# “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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## 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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