2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/97276It is proved that given a divergence operator on the structural sheaf of graded commutative algebras of a supermanifold, it is possible to construct a generating operator for the Krashil'shchik-Schouten bracket. This is a particular case of the construction of generating operators for a special class of bigraded Gerstenhaber algebras. Also, some comments on the generalization of these results to the context of $n-$graded Jacobi algebras are included.Mathematical Physics13Nxx, 53Dxx (Primary) 14F05, 53DA7 (Secondary)text