“Representability” II

As we have seen in the last problem, we can identify a compact smooth manifold M with \mathrm{hom}_{\Bbb{R}\mathrm{-alg}}(C^\infty(M),\Bbb{R}). Give a similar identification between the tangent bundle TM and  \mathrm{hom}_{\Bbb{R}\mathrm{-alg}}(C^\infty(M),\Bbb{R}[x]/(x^2)).

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“Representability” II

One thought on ““Representability” II

  1. nachodarago says:

    We know that TM is identified with the set of \mathbb R-linear derivations \mathrm{Der}_{\mathbb{R}}(C^{\infty}M). Now, to identify derivations with \mathbb{R}-algebra morphisms C^\infty M\to \mathbb{R}[t]/(t^2) just consider, for any derivation \delta:C^\infty M\to \mathbb{R} the map f\mapsto f + \delta(f)t. Conversely, given a map C^\infty M\to \mathbb{R}[t]/(t^2) consider the term that is with t and notice that it is a derivation. This two maps are the inverse of each other.

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