On the extensions of \(\widehat{\mathcal G}^{(\lambda)}\) by \(\widehat{\mathbf G}_a\) over a \({\mathbb{Z}}_{(p)}\)-algebra.
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)].
scientific article; zbMATH DE number 1799655