Definition

A local $L_\infty$-algebra on a smooth manifold $X$ is defined by the following data:

1. A graded vector bundle $L\rightarrow X$,

2. A square $0$ differential operator $\mathrm{d}:\Gamma(X,L)\rightarrow\Gamma(X,L)$ of cohomological degree $1$,

3. A collection $\{ l_n \}_{n \geq 2}$ of poly-differential operators

   $$l_n \,:\; \Gamma(X,L)^{\otimes n} \rightarrow \Gamma(X,L),$$

   which are alternating, of cohomological degree $2-n$ and such that they endow $\Gamma(X,L)$ with an $L_\infty$ structure.

Definition

The Chevalley-Eilenberg cochain complex $\mathrm{CE}(\mathcal{L})$ of the local $L_\infty$-algebra $\mathcal{L}$ is defined by

equipped with the usual Chevalley-Eilenberg differential; where $\hat{\otimes}$ denotes the completed projective tensor product, $\mathrm{Hom}$ denotes the dg-vector space of continuous linear maps and the subscript $S_n$ denotes the coinvariants respect to the action of the symmetric group $S_n$.

Definition

The reduced Chevalley-Eilenberg cochain complex $\mathrm{CE}_{\mathrm{red}}(\mathcal{L})$ of the local $L_\infty$-algebra $\mathcal{L}$ is defined by the kernel

of the natural augmentation map $\mathrm{CE}\left(\mathcal{L}(U)\right) \rightarrow \mathbb{R}$.

Definition

The local Chevalley-Eilenberg cochain complex $\mathrm{CE}_{\mathrm{loc}}(\mathcal{L})$ of the local $L_\infty$-algebra $\mathcal{L}$ is defined by

where $\mathscr{Jet}(L)$ is the local $L_\infty$-algebra, in the category of $D_X$-modules, of sections of the jet bundle of $L$ and $mathscr{Dens}_X$ is the right $D_X$-module of densities on $X$.