External direct product of formal group laws

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Definition

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.