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 Post subject: Algebraic Tangent Space Posted: Sun Jul 24, 2016 11:34 pm
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Let $M$ be a smooth manifold. For a point $p \in M$, set $\mathfrak{m}_{p} = \left\{ \ f \in C^{\infty}(M) \ \big| \ f(p) = 0 \ \right\}$Show that
1. $\mathfrak{m}_{p}$ is a maximal ideal of $C^{\infty}(M)$.
2. Any derivation on $C^{\infty}(M)$ is determined by its values of $\mathfrak{m}_{p}$.
3. There is an $\mathbb{R}$-linear isomorphism $T_{p}M \longrightarrow \text{Hom} \left( \mathfrak{m}_{p} / \mathfrak{m}_{p}^{2}, \mathbb{R} \right)$, where $T_{p}M$ is the (usual/geometric) tangent space of $M$ at $p$.
[You may have to use the isomorphism $T_{p}M \cong \mathcal{Der} \left( C^{\infty}(M) _{p} \right)$ ]

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