Representations of reductive group schemes (Q1192567)

Let \(S\) be a normal, locally Noetherian scheme, \(G\) a Chevalley-Demazure group scheme with ``épinglage'' over \(S\). Fix a generic geometric point \(\overline\eta:\text{Spec} K\to S\), and let \(\rho_ K:G_ K \to Gl(V_ K) \subset \text{End}(V_ K)\) be a representation of \(G_ K\). An \(S\)- form of \(V_ K\) is a pair \(\bigl( V({\mathcal O}_ S),i \bigr)\), where \(V({\mathcal O}_ S)\) is a vector bundle over \(S\) equipped with a \(G\)-action and \(i:V({\mathcal O}_ S)_ K \to V_ K\) is an isomorphism respecting the action of the group scheme \(G\). The author classifies all \(S\)-forms of \(V_ K\) by constructing a mapping from the set of such forms to \(\text{Pic} S\) and then classifying the elements in the fibre of the map as locally free \({\mathcal O}_ S\)-modules equipped with a graded structure defined in terms of a Chevalley system of \(G_ K\). This characterization is extended to nonsplit group schemes \(G\) by using the rigidity of the épinglage structure to construct descent data. Here the fibres are classified by \(H^ 1\bigl(S_{\text{ét}},\Aut(\rho)\bigr)\). The author finishes by describing \(\Aut(\rho)\) as an extension of \(G_ m\) by \(H\), where \(H\) is a semidirect product of \(\text{ad}(G)\) by the discrete group of automorphisms of the épinglage structure.