Batchelor theorem - Knowlpedia

Statement

Let $X$ be a finite-dimensional smooth real supermanifold whose reduced manifold is Hausdorff and paracompact. The Batchelor theorem states that there is a vector bundle $E\to X_{\mathrm{red}}$ and an isomorphism

X\cong \Pi E

of supermanifolds. Equivalently,

\mathcal O_X\cong \Lambda_{\mathcal O_{X_{\mathrm{red}}}} (\mathcal J_X/\mathcal J_X^2),

after identifying the locally free module $\mathcal J_X/\mathcal J_X^2$ with sections of $E^*$ .

Noncanonical is essential

The theorem is an existence theorem. A splitting requires choices, commonly constructed using partitions of unity, and is generally neither unique nor natural with respect to supermanifold morphisms. Thus the smooth category is not obtained by simply declaring that vector bundles and their exterior algebras are the morphism-free data of all supermanifolds.

The result is special to the smooth real setting. Complex-analytic and algebraic supermanifolds can have obstruction classes preventing a global splitting, so the statement must not be transported to those categories without additional hypotheses.

References

M. Batchelor, “The structure of supermanifolds,” Transactions of the American Mathematical Society 253, 1979, 329–338. Article.
K. Gawędzki, “Supersymmetries—mathematics of supergeometry,” Annales de l’Institut Henri Poincaré A 27(4), 1977, 335–366. Numdam record.