathbb{P}^1_U \smallsetminus \mathbb{A}^1_U\) is replaced by any finite étale \(U\)-subscheme \(Y \subset \mathbb{P}^1_U\).NEWLINENEWLINEThe main Theorem 2 states that when \(Z \to U\) is finite étale, \(\mathbf{G}\) admits a proper parabolic subgroup scheme when pulled back to each connected component of \(Z\), and that the special fiber \(Z_u\) has a \(k(u)\)-point, then for any \(\mathbf{G}_u\)-torsor \(E\) over \(\mathbb{P}^1_u\) that is trivial at the generic point, \(E\) extends to a \(\mathbf{G}\)-torsor \(\mathcal{E}\) over \(\mathbb{P}^1_U\) which is trivial off \(Z\).NEWLINENEWLINEFinally, Theorem 3 states that for algebraically closed \(k\), assume that \(G\) is a simple and simply connected reductive group \(G\) over \(k\), and that there exists a \(G\)-torsor over \(U\) that admits no proper parabolic reduction at the generic point of \(U\). Then: (i) There exists a \(G\)-torsor over \(\mathbb{A}^1_U\) that is not a pull-back of any \(\mathbf{G}\)-torsor over \(U\). (ii) There is an \(U\)-group scheme \(\mathbf{G}\) that is a strongly inner form (see \S 3.1) of \(G\), as well as a \(\mathbf{G}\)-torsor \(\mathcal{E}\) over \(\mathbf{A}^1_U\), such that \(\mathcal{E}\) is non-trivial, yet trivial off a finite étale \(U\)-subscheme \(Y \subset \mathbb{A}^1_U\).NEWLINENEWLINEThe last theorem furnishes a negative answer to the Bass--Quillen problem on pull-backs of torsors on the affine line. With hindsight, one can say that such questions on trivializibility of torsors lead naturally to affine Grassmannians, whose use is the main innovation of this article. The \S\S 4---5 of the paper include a highly helpful review of the theory of affine Grassmannians. Some results therein, such as the Bruhat decomposition for non-split groups, seem to be new.