| Abstrakt: |
Let |$A=\boldsymbol{k}[x_1,x_2,\dots ,x_n]/I$| 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. [ABSTRACT FROM AUTHOR] |
