Exercise 24, § 1

Let $K$ be a (commutative) field of (arbitrary) characteristic $p$ and $X_i, Y_i, Z_i$ $3n$ distinct indeterminates; to abbreviate we shall denote the systems $(X_i)_{1 \leqslant i \leqslant n}, (Y_i)_{1 \leqslant i \leqslant n}, (Z_i)_{1 \leqslant i \leqslant n}$ by $\mathbf{x}, \mathbf{y}, \mathbf{z}$ respectively. Let $A_{\mathbf{x}, \mathbf{y}, \mathbf{z}}$ be the algebra of formal power series in the $3n$ indeterminates $X_i, Y_i, Z_i$ with coefficients in $K$; we shall denote by $A_{\mathbf{x}, \mathbf{y}}$ (resp. $A_{\mathbf{x}}$) the subalgebra of $A_{\mathbf{x}, \mathbf{y}, \mathbf{z}}$ consisting of the power series not containing $\mathbf{z}$ (resp. $\mathbf{y}, \mathbf{z}$). An element of $A_{\mathbf{x}, \mathbf{y}, \mathbf{z}}$ (resp. $A_{\mathbf{x}, \mathbf{y}}, A_{\mathbf{x}}$) will be denoted by $u(\mathbf{x}, \mathbf{y}, \mathbf{z})$ (resp. $u(\mathbf{x}, \mathbf{y}), u(\mathbf{x})$);

a system $(u_i(\mathbf{x}, \mathbf{y}, \mathbf{z}))_{1 \leq i \leq n}$ of $n$ elements of $A_{\mathbf{x}, \mathbf{y}, \mathbf{z}}$ will be denoted by $u(\mathbf{x}, \mathbf{y}, \mathbf{z})$; analogous notation will be used for formal power series containing only one or two of the three systems $\mathbf{x}, \mathbf{y}, \mathbf{z}$. If $u(\mathbf{x}, \mathbf{y}, \mathbf{z}) \in A_{\mathbf{x}, \mathbf{y}, \mathbf{z}}$ and $\mathbf{f} = (f_i), \mathbf{g} = (g_i), \mathbf{h} = (h_i)$ are three systems of $n$ elements of $A_{\mathbf{x}, \mathbf{y}, \mathbf{z}}$ which are formal power series with no constant term, we denote by $u(\mathbf{f}(\mathbf{x}, \mathbf{y}, \mathbf{z}), \mathbf{g}(\mathbf{x}, \mathbf{y}, \mathbf{z}), \mathbf{h}(\mathbf{x}, \mathbf{y}, \mathbf{z}))$ the formal power series obtained by substituting $f_i$ for $X_i$, $g_i$ for $Y_i$, $h_i$ for $Z_i$ ($1 \leq i \leq n$) in $u$. For every system $\alpha = (\alpha_1, \ldots, \alpha_n)$ of $n$ integers $\geq 0$, let $\mathbf{x}^\alpha$ denote the monomial $X_1^{\alpha_1} \ldots X_n^{\alpha_n}$; $\mathbf{y}^\alpha$ and $\mathbf{z}^\alpha$ are defined similarly. Let $\varepsilon_i$ denote the system $\alpha$ for which $\alpha_j = 0$ if $j \neq i$, $\alpha_i = 1$. We denote by $\mathbf{e}$ the system $(0, \ldots, 0)$ of $n$ elements of $A_{\mathbf{x}, \mathbf{y}, \mathbf{z}}$.

(a) A formal group law over $K$ (or, by an abuse of language, a formal group over $K$) of dimension $n$ is a system $G = \mathbf{f}(\mathbf{x}, \mathbf{y})$ of $n$ elements of $A_{\mathbf{x}, \mathbf{y}}$ with the following properties: (1) $\mathbf{f}(\mathbf{x}, \mathbf{f}(\mathbf{y}, \mathbf{z})) = \mathbf{f}(\mathbf{f}(\mathbf{x}, \mathbf{y}), \mathbf{z})$; (2) $\mathbf{f}(\mathbf{e}, \mathbf{y}) = \mathbf{y}$, $\mathbf{f}(\mathbf{x}, \mathbf{e}) = \mathbf{x}$. Show that necessarily $f_i(\mathbf{x}, \mathbf{y}) = X_i + Y_i + g_i(\mathbf{x}, \mathbf{y})$, where $g_i$ contains only monomials of degree $\geq 2$, each of which contains at least one $X_j$ and at least one $Y_j$. Show that there exists one and only one system $\mathbf{h}(\mathbf{x})$ of $n$ elements of $A_{\mathbf{x}}$ such that $\mathbf{f}(\mathbf{x}, \mathbf{h}(\mathbf{x})) = \mathbf{f}(\mathbf{h}(\mathbf{x}), \mathbf{x}) = \mathbf{e}$ (Algebra, Chapter IV, § 5, no. 9, Proposition 10). $G$ is called commutative if $\mathbf{f}(\mathbf{y}, \mathbf{x}) = \mathbf{f}(\mathbf{x}, \mathbf{y})$.

