Effective models of group schemes (Q2913219)

Let \(R\) be a discrete valuation ring with residue field \(k\) and field of fractions \(K\). If \(G\) is an \(R\)-group scheme acting faithfully on a \(K\)-scheme \(X\), then often there exists a model \(X_R\) of \(X\), such that the \(G\)-action on \(X\) is induced by a \(G\)-action on \(X_R\). For example this happens for classes of schemes \(X\) for which a model exists which satisfies some sort of universal property, e.g.~if \(X\) is an abelian variety with faithful \(G\)-action, typically the action extends to its the Néron model. In general, however, the \(G\)-action on the special fiber of \(X_R\) will not be faithful anymore. A basic example (taken from the introduction of the article under review) in the situation where \(R\) is of mixed characterstic with \(\text{char}{k}=p\), comes from an abelian scheme \(A\) on \(R\), for which the \(p\)-torsion points of \(A_K\) are rational. The constant group scheme \(G=\mathbb{Z}/p^{2\dim A_K}\mathbb{Z}\) acts faithfully on \(A_K\) by translation, and this action extends to \(A\), but it is not faithful on the special fiber.NEWLINENEWLINEThe article under review is concerned with the question under which circumstances one can find a model of \(G\) which does act faithfully on the special fiber. A good candidate for such a model would be the schematic image of \(G\rightarrow \Aut_R(X)\). In many situations the fppf-sheaf \(\Aut_R(X)\) is not representable, but there still is a notion of schematic image for the morphism \(G\rightarrow \Aut_R(X)\) due to \textit{M. Raynaud} [Publ.~Math., Inst.~Hautes Étud.~Sci.~38, 27--76 (1970; Zbl 0207.51602)], and the main results of the present article establish conditions under which this schematic image is representable by a group scheme, which is called the \textit{effective model} of \(G\).NEWLINENEWLINEThe main theorem states that, if \(X\) is a separated, flat, pure \(R\)-scheme which is locally of finite type, and \(G\) a proper flat \(R\)-group scheme acting on \(X\) and faithfully on \(X_K\), then the schematic image of \(G\) in \(\Aut_R(X)\) is representable by a flat, finite type group scheme, if and only if \(\ker(G\rightarrow \Aut_R(X))\) is a finite group scheme. There is also a version of this statement for formal schemes, and there is a version for a class of affine schemes \(X\), which works under weaker assumptions.NEWLINENEWLINEThe notion of a pure morphism (defined in [\textit{M.~Raynaud} and \textit{L.~Gruson}, Invent.~Math.~13, 1--89 (1971; Zbl 0227.14010)]) is of central importance for the arguments; examples are proper morphisms and faithfully flat morphisms with geometrically irreducible fibers without embedded components. In fact the above theorem is deduced from a structure theorem for flat, pure schemes locally of finite type over a henselian discrete valuation ring \(R\), saying that if \(X\) is such an \(R\)-scheme, then the family of closed subschemes which are finite and flat over \(R\) is \(R\)-universally dense in \(X\), and that \(X\) is the amalgamated sum of its generic fiber and this family of closed subschemes.