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.
Graded manifolds A graded manifold of dimension ( n , m ) {\displaystyle (n,m)} is defined as a locally ringed space ( Z , A ) {\displaystyle (Z,A)} where Z {\displaystyle Z} is an n {\displaystyle n} -dimensional smooth manifold and A {\displaystyle A} is a C Z ∞ {\displaystyle C_{Z}^{\infty }} -sheaf of Grassmann algebras of rank m {\displaystyle m} where C Z ∞ {\displaystyle C_{Z}^{\infty }} is the sheaf of smooth real functions on Z {\displaystyle Z} . The sheaf A {\displaystyle A} is called the structure sheaf of the graded manifold ( Z , A ) {\displaystyle (Z,A)} , and the manifold Z {\displaystyle Z} is said to be the body of ( Z , A ) {\displaystyle (Z,A)} . Sections of the sheaf A {\displaystyle A} are called graded functions on a graded manifold ( Z , A ) {\displaystyle (Z,A)} . They make up a graded commutative C ∞ ( Z ) {\displaystyle C^{\infty }(Z)} -ring A ( Z ) {\displaystyle A(Z)} called the structure ring of ( Z , A ) {\displaystyle (Z,A)} . The well-known Batchelor theorem and Serre–Swan theorem characterize graded manifolds as follows.
Serre–Swan theorem for graded manifolds Let ( Z , A ) {\displaystyle (Z,A)} be a graded manifold. There exists a vector bundle E → Z {\displaystyle E\to Z} with an m {\displaystyle m} -dimensional typical fiber V {\displaystyle V} such that the structure sheaf A {\displaystyle A} of ( Z , A ) {\displaystyle (Z,A)} is isomorphic to the structure sheaf of sections of the exterior product Λ ( E ) {\displaystyle \Lambda (E)} of E {\displaystyle E} , whose typical fibre is the Grassmann algebra Λ ( V ) {\displaystyle \Lambda (V)} . Let Z {\displaystyle Z} be a smooth manifold. A graded commutative C ∞ ( Z ) {\displaystyle C^{\infty }(Z)} -algebra is isomorphic to the structure ring of a graded manifold with a body Z {\displaystyle Z} if and only if it is the exterior algebra of some projective C ∞ ( Z ) {\displaystyle C^{\infty }(Z)} -module of finite rank.
Graded functions 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 ( U ; z A , y a ) {\displaystyle (U;z^{A},y^{a})} of the vector bundle E → Z {\displaystyle E\to Z} yields a splitting domain ( U ; z A , c a ) {\displaystyle (U;z^{A},c^{a})} of a graded manifold ( Z , A ) {\displaystyle (Z,A)} , where { c a } {\displaystyle \{c^{a}\}} is the fiber basis for E {\displaystyle E} . Graded functions on such a chart are Λ ( V ) {\displaystyle \Lambda (V)} -valued functions
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