Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Effective models of group schemes

Author: Matthieu Romagny
Journal: J. Algebraic Geom. 21 (2012), 643-682
Published electronically: October 26, 2011
MathSciNet review: 2957691
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Abstract: Let $ R$ be a discrete valuation ring with fraction field $ K$ and let $ X$ be a flat $ R$-scheme. Given a faithful action of a $ K$-group scheme $ G_K$ over the generic fibre $ X_K$, we study models $ G$ of $ G_K$ 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 $ \operatorname {Aut}_R(X)$; the major difficulty is to prove that it is representable by a scheme. For example, this holds if $ X$ is locally of finite type, separated, flat and pure and $ G$ is finite flat. 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 fibre and the family of their finite flat closed subschemes. We also provide versions of our results in the setting of formal schemes.

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Matthieu Romagny
Affiliation: Institut de Mathématiques, Théorie des Nombres, Université Pierre et Marie Curie, Case 82, 4, place Jussieu, F-75252 Paris Cedex 05

Received by editor(s): September 30, 2009
Received by editor(s) in revised form: February 3, 2010
Published electronically: October 26, 2011

American Mathematical Society