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  "body": "\n## Definition\n\nLet $n, m \\ge 0$ 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]].\nLet $\\Lambda_m = \\Lambda(\\R^m)$ be the real [[exterior algebra|0MDMZ965VE]] on $m$ generators, with its $\\Z/2$-grading into even and odd parts.\nA *graded manifold* of dimension $(n, m)$ with body $Z$ is a [[ringed space|KNDXDE7K8M]] $(Z, \\mathcal{A})$ such that:\n\n1. $\\mathcal{A}$ is a [[sheaf|TB6TM6R4MD]] of $\\Z/2$-graded commutative real algebras on $Z$.\n2. 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.\n3. 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 $C^\\infty_U \\otimes_\\R \\Lambda_m$.\n\nHere $\\Z/2$-graded commutative means $ab = (-1)^{\\abs{a}\\abs{b}} ba$ for homogeneous elements $a, b$ of degrees $\\abs{a}, \\abs{b} \\in \\set{0, 1}$.\n\n## Remarks\n\nExample: 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 $\\Lambda E^*$.\nOver 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.\nThe 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.\nFor $E = T^*Z$ this is the graded manifold of differential forms on $Z$.\nBatchelor's theorem says that every smooth graded manifold is isomorphic to one of this form, though not canonically.\n\nNon-example: let $\\mathcal{A} = C^\\infty_Z[\\varepsilon] / (\\varepsilon^2)$, with $\\varepsilon$ declared even, so that all of $\\mathcal{A}$ is even.\nClauses 1 and 2 hold, with nilpotent ideal $\\varepsilon C^\\infty_Z$.\nClause 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.\n\nThe local odd coordinates are the images $\\theta_1, \\dots, \\theta_m$ of a basis of $\\R^m$ under clause 3; they satisfy $\\theta_i \\theta_j = -\\theta_j \\theta_i$.\n",
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