The thought process that led me to this question is that the identity $$\left(\prod_i \frac1{1-x_i}\right)\left(\prod_i {1-x_i}\right)=1$$ can be understood as expressing exactness of the Koszul complex. This identity is rewritten by taking $\left(\prod_i \frac1{1-x_i}\right)$ as the generating function for the complete symmetric functions $h_n$ and $\left(\prod_i {1+x_i}\right)$ as the generating function for the elementary symmetric functions $e_n$.