Higher Koszul brackets on the cotangent complex

Abstract

Let $n\ge 1$ and $A$ be a commutative algebra of the form $\mathbf{k}[x_1,x_2,\dots ,x_n]/I$ where $\mathbf{k}$ is a field of characteristic $0$ and $I\subseteq \mathbf{k}[x_1,x_2,\dots ,x_n]$ 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 ${\Omega }_{A|\mathbf{k}}$ of Kähler differentials making $(A,{\Omega }_{A|\mathbf{k}})$ a Lie-Rinehart pair. Recall that $A$ is regular if and only if ${\Omega }_{A|\mathbf{k}}$ is projective as an $A$-module. If $A$ is not regular, the cotangent complex ${\mathbb{L}}_{A|\mathbf{k}}$ may serve as a replacement for the $A$-module ${\Omega }_{A|\mathbf{k}}$. We prove that there is a structure of an $L_{\infty }$-algebroid on ${\mathbb{L}}_{A|\mathbf{k}}$, compatible with the Lie-Rinehart pair $(A,{\Omega }_{A|\mathbf{k}})$. The $L_{\infty }$-algebroid on ${\mathbb{L}}_{A|\mathbf{k}}$ 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 $\mathbf{k}[x_1,x_2,\dots ,x_n]$ carries a (possibly nonstandard) ${\mathbb{Z}}_{\ge 0}$-grading and both $I$ and $\{,\}$ are homogeneous.

32 pages, 2 tables. V2: We corrected an error in the bracket table of the invariants of the Kleinian singularity E_7 (which does not appear in the published version in IMRN, https://doi.org/10.1093/imrn/rnac170) that was identified by William Osnayder Clavijo Esquivel; we express our appreciation for identifying this error

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• The symplectic form associated to a singular Poisson algebra