Graded manifold

In algebraic geometry, graded manifolds are extensions of the concept of manifolds 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. A graded manifold of dimension is defined as a locally ringed space where is an -dimensional smooth manifold and is a -sheaf of Grassmann algebras of rank where is the sheaf of smooth real functions on . The sheaf is called the structure sheaf of the graded manifold , and the manifold is said to be the body of . Sections of the sheaf are called graded functions on a graded manifold . They make up a graded commutative -ring called the structure ring of . The well-known Batchelor theorem and Serre–Swan theorem characterize graded manifolds as follows. Let be a graded manifold. There exists a vector bundle with an -dimensional typical fiber such that the structure sheaf of is isomorphic to the structure sheaf of sections of the exterior product of , whose typical fibre is the Grassmann algebra . Let be a smooth manifold. A graded commutative -algebra is isomorphic to the structure ring of a graded manifold with a body if and only if it is the exterior algebra of some projective -module of finite rank. Note that above mentioned Batchelor's isomorphism fails to be canonical, but it often is fixed from the beginning. In this case, every trivialization chart of the vector bundle yields a splitting domain of a graded manifold , where is the fiber basis for . Graded functions on such a chart are -valued functions where are smooth real functions on and are odd generating elements of the Grassmann algebra . Given a graded manifold , graded derivations of the structure ring of graded functions are called graded vector fields on . They constitute a real Lie superalgebra with respect to the superbracket where denotes the Grassmann parity of .