Models of the group schemes of roots of unity

Abstract : Let O_K be a discrete valuation ring of mixed characteristics (0,p), with residue field k. Using work of Sekiguchi and Suwa, we construct some finite flat O_K-models of the group scheme \mu_{p^n,K} of p^n-th roots of unity, which we call Kummer group schemes. We set carefully the general framework and algebraic properties of this construction. When k is perfect and O_K is a complete totally ramified extension of the ring of Witt vectors W(k), we provide a parallel study of the Breuil-Kisin modules of finite flat models of \mu_{p^n,K}, in such a way that the construction of Kummer groups and Breuil-Kisin modules can be compared. We compute these objects for n < 4. This leads us to conjecture that all finite flat models of \mu_{p^n,K} are Kummer group schemes.
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Submitted on : Wednesday, February 22, 2012 - 10:04:07 AM
Last modification on : Friday, May 24, 2019 - 5:26:42 PM

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Ariane Mézard, Matthieu Romagny, Dajano Tossici. Models of the group schemes of roots of unity. Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2013, 63 (3), pp.1055-1135. ⟨10.5802/aif.2784⟩. ⟨hal-00672822⟩

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