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  "body": "\n## Definition\n\nLet $p, q \\ge 0$ be integers.\nLet $\\Lambda_q = \\Lambda[\\theta_1, \\dots, \\theta_q]$ be the real [[exterior algebra|0MDMZ965VE]] on $q$ generators.\nIt is graded by parity: a product of an even number of generators is even, and a product of an odd number is odd.\nFor an open set $U \\subseteq \\R^p$, let $\\mathcal{O}_{p|q}(U) = C^\\infty(U) \\otimes_\\R \\Lambda_q$.\nThe pair $\\R^{p|q} = (\\R^p, \\mathcal{O}_{p|q})$ is the *superdomain* of dimension $p|q$.\n\nA smooth *supermanifold* of dimension $p|q$ is a pair $\\mathcal{M} = (\\abs{\\mathcal{M}}, \\mathcal{O}_\\mathcal{M})$ with these properties.\n\n1. $\\abs{\\mathcal{M}}$ is a Hausdorff, second-countable [[topological space|0CQHFPGJ45]].\n2. $\\mathcal{O}_\\mathcal{M}$ 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 $\\abs{a}, \\abs{b}$.\n3. 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 $U \\subseteq \\R^p$.\n\nA morphism of supermanifolds is a morphism of [[locally ringed spaces|B7FPT8JTXR]] whose map on sheaves preserves parity.\n\n## Remarks\n\nLet $\\mathcal{J} \\subseteq \\mathcal{O}_\\mathcal{M}$ be the ideal generated by odd elements.\nThen $(\\abs{\\mathcal{M}}, \\mathcal{O}_\\mathcal{M}/\\mathcal{J})$ is an ordinary [[smooth manifold|F45VQ47GPV]] of dimension $p$, the underlying manifold.\nLet $E$ 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.\nBy Batchelor's theorem, every smooth supermanifold is isomorphic, not canonically, to some $(M, \\Gamma(\\Lambda E^*))$.\n\nExample: $\\R^{p|q}$ itself, and more generally $(M, \\Gamma(\\Lambda E^*))$ for a vector bundle $E \\to M$ of rank $q$.\nOver 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.\n\nNon-example: $(\\R, \\mathcal{O})$ with $\\mathcal{O}(U) = C^\\infty(U)[\\varepsilon]/(\\varepsilon^2)$ and $\\varepsilon$ declared even.\nEvery element is even, so the sheaf is not locally isomorphic to $C^\\infty \\otimes \\Lambda_q$ with $q \\ge 1$, which has odd elements.\nIt is not locally isomorphic to $C^\\infty$, the case $q = 0$, either, since $C^\\infty$ has no nonzero nilpotents.\nClause 3 fails.\n",
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