# External direct product of formal group laws

Suppose $R$ is a commutative unital ring. Suppose $F_1$ and $F_2$ are formal group laws over $R$ of dimensions $d_1$ and $d_2$ respectively. The external direct product of $F_1$ and $F_2$, which we denote $F_1 \times F_2$, is a formal group law of dimension $d_1 + d_2$, given by:
$(F_1 \times F_2)(x_1,x_2,\dots,x_{d_1},x_{d_1 + 1},\dots,x_{d_1+d_2},y_1,y_2,\dots,y_{d_1},y_{d_1+1},\dots,y_{d_1+d_2})$
$= (F_1(x_1,x_2,\dots,x_{d_1},y_1,y_2,\dots,y_{d_1}),F_2(x_{d_1 + 1},\dots,x_{d_1+d_2},y_{d_1+1},\dots,y_{d_1+d_2}) )$
In other words, the first $d_1$ coordinates of $F_1 \times F_2$ are obtained by applying $F_1$ to the first $d_1$ coordinates of the two inputs, and the last $d_2$ coordinates of $F_1 \times F_2$ are obtained by applying $F_2$ to the last $d_2$ coordinates of the two inputs.