E ective models of group schemes

Let R be a discrete valuation ring with fraction eld K and X a at R-scheme. Given a faithful action of a K-group scheme GK over the generic bre XK , we study models G of GK acting on X. In various situations, we prove that if such a model G exists, then there exists another model G′ that acts faithfully on X. This model is the schematic closure of G inside the fppf sheaf AutR(X) ; the major di culty is to prove that it is representable by a scheme. For example, this holds if X is locally of nite type, separated, at and pure and G is nite at. Pure schemes (a notion recalled in the text) have many nice properties : in particular, we prove that they are the amalgamated sum of their generic bre and the family of their nite at closed subschemes. We also provide versions of our results in the setting of formal schemes.