The homotopy Lie algebra of a Torindependent 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 Torindependent 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 AndréQuillen 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 $S$.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.01003
 Bibcode:
 2021arXiv210901003F
 Keywords:

 Mathematics  Commutative Algebra;
 13D02;
 13D07;
 16E45
 EPrint:
 20 pages. Corrected a mistake in 1.7