Higher Koszul brackets on the cotangent complex

Abstract

Let $n\ge 1$ and $A$ be a commutative algebra of the form $\boldsymbol k[x_1,x_2,\dots, x_n]/I$ where $\boldsymbol k$ is a field of characteristic $0$ and $I\subseteq \boldsymbol k[x_1,x_2,\dots, x_n]$ is an ideal. Assume that there is a Poisson bracket $\{\:,\:\}$ on $S$ such that $\{I,S\}\subseteq I$ and let us denote the induced bracket on $A$ by $\{\:,\:\}$ as well. It is well-known that $[\mathrm d x_i,\mathrm d x_j]:=\mathrm d\{x_i,x_j\}$ defines a Lie bracket on the $A$-module $Ω_{A|\boldsymbol k}$ of Kähler differentials making $(A,Ω_{A|\boldsymbol k})$ a Lie-Rinehart pair. Recall that $A$ is regular if and only if $Ω_{A|\boldsymbol k}$ is projective as an $A$-module. If $A$ is not regular, the cotangent complex $\mathbb L_{A|\boldsymbol k}$ may serve as a replacement for the $A$-module $Ω_{A|\boldsymbol k}$. We prove that there is a structure of an $L_\infty$-algebroid on $\mathbb L_{A|\boldsymbol k}$, compatible with the Lie-Rinehart pair $(A,Ω_{A|\boldsymbol k})$. The $L_\infty$-algebroid on $\mathbb L_{A|\boldsymbol k}$ actually comes from a $P_\infty$-algebra structure on the resolvent of the morphism $k[x_1,x_2,\dots, x_n]\to A$. We identify examples when this $L_\infty$-algebroid simplifies to a dg Lie algebroid. For aesthetic reasons we concentrate on cases when $ \boldsymbol k[x_1,x_2,\dots, x_n]$ carries a (possibly nonstandard) $\mathbb Z_{\ge 0}$-grading and both $I$ and $\{\:,\:\}$ are homogeneous.