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 $\pi (S)$, 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 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 $S$.

20 pages. Corrected a mistake in 1.7; simplified and reorganized Sections 4 and 5