Differential calculus on \(\mathbb{N}\)-graded manifolds (Q2421759)

Summary: The differential calculus, including formalism of linear differential operators and the Chevalley-Eilenberg differential calculus, over \(\mathbb{N}\)-graded commutative rings and on \(\mathbb{N}\)-graded manifolds is developed. This is a straightforward generalization of the conventional differential calculus over commutative rings and also is the case of the differential calculus over Grassmann algebras and on \(\mathbb{N}_2\)-graded manifolds. We follow the notion of an \(\mathbb{N}\)-graded manifold as a local-ringed space whose body is a smooth manifold \(Z\). A key point is that the graded derivation module of the structure ring of graded functions on an \(\mathbb{N}\)-graded manifold is the structure ring of global sections of a certain smooth vector bundle over its body \(Z\). Accordingly, the Chevalley-Eilenberg differential calculus on an \(\mathbb{N}\)-graded manifold provides it with the de Rham complex of graded differential forms. This fact enables us to extend the differential calculus on \(\mathbb{N}\)-graded manifolds to formalism of nonlinear differential operators, by analogy with that on smooth manifolds, in terms of graded jet manifolds of \(\mathbb{N}\)-graded bundles.