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Algebraic Tangent Space

Differential Geometry
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Tsakanikas Nickos
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Algebraic Tangent Space

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Post by Tsakanikas Nickos » Sun Jul 24, 2016 11:34 pm

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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