Covering groups and their integral models (Q2790605)

Let \(S\) be a scheme and \(\mathbf{G}\) a smooth group scheme over \(S\) whose geometric fibres are connected reductive groups. Assume further that \(\mathbf{G}\) has a maximal torus \(\mathbf{T}\) defined over \(S\), i.e. a smooth subgroup scheme whose geometric fibres are algebraic tori. If \(\mathbf{X} \to S\) is a scheme over \(S\) and \(\mathbf{U} \subset \mathbf{X}\) affine and open, \(\mathbf{U} = \mathrm{Spec}(R)\), we define \(K_i(\mathbf{U}) = K_i(A)\) for \(i \geq 0\) where \(K_i(A)\) are Quillen's \(K\)-groups. Then we write \(\mathbf{K}\) for the Zariski sheaf associated to the presheaf \(\mathbf{U} \to K_i(\mathbf{U})\). As \(\mathbf{X}\) varies over schemes of finite type over \(S\), the \(\mathbf{K}_i\) form a sheaf of abelian groups on the big Zariski site \(S_{\mathrm{Zar}}\). We also view \(\mathbf{G}\) as a sheaf of groups on \(S_{\mathrm{Zar}}\).NEWLINENEWLINEIn [Publ. Math., Inst. Hautes Étud. Sci. 94, 5--85 (2001; Zbl 1093.20027)], \textit{J.-L. Brylinski} and \textit{P. Deligne} studied the category of central extensions \(\mathbf{K}_2 \to \mathbf{G}' \to \mathbf{G}\) of \(\mathbf{G}\) by \(\mathbf{K_2}\).NEWLINENEWLINEThe author extends in the present paper results of Brylinski and Deligne who had classified these extensions by triples \((Q,D,f)\) in the case where \(S = \mathrm{Spec}(F)\) is a field to the case where \(S = \mathrm{Spec}(\mathcal{O})\) is the spectrum of a Dedekind domain (e.g. the ring of integers of \(F\)).NEWLINENEWLINEWhen \(S = \mathrm{Spec}(F)\), \(F\) a field, there is an equivalence of categories between the category of central extensions of \(\mathbf{G}\) by \(\mathbf{K_2}\) and a category of triples \((Q,D,f)\) which are obtained as follows. For a fixed maximal torus \(\mathbf{T}\) we denote by \(\mathcal{Y} = \mathcal{H}\mathrm{om}(\mathbf{G}_m,\mathbf{T})\) the cocharacters viewed as a local system on the étale site \(S_{\mathrm{\'et}}\). Similarly we can view the Weyl group as a sheaf \(\mathcal{W}\) of finite groups on \(S_{\mathrm{\'et}}\). To any central extension NEWLINE\[NEWLINE \mathbf{K}_2 \to \mathbf{G}' \to \mathbf{G} NEWLINE\]NEWLINE Brylinski and Deligne associated 3 invariants:NEWLINENEWLINE1) A Weyl-invariant quadratic form \(Q:\mathcal{Y} \to \mathbf{Z}\).NEWLINENEWLINE2) A central extension \(\mathcal{G}_m \to \mathcal{D} \to \mathcal{Y}\) of sheaves of groups on \(F_{\mathrm{\'et}}\) of \(\mathcal{Y}\) by \(\mathcal{G}_m\) where \(\mathcal{G}_m\) is the multiplicative group viewed as a sheaf on \(S_{\mathrm{\'et}}\).NEWLINENEWLINENow if \(\mathbf{G}\) is simply connected and semisimple, the central extensions \(\mathbf{K}_2 \to \mathbf{G}' \to \mathbf{G}\) are classified by Weyl-invariant quadratic forms \(Q:\mathcal{Y} \to \mathbf{Z}\). For such a group \(\mathbf{G}\) we denote by \(\mathbf{G}_Q\) the unique central extension of \(\mathbf{G}\) by \(\mathbf{K}_2\) associated to \(Q\).NEWLINENEWLINE3) The third invariant \(f\) is then a homomorphism of sheaves of groups on \(F_{\mathrm{\'et}}\) \(f:\mathcal{D}_{Q} \to \mathcal{D}\) where \(\mathcal{D}_Q\) is defined for arbitrary \(\mathbf{G}\) by passing to the simply connected cover of the derived subgroup.NEWLINENEWLINEThe main theorem of [loc. cit.] shows that these 3 invariants classify the central extensions up to isomorphism in the sense that there is an equivalence of categories between the category of central extensions of \(\mathbf{G}\) by \(\mathbf{K}_2\) and a category of triples \((Q,D,f)\) where \(Q\), \(D\) and \(f\) mimic the known properties of the three invariants attached to a central extension in 1), 2) and 3).NEWLINENEWLINEThe three invariants of [loc. cit.] can also be defined in the more general case where \(F\) is replaced by a Dedekind domain \(\mathcal{O}\) (i.e. \(S = \mathrm{Spec}(\mathcal{O}))\) of the following sort:NEWLINENEWLINE1) \(\mathcal{O}\) is a discrete valuation ring (DVR) with finite residue field.NEWLINENEWLINE2) \(\mathcal{O}\) is a DVR containing a field.NEWLINENEWLINE3) \(\mathcal{O} = \mathcal{O}_{\mathcal{S}}\) the ring of of \(\mathcal{S}\)-integers in a global field of characteristic \(p\) with \(\mathcal{S}\) sufficiently large so that \(\mathbf{Pic}(\mathcal{O}) = 0\)NEWLINENEWLINE4) \(\mathcal{O} = \mathcal{O}_{\mathcal{S}}\) the ring of of \(\mathcal{S}\)-integers in a function field with \(\mathcal{S}\) sufficiently large so that \(\mathbf{Pic}(\mathcal{O}) = 0\), assuming Gersten's conjecture holds in weight two for smooth schemes of finite type over \(\mathcal{O}\).NEWLINENEWLINEThe main theorem of the article shows that there is a similar classification of central extensions of \(\mathbf{G}\) by \(\mathbf{K_2}\) in terms of triples \((Q,D,f)\) in the cases 1) and 2). It is necessary for the proof that both categories have the structure of a Picard category. As a motivation the author explains why it is important to look at central extensions over the ring of integers \(\mathcal{O} \subset F\) in a \(p\)-adic field \(F\) in the study of unramified representations of a cover of a reductive \(p\)-adic group.