(b) For all $u \in A_{\mathbf{x}}$, let $L_y u$ denote the element $u(\mathbf{f}(\mathbf{y}, \mathbf{x}))$ of $A_{\mathbf{x}, \mathbf{y}}$. A derivation $D$ of $A_{\mathbf{x}}$ can be canonically extended to a derivation of $A_{\mathbf{x}, \mathbf{y}}$ (also denoted $D$ by an abuse of notation) by the condition that $D(Y_i) = 0$ for all $i$ (Algebra, Chapter IV, § 5, no. 8, Proposition 6). $D$ is said to be left invariant for the formal group $G$ in question if $L_y D = D L_y$. We denote by $D^{(\mathbf{e})}$ the linear mapping of $A_{\mathbf{x}, \mathbf{y}}$ into $A_{\mathbf{y}}$ defined by $D^{(\mathbf{e})}(u) = (D u)(\mathbf{e}, \mathbf{y})$; it maps $A_{\mathbf{x}}$ into $K$ and is determined by its restriction $A_{\mathbf{x}}$. Show that, for $D$ to be left invariant, it is necessary and sufficient that for all $v \in A_{\mathbf{x}}$, $D^{(\mathbf{e})}(L_y v) = (D v)(\mathbf{y})$.

Let $D_i$ be the derivation $\partial / \partial X_i$ of $A_{\mathbf{x}}$ ($1 \leq i \leq n$); there exists one and only one left invariant derivation $T_i$ such that $T_i^{(\mathbf{e})} = D_i^{(\mathbf{e})}$. Show that the $T_i$ are linearly independent over $K$ and that every left invariant derivation is a linear combination of the $T_i$ with coefficients in $K$. Deduce that, for the bracket $[D, D']$ and the $p$-mapping $D \mapsto D^p$ (when $p > 0$), the set $g$ of left invariant derivations is a Lie algebra (a Lie p-algebra if $p > 0$) called the Lie algebra of the formal Lie group $G$.

(c) Show that for every formal power series $u \in A_{\mathbf{x}}$:

where in the series $o_2$ all the monomials are of degree $\geq 2$ in the $Y_j$. Deduce that

where in the series $o_3$ all the monomials are of degree $\geq 3$ in the set of $Y_i$ and $Z_i$. Deduce that:

where all the terms of $o'_3$ are of total degree $\geq 3$. Show that, if G is commutative, g is commutative.

(d) Let $G'$ be another formal group of dimension m, whose group law is $f'$. A formal homomorphism of G into $G'$ is a system $F = (F_j(x))_{1 \leq j \leq m}$ of elements of $A_x$ such that $f'(F(x), F(y)) = F(f(x, y))$. For every left invariant derivation $D \in g$, show that $v \mapsto D(v \circ F)$ is a left invariant derivation belonging to the Lie algebra $g'$ of $G'$. If it is denoted by $F^*(D)$, $F^*$ is a homomorphism (a $p$-homomorphism for $p > 0$) of $g$ into $g'$. If $(T_i)_{1 \leq i \leq n}$, $(T'_j)_{1 \leq j \leq m}$ are the bases of $g$ and $g'$ such that $T_i^{(e)} = D_i^{(e)}$ and $T_{j'}^{(e)} = D_{j'}^{(e)}$, show that the matrix of $F^*$ relative to these bases is the matrix $(\partial F_j / \partial X_i)_0$ (constant term of $\partial F_j / \partial X_i$, where i is the index of the rows and j that of the columns).