The homotopy Lie algebra of a Tor-independent tensor product

Abstract

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 $\mathrm{\pi }(S)$, in terms of those of R, $S_1$, and $S_2$; namely, $\mathrm{\pi }(S)$ is the pullback of (adjusted) Lie algebras along the maps $\mathrm{\pi }(S_i)\to \mathrm{\pi }(R)$ in various cases, including when the maps above have residual characteristic zero. Consequences to the main theorem include structural results on André–Quillen cohomology, stable cohomology, and Tor algebras, as well as an equality relating the Poincaré series of the common residue field of R, $S_1$, $S_2$, and S.