{{ safesubst:#invoke:Unsubst||$N=Technical |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} In algebraic geometry, Graded manifolds are extensions of the manifold concept based on ideas coming from supersymmetry and supercommutative algebra. Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutative algebras. However, graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves of supervector spaces.

### Serre-Swan theorem for graded manifolds

Let $Z$ be a smooth manifold. A graded commutative $C^{\infty }(Z)$ -algebra is isomorphic to the structure ring of a graded manifold with a body $Z$ if and only if it is the exterior algebra of some projective $C^{\infty }(Z)$ -module of finite rank.

Provided with the graded exterior product

graded one-forms generate the graded exterior algebra $O^{*}(Z)$ of graded exterior forms on a graded manifold. They obey the relation

where $|\phi |$ denotes the form degree of $\phi$ . The graded exterior algebra $O^{*}(Z)$ is a graded differential algebra with respect to the graded exterior differential