The Schouten-Nijenhuis bracket in infinite dimensions

The Schouten-Nijenhuis bracket on smooth infinite-dimensional manifolds $M$ is developed in two steps: For summable multivector fields whose pointwise dual are all differential form, and in an extended form for multivector fields which are sections of $L_{\text{skew}}^{\bullet }(T^{\ast }M,\mathbb{R})$. We need to either assume that $C^{\mathrm{\infty }}(M)$ separates points on $TM$, or consider sheaves of local sections.