A note on extensions of algebraic and formal groups. I (Q1814114)

Let \(A\) denote a commutative ring of characteristic \(p>0\). The commutative extensions \(\hbox{Ext}^ 1_ A(\mathbb{G}_{a,A},\mathbb{G}_{m,A})\) and \(\hbox{Ext}^ 1_ A(\hat \mathbb{G}_{a,A},\hat \mathbb{G}_{m,A})\) are explicitly described when \(A\) is a \(\mathbb{Z}_{(p)}\)-algebra. Here \(\mathbb{G}_{a,A}\) (respectively \(\mathbb{G}_{m,A})\) denotes the additive (respectively multiplicative) group over \(A\), and \(W_ A\) the group scheme of Witt vectors over \(A\). The corresponding formal groups are indicated by putting a hat (or roof) on algebraic objects. Let \(W(A)\) denote the ring of Witt vectors over \(A\), and let \(\hat W(A)\) be a subset of \(W(A)\) consisting of Witt vectors \({\mathbf a}=(a_ r)_{r\geq 0}\) such that \(a_ r\) is nilpotent for all \(r\) and \(a_ r=0\) for almost all \(r\). \(F\) stands for the Frobenius endomorphism of \(W(A)\). -- Let \[ F_ p(U;X,Y)=\exp\left(\sum_{i\geq1}U^{p^ i-1}{X^{p^ i}+Y^{p^ i}-(X+Y)^{p^ i}\over p^ i}\right)\in\mathbb{Z}_{(p)}[U][[X,Y]] \] be a formal power series. Associated to \({\mathbf a}=(a_ r)_{r\geq 0}\in W(A)\), define a formal power series \[ F_ p({\mathbf a};X,Y)=\prod_{r\geq 0}F_ p(a_ r;X,Y)\in A[[X,Y]]. \] Theorem. Let \(A\) be an \(\mathbb{F}_ p\)-algebra. Then the map \({\mathbf a}\mapsto F_ p({\mathbf a};X,Y)\) induces bijective homomorphisms \(W(A)/F\cong H^ 2_ 0(\hat \mathbb{G}_{a,A},\hat \mathbb{G}_{m,A})\) and \(\hat W(A)/F\cong H^ 2_ 0(\mathbb{G}_{a,A},\mathbb{G}_{m,A})\). As \(H^ 2_ 0(\hat \mathbb{G}_{a,A},\hat \mathbb{G}_{m,A})\) (respectively \(H^ 2_ 0(\mathbb{G}_{a,A},\mathbb{G}_{a,A}))\) is isomorphic to the group of classes of commutative extensions of \(\hat \mathbb{G}_{a,A}\) by \(\hat \mathbb{G}_{m,A}\) (respectively \(\mathbb{G}_{a,A}\) by \(\mathbb{G}_{m,A})\), which split as extensions of formal \(A\)-schemes (respectively \(A\)-schemes), the above theorem gives an explicit description of \(\hbox{Ext}^ 1_ A(\hat \mathbb{G}_{a,A},\hat \mathbb{G}_{m,A})\) (respectively \(\hbox{Ext}^ 1_ A(\mathbb{G}_{a,A},\mathbb{G}_{m,A}))\).