Higher Koszul Brackets on the Cotangent Complex

Abstract

AbstractLet $A=\boldsymbol{k}[x_1,x_2,\dots ,x_n]/I$ be a commutative algebra where $\boldsymbol{k}$ is a field, $\operatorname{char}(\boldsymbol{k})=0$, and $I\subseteq S:=\boldsymbol{k}[x_1,x_2,\dots , x_n]$ a Poisson ideal. It is well known that $[\textrm{d} x_i,\textrm{d} x_j]:=\textrm{d}\{x_i,x_j\}$ defines a Lie bracket on the $A$-module $\Omega _{A|\boldsymbol{k}}$ of Kähler differentials, making $(A,\Omega _{A|\boldsymbol k})$ a Lie–Rinehart pair. If $A$ is not regular, that is, $\Omega _{A|\boldsymbol{k}}$ is not projective, the cotangent complex $\mathbb{L}_{A|\boldsymbol{k}}$ serves as a replacement for $\Omega _{A|\boldsymbol k}$. We prove that $\mathbb{L}_{A|\boldsymbol{k}}$ is an $L_\infty $-algebroid compatible with the Lie–Rinehart pair $(A,\Omega _{A|\boldsymbol{k}})$. The $L_\infty $-algebroid structure comes from a $P_\infty $-algebra structure on the resolvent of the morphism $S\to A$. We identify examples when this $L_\infty $-algebroid simplifies to a dg Lie algebroid, concentrating on cases where $S$ is $\mathbb{Z}_{\ge 0}$-graded and $I$ and $\{\:,\:\}$ are homogeneous.