---
tag: TQNG7T2RQ6
kind: definition
name: Supermanifold
msc: [58A50]
defines: Z0SB9HQBAJ
uses: [0CQHFPGJ45, TB6TM6R4MD, 6JZW4JC68Z, 0MDMZ965VE, B7FPT8JTXR, F45VQ47GPV, P4DZM975BV]
sources:
  - url: https://en.wikipedia.org/wiki/Supermanifold
---

## Definition

Let $p, q \ge 0$ be integers.
Let $\Lambda_q = \Lambda[\theta_1, \dots, \theta_q]$ be the real [[exterior algebra|0MDMZ965VE]] on $q$ generators.
It is graded by parity: a product of an even number of generators is even, and a product of an odd number is odd.
For an open set $U \subseteq \R^p$, let $\mathcal{O}_{p|q}(U) = C^\infty(U) \otimes_\R \Lambda_q$.
The pair $\R^{p|q} = (\R^p, \mathcal{O}_{p|q})$ is the *superdomain* of dimension $p|q$.

A smooth *supermanifold* of dimension $p|q$ is a pair $\mathcal{M} = (\abs{\mathcal{M}}, \mathcal{O}_\mathcal{M})$ with these properties.

1. $\abs{\mathcal{M}}$ is a Hausdorff, second-countable [[topological space|0CQHFPGJ45]].
2. $\mathcal{O}_\mathcal{M}$ is a [[sheaf|TB6TM6R4MD]] of supercommutative [[superalgebras|6JZW4JC68Z]] on $\abs{\mathcal{M}}$; supercommutative means $ab = (-1)^{\abs{a}\abs{b}} ba$ for homogeneous $a, b$ of parities $\abs{a}, \abs{b}$.
3. Every point of $\abs{\mathcal{M}}$ has an open neighbourhood $V$ with the following property. As a space with a sheaf of superalgebras, $(V, \mathcal{O}_\mathcal{M}|_V)$ is isomorphic to $(U, \mathcal{O}_{p|q}|_U)$ for some open $U \subseteq \R^p$.

A morphism of supermanifolds is a morphism of [[locally ringed spaces|B7FPT8JTXR]] whose map on sheaves preserves parity.

## Remarks

Let $\mathcal{J} \subseteq \mathcal{O}_\mathcal{M}$ be the ideal generated by odd elements.
Then $(\abs{\mathcal{M}}, \mathcal{O}_\mathcal{M}/\mathcal{J})$ is an ordinary [[smooth manifold|F45VQ47GPV]] of dimension $p$, the underlying manifold.
Let $E$ be a [[vector bundle|P4DZM975BV]] of rank $q$ over a manifold $M$, and let $\Gamma(\Lambda E^*)$ be the sheaf of sections of its exterior algebra bundle.
By Batchelor's theorem, every smooth supermanifold is isomorphic, not canonically, to some $(M, \Gamma(\Lambda E^*))$.

Example: $\R^{p|q}$ itself, and more generally $(M, \Gamma(\Lambda E^*))$ for a vector bundle $E \to M$ of rank $q$.
Over a trivialising open set $U$ of $E$ with $U$ a chart domain, $\Gamma(\Lambda E^*)|_U \cong C^\infty(U) \otimes \Lambda_q$, which is clause 3.

Non-example: $(\R, \mathcal{O})$ with $\mathcal{O}(U) = C^\infty(U)[\varepsilon]/(\varepsilon^2)$ and $\varepsilon$ declared even.
Every element is even, so the sheaf is not locally isomorphic to $C^\infty \otimes \Lambda_q$ with $q \ge 1$, which has odd elements.
It is not locally isomorphic to $C^\infty$, the case $q = 0$, either, since $C^\infty$ has no nonzero nilpotents.
Clause 3 fails.