aa r X i v : . [ m a t h - ph ] A p r Remark on the Serre – Swan theorem for gradedmanifolds
G. SARDANASHVILYDepartment of Theoretical Physics, Moscow State University, 117234 Moscow,Russia
Abstract
Combining the Batchelor theorem and the Serre – Swan theorem, wecome to that, given a smooth manifold X , a graded commutative C ∞ ( X )-algebra A is isomorphic to the structure ring of a graded manifold witha body X iff it is the exterior algebra of some projective C ∞ ( X )-moduleof finite rank. In particular, it follows that odd fields in field theory ona smooth manifold X can be represented by graded functions on somegraded manifold with body X . In classical field theory, there are different descriptions of odd fields on gradedmanifolds [3, 7, 8, 12] and supermanifolds [4, 5]. Both graded manifolds andsupermanifolds are phrased in terms of sheaves of graded commutative algebras[1, 7, 11]. However, graded manifolds are characterized by sheaves on smoothmanifolds, while supermanifolds are constructed by gluing sheaves on super-vector spaces. Treating odd fields on a smooth manifold X , one can followforthcoming Theorem 3. It states that, if a graded commutative algebra is anexterior algebra of some projective C ∞ ( X )-module of finite rank, it is isomor-phic to the algebra of graded functions on a graded manifold whose body is X .By virtue of this theorem, odd fields can be represented by generating elementsof the structure ring of a graded manifold whose body is X [6, 7, 12].Let X be a smooth manifold which is assumed to be real, finite-dimensional,Hausdorff, second-countable, and connected. The well-known Serre – Swantheorem establishes the following. Theorem 1. A C ∞ ( X ) -module P is isomorphic to the structure module ofsections of a smooth vector bundle over X iff it is a projective module of finiterank. Originally proved for smooth bundles over a compact base X , this theoremhas been extended to an arbitrary X [7, 9, 10]Turn now to graded manifolds [1, 7, 11]. A graded manifold of dimension( n, m ) is defined as a local-ringed space ( X, A ) where X is an n -dimensionalsmooth manifold X and A = A ⊕ A is a sheaf of graded commutative algebrasof rank m such that: 1 there is the exact sequence of sheaves0 → R → A σ → C ∞ Z X → , R = A + ( A ) , where C ∞ X is the sheaf of smooth real functions on X ; • R / R is a locally free sheaf of C ∞ X -modules of finite rank (with respectto pointwise operations), and the sheaf A is locally isomorphic to the exteriorproduct ∧ C ∞ X ( R / R ).A sheaf A is called the structure sheaf of a graded manifold ( X, A ), anda manifold X is said to be the body of ( X, A ). Sections of the sheaf A arecalled graded functions on a graded manifold ( X, A ). They make up a gradedcommutative C ∞ ( X )-ring A ( X ) called the structure ring of ( X, A ).The above mentioned Batchelor theorem states the following [1, 7]. Theorem 2.
Let ( X, A ) be a graded manifold. There exists a vector bundle E → X with an m -dimensional typical fibre V such that the structure sheaf A of ( X, A ) is isomorphic to the structure sheaf A E = S ∧ E ∗ of germs of sectionsof the exterior product ∧ E ∗ = ( X × R ) ⊕ X E ∗ ⊕ X ∧ E ∗ ⊕ X · · · ⊕ X ∧ m E ∗ (1) of the dual E ∗ of E . Its typical fibre is the Grassmann algebra ∧ V ∗ = R ⊕ V ⊕ ∧ V ⊕ · · · ⊕ m ∧ V. In particular, it follows that the structure ring A ( X ) of a graded manifold( X, A ) is isomorphic to the ring of sections of the exterior product (1).It should be emphasized that Batchelor’s isomorphism in Theorem 2 fails tobe canonical, but in field models it usually is fixed from the beginning. We agreeto call a graded manifold ( Z, A E ) whose structure sheaf is the sheaf of germs ofsections of some exterior bundle ∧ E ∗ the simple graded manifold modelled overa vector bundle E → ZX ,Combining Batchelor Theorem 2 and Serre – Swan Theorem 1, we come tothe following Serre – Swan theorem for graded manifolds [2, 7, 12]. Theorem 3.
Let X be a smooth manifold. A graded commutative C ∞ ( X ) -algebra A is isomorphic to the structure ring of a graded manifold with a body X iff it is the exterior algebra of some projective C ∞ ( X ) -module of finite rank.Proof. By virtue of Batchelor Theorem 2, any graded manifold is isomorphicto a graded manifold ( X, A E ) modelled over some vector bundle E → X . Itsstructure ring A E of graded functions consists of sections of the exterior bundle ∧ E ∗ (1) and, thus, it is generated by a C ∞ ( X )-module E ∗ ( X ) of sections of E ∗ → X . By virtue of Serre – Swan Theorem 1, this module is a projectivemodule of finite rank. Conversely, let a graded commutative C ∞ ( X )-algebra A
2e generated by some projective C ∞ ( X )-module of finite rank. In accordancewith the Serre – Swan Theorem 1, this module is isomorphic to a module of sec-tions of some vector bundle E → X and, thus, A is isomorphic to the structurering of a simple graded manifold modelled over E .As a consequence, a graded commutative algebra in Theorem 3 possessesa number of particular properties. For instance, the Chevalley – Eilenbergdifferential calculus of such an algebra is minimal, and the cohomology of its deRham complex equals the de Rham cohomology of a manifold X [7].As physical outcome, let us mention that higher-stage Noether identitiesand gauge symmetries of a reducible degenerate Lagrangian system on a fiberbundle over X are parameterized by odd fields, called the antifields and ghosts,respectively [2, 7]. References [1] Bartocci, C., Bruzzo, U. and Hern´andez Ruip´erez, D.(1991).
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