Higher Koszul Brackets on the Cotangent Complex

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