2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67529Generalized symmetries and supersymmetries depending on derivatives of dynamic variables are treated in a most general setting. Studding cohomology of the variational bicomplex, we state the first variational formula and conservation laws for Lagrangian systems on fiber bundles and graded manifolds under generalized symmetries and supersymmetries of any order. Cohomology of nilpotent generalized supersymmetries are obtained.29 pagesAlgebraic GeometryMathematical Physics58A20; 58A50; 81T60text