Graded manifold

Remarks

Example: let $E \to Z$ be a smooth real vector bundle of rank $m$, and let $\mathcal{A}$ be the sheaf of smooth sections of the exterior algebra bundle $\Lambda E^*$. Over an open set $U$ where $E$ is trivial, a frame gives $\mathcal{A}|_U \cong C^\infty_U \otimes \Lambda_m$, so clause 3 holds. The kernel of the map $\mathcal{A} \to C^\infty_Z$ that keeps the degree-zero component is generated by sections of $E^*$, which are nilpotent, so clause 2 holds. For $E = T^*Z$ this is the graded manifold of differential forms on $Z$. Batchelor's theorem says that every smooth graded manifold is isomorphic to one of this form, though not canonically.

Non-example: let $\mathcal{A} = C^\infty_Z[\varepsilon] / (\varepsilon^2)$, with $\varepsilon$ declared even, so that all of $\mathcal{A}$ is even. Clauses 1 and 2 hold, with nilpotent ideal $\varepsilon C^\infty_Z$. Clause 3 fails: for $m = 0$ the local model has no non-zero nilpotents, and for $m \ge 1$ it has non-zero odd elements, while $\mathcal{A}$ has nilpotents and no odd part.

The local odd coordinates are the images $\theta_1, \dots, \theta_m$ of a basis of $\R^m$ under clause 3; they satisfy $\theta_i \theta_j = -\theta_j \theta_i$.