On non-commutative extensions of \(G_a\) by \(G_m\) over an \(\mathbb F_p\)-algebra
Let \(A\) be a ring of characteristic \(p\). Let \(G_{a}\) and \(G_{m}\) be the additive and mulitplicative group schemes respectively, and let \(\widehat{G} _{a}\) and \(\widehat{G}_{m}\) be the corresponding formal group schemes. The second symmetric Hochschild cohomology groups \(H_{0}^{2}(G_{a,A},G_{m,A}) \) and \(H_{0}^{2}(\widehat{G}_{a,A},\widehat{G} _{m,A}) \) are easily computed -- they arise as the cokernel of the Frobenius map on \(\widehat{W}(A) \) and \(W(A) \) respectively. These groups describe the commutative extensions of the first (formal) group scheme by the second. Using the Artin-Hasse exponential series, the author describes the non-commutative extensions of \(\widehat{G} _{a}\) by \(\widehat{G}_{m}.\) Specifically, the factor groups \(H^{2}(\widehat{G}_{a,A},\widehat{G}_{m,A}) /H_{0}^{2}(\widehat{G} _{a,A},\widehat{G}_{m,A})\) and \(H^{2}(G_{a,A},G_{m,A}) /H_{0}^{2}(G_{a,A},G_{m,A})\) are computed, which completely determines the classification since \(H^{2}(-,-)\) gives the central extensions of the first group by the second. Let \(E_{p}(T)\) denote the Artin-Hasse exponential series. For an infinite vector \(\mathbf{U}=(U_{r}), r\geq 0\) we let \(E_{p}(\mathbf{U};T) =\prod E_{p}(U_{r}T^{p^{r}}).\) The statement of the main theorem states that if \(K\) (resp. \(\widehat{K}\)) is the kernel of \(F\) on \(W(A)\) (resp. \(\widehat{W}(A)\)) then the map \((\mathbf{a}_{r})_{r\geq 1}\mapsto \prod_{r\geq 1}E_{p}(\mathbf{a}_{r};XY^{p^{r}})\) induces isomorphisms \[ \bigoplus_{i\in \mathbb{N}}K\to H^{2}(G_{a,A},G_{m,A}) /H_{0}^{2}(G_{a,A},G_{m,A}) \] and \[ \prod_{i\in \mathbb{N}}\widehat{K}\to H^{2}(\widehat{G}_{a,A},\widehat{G}_{m,A}) /H_{0}^{2}(\widehat{G}_{a,A},\widehat{G} _{m,A}). \]
scientific article; zbMATH DE number 5017533