Additive extensions of a Barsotti-Tate group
type: Item
instance of: scholarly article
title: Additive extensions of a Barsotti-Tate group
zbMATH Open document ID: 0919.14028
author: Elena Mantovan
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1998-10-27T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108470
review text: Let \(k\) be a perfect field of characteristic \(p\) and \(W(k)\) the ring of Witt vectors over \(k\). If \(G\) is a smooth formal group over \(A=W(k)\) then an additive extension of \(G\) is a pair \((H,\pi)\) with \(H\) a formal group over \(A\) and \(\pi:H\rightarrow G\) an epimorphism whose kernel is isomorphic to \({\mathbb G}_a^n\) where \({\mathbb G}_a\) is the additive formal group over \(A\). The author shows (proposition 13) that if \(G\) is a Barsotti-Tate group over \(A\) then to each additive extension there corresponds a submodule of \(M^{(1)}\), where \(M\) is the Dieudonné module of \(G\), and that these classify such extensions (theorem 17). Using this an explicit construction of the universal extension of \(G\) over \(W(k)\) is given (theorems 19, 23). If G is a Barsotti-Tate group over \(k\) and \(G_L\) is its lifting over \(W(k)\), then additive extensions of \(G\) are shown to be isomorphic to the special fibres of additive extensions of \(G_L\). From this an explicit construction of the universal additive extension for Barsotti-Tate groups over \(k\) follows (theorem 27). The paper concludes with a result relating non-decomposable additive extensions of \(G\) over \(k\) and sub-\(k\)-bialgebras of the Barsotti algebra of \(G\) (theorem 31).
reviewed by: Keith Johnson
Mathematics Subject Classification ID: 14L05
zbMATH DE Number: 1183137
zbMATH Keywords: \(p\)-divisible group
zbMATH Keywords: smooth formal group
zbMATH Keywords: Dieudonné module
zbMATH Keywords: additive extension for Barsotti-Tate groups
MaRDI profile type: Publication
cites work: Q5732269
cites work: Q4322755
cites work: Q4156867
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cites work: Universal extensions and one dimensional crystalline cohomology
cites work: The crystals associated to Barsotti-Tate groups: with applications to Abelian schemes
description: scientific article; zbMATH DE number 1183137