The homotopy Lie algebra of a Tor-independent tensor product

TL;DR

This paper explores the structure of the homotopy Lie algebra of a Tor-independent tensor product of local rings, revealing how it relates to the individual algebras of the rings involved, with implications for cohomology and Tor algebra structures.

Contribution

It establishes a structural description of the homotopy Lie algebra of a tensor product of local rings in terms of the algebras of the factors, extending understanding of their algebraic and homotopical properties.

Findings

Homotopy Lie algebra of the tensor product is a pullback of the individual algebras.

Structural results on Andre9-Quillen cohomology and stable cohomology.

An equality relating Poincare9 series of residue fields.

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 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 $S$.