On the extensions of \(\widehat{\mathcal G}^{(\lambda)}\) by \(\widehat{\mathbf G}_a\) over a \({\mathbb{Z}}_{(p)}\)-algebra.
type: Item
instance of: scholarly article
title: On the extensions of \(\widehat{\mathcal G}^{(\lambda)}\) by \(\widehat{\mathbf G}_a\) over a \({\mathbb{Z}}_{(p)}\)-algebra.
zbMATH Open document ID: 1051.14054
author: Mitsuaki Yato
published in: Tokyo Journal of Mathematics
publication date: 2002-01-01T00:00:00Z
review text: Let \(A\) be a ring, \(\lambda \in A\), and let \(G_{a,A}\) and \(G_{m, A}\) (resp. \(\widehat{G}_{a, A}\) and \(\widehat{G}_{m,A}\)) denote the additive and multiplicative group schemes over \(A\) (resp. the additive and multiplicative formal group schemes over \(A\)). Let \({\mathcal G}^{(\lambda)}\) be the group scheme Spec\(A[T, (1+\lambda T)^{-1}]\) with multiplication \(1 \otimes T + T \otimes 1 + \lambda T \otimes T\) which gives a deformation from \(G_{a,A}\) to \(G_{m,A}\), and \(\widehat{\mathcal G}^{(\lambda)}\) the corresponding formal group scheme. This paper computes Hom(\({\mathcal G}^{(\lambda)}, G_{a,A})\) and \(H^2_0 ({\mathcal G}^{(\lambda)}, G_{a,A})\) (resp. Hom\((\widehat{\mathcal G}^{(\lambda)}, \widehat{G}_{a,A})\) and \(H^2_0(\widehat{\mathcal G}^{(\lambda)}, \widehat{G}_{a,A}))\) where \(H^2\) denotes Hochschild cohomology, and identifies these groups as the kernel and cokernel of a certain endomorphism of \(A^{({\mathbb N})}\) (resp. \(A^{\mathbb N}\)) if \(A\) is a \({\mathbb Z}_{(p)}\) algebra. Two proofs of this result are given, one elementary and computational, and the other using a generalization of the Artin-Hasse exponential series and some results of \textit{T. Sekiguchi} and \textit{N. Suwa} [Tôhoku Math. J. (2) 53, 203--240 (2001; Zbl 1073.14546)].
reviewed by: Keith Johnson
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 16E40
zbMATH DE Number: 1799655
zbMATH Keywords: formal group schemes
zbMATH Keywords: group scheme extensions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1244208940
OpenAlex ID: W2088822799
cites work: Sur les groupes de Lie formels à un paramètre
cites work: Q3979762
cites work: A note on extensions of algebraic and formal groups. IV: Kummer-Artin-Schreier-Witt theory of degree \(p^2\)
cites work: On a Case of Extensions of Group Schemes
cites work: Q5602064
DOI: 10.3836/TJM/1244208940
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description: scientific article; zbMATH DE number 1799655