Additive extensions of a Barsotti-Tate group
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).
additive extension for Barsotti-Tate groups
scientific article; zbMATH DE number 1183137