The homotopy Lie algebra of a Tor-independent tensor product

In this article we investigate a pair of surjective local ring maps $S_1\leftarrow R\to S_2$ and their relation to the canonical projection $R\to S_1\otimes_R S_2$, where $S_1,S_2$ are Tor-independent over $R$. Our main result asserts a structural connection between the homotopy Lie algebra of $S:=S_1\otimes_R S_2$, denoted $\pi(S)$, in terms of those of $R,S_1$ and $S_2$. Namely, $\pi(S)$ is the pullback of (adjusted) Lie algebras along the maps $\pi(S_i)\to \pi(R)$ in various cases, including when the maps above have residual characteristic zero. Consequences to the main theorem include structural results on Andr\'{e}-Quillen cohomology, stable cohomology, and Tor algebras, as well as an equality relating the Poincar\'{e} series of the common residue field of $R,S_1,S_2$ and $S$.