Remark on the Serre-Swan theorem for graded manifolds

TL;DR

This paper extends the Serre-Swan theorem to graded manifolds, showing that certain graded algebras correspond to graded manifolds with a given base, which has implications for representing odd fields in field theory.

Contribution

It combines the Batchelor and Serre-Swan theorems to characterize graded commutative algebras as exterior algebras of projective modules, linking algebraic structures to geometric graded manifolds.

Findings

A graded commutative algebra is isomorphic to a graded manifold's structure ring iff it is an exterior algebra of a finite rank projective module.

Odd fields in field theory can be modeled as graded functions on graded manifolds with the same base manifold.

Provides a criterion for when a graded algebra corresponds to a graded manifold structure.

Abstract

Combining the Batchelor theorem and the Serre-Swan theorem, we come to that, given a smooth manifold $X$, a graded commutative $C^\infty(X)$-algebra $\cA$ is isomorphic to the structure ring of a graded manifold with a body $X$ iff it is the exterior algebra of some projective $C^\infty(X)$-module of finite rank. In particular, it follows that odd fields in field theory on a smooth manifold $X$ can be represented by graded functions on some graded manifold with body $X$.