---
tag: 9X3B0EX04F
kind: definition
name: Graded manifold
msc: [58A50, 58C50]
defines: E1CA8QFEHB
uses: [F45VQ47GPV, KNDXDE7K8M, TB6TM6R4MD, 0MDMZ965VE, 0V33BNKZ9S, 0ZB2JG1M9C]
sources:
  - url: https://en.wikipedia.org/wiki/Graded_manifold
---

## Definition

Let $n, m \ge 0$ be integers, let $Z$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, and let $C^\infty_Z$ be its sheaf of real [[smooth functions|0V33BNKZ9S]].
Let $\Lambda_m = \Lambda(\R^m)$ be the real [[exterior algebra|0MDMZ965VE]] on $m$ generators, with its $\Z/2$-grading into even and odd parts.
A *graded manifold* of dimension $(n, m)$ with body $Z$ is a [[ringed space|KNDXDE7K8M]] $(Z, \mathcal{A})$ such that:

1. $\mathcal{A}$ is a [[sheaf|TB6TM6R4MD]] of $\Z/2$-graded commutative real algebras on $Z$.
2. There is a surjective morphism of sheaves of algebras $\mathcal{A} \to C^\infty_Z$ whose kernel is the ideal $\mathcal{N}$ of [[nilpotent|0ZB2JG1M9C]] sections.
3. Every point of $Z$ has an open neighbourhood $U$ such that $\mathcal{A}|_U$ is isomorphic, as a sheaf of graded algebras, to the sheaf $C^\infty_U \otimes_\R \Lambda_m$.

Here $\Z/2$-graded commutative means $ab = (-1)^{\abs{a}\abs{b}} ba$ for homogeneous elements $a, b$ of degrees $\abs{a}, \abs{b} \in \set{0, 1}$.

## 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 $\theta_i \theta_j = -\theta_j \theta_i$.