Some cases of extensions of twisted group schemes over a discrete valuation ring
Let \(A\) be a discrete valuation ring with residue field \(k\), and let \(B\) be an unramified quadratic extension of \(A\). Let \(G\) be the nontrivial \(B/A\) -form of the multiplicative group scheme \(\mathbb{G}_{m}.\) Then \(G=\text{ Spec}A[\xi ,\tau]/(n\xi ^{2}-m\xi \tau +\tau ^{2}+m^{2}-4n),\) where \(B=A[\theta]\) and \(\theta ^{2}-m\theta +n=0.\) If we let \(\mathcal{G}\) be the model of \(G\), then \(\mathcal{G}=\text{ Spec}A[v,w]/(\lambda (nv^{2}-mvw+w^{2})-(m^{2}-4n)v)\) for some nonzero \(\lambda \in A.\) The object of this paper is to compute some specific extension groups for \( \mathcal{G}\) and \(G\) using the fppf topology. The most concrete results are as follows: \[ H^{1}(\mathcal{G},\mathbb{G}_{m,A}) \cong \mathbb{Z}/2\mathbb{Z}, \text{ Ext}^{1}(\mathcal{G},\mathbb{G}_{m,A}) \cong \mathbb{Z}/2\mathbb{Z} \] \[ H^{1}(\mathcal{G}^{(\lambda)},G) \cong A^{\times}/\text{Norm}(B^{\times}) \times \mathbb{Z}/2\mathbb{Z}, \text{ Ext}^{1}(\mathcal{G}^{(\lambda)},G) \cong \mathbb{Z}/2\mathbb{Z} \] \[ H^{1}(\mathcal{G},G) \cong A^{\times}/\text{Norm}(B^{\times}), \text{ Ext}^{1}(\mathcal{G},G) =0 \] \[ H^{1}(\mathbb{G}_{a,A},G) \cong A^{\times}/\text{Norm}(B^{\times}) \] where \(\mathbb{G}_{a,A}\) is the additive group scheme over \(A,\) \(( \,) ^{\times}\) represents the group of units, and Norm is the norm map \(B^{\times }\to A^{\times}.\) There is little here to motivate the desire to compute such groups, except the remark that Sekiguchi and Suwa have done similar calculations (such as Ext\(^{1}(\mathcal{G}^{(\lambda)}, \mathcal{G}^{(\mu)})\)). The proofs of these facts require extensive use of cohomological techniques which would be difficult to summarize in this review. In addition to the facts above, the authors obtain the following three results: The model of the \(\prod_{B/A}\mathbb{G}_{m}\) which is the Néron blowup of \(\mathbb{G}_{m,k}\) is the nontrivial extension of \(\mathcal{G}\) by \(\mathbb{G}_{m}.\) The model of the \(\prod_{B/A}\mathbb{G} _{m}\) which is the Néron blowup of \(G_{k}\) is the nontrivial extension of \(\mathcal{G}^{( \lambda ) }\) by \(\mathbb{G}_{m}.\) Any element of Ext\(^{1}( \mathbb{G}_{a,A},\mathcal{G}) \) corresponds to a polynomial \(F( a_{1},a_{p},a_{p^{2},}\ldots ;X) \) (whose explicit form given in the text is eight displayed lines long).
scientific article; zbMATH DE number 1471028