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I am trying to understand smooth real manifolds using sheaves and finding some trouble with differential of smooth maps.

Some notation: Let $M$ be manifold, I denote by $\mathcal{O}_M$ it's sheaf of smooth $\mathbb{R}-$functions and by $\mathcal{T}M = \underline{Der}_{\mathbb{R}}(\mathcal{O}_M)$ it's tangent sheaf, where $(\mathcal{T}M)(U) = Der_{\mathbb{R}}(\mathcal{O}_M|_U)$ the set of derivations of the restriction of structure sheaf to open subsets.

Let $F: M \to N$ be a smooth map, we have a morphism of sheaves of rings $F^\#: \mathcal{O}_N \to F_*\mathcal{O}_M$ given by precomposition: $F^\#_U(g) = g \circ F$. Given $X \in \mathcal{T}M$, we can look to $$ F_*X \circ F^\#: \quad \quad \mathcal{O}_N \to F_* \mathcal{O}_M \to F_*\mathcal{O}_M $$ Consider the map associated to it under the $F^{-1} \dashv F_*$ adjunction, $F'X: F^{-1} \mathcal{O}_N \to \mathcal{O}_M$, we have $F'X \in \underline{Der}_{\mathbb{R}}(F^{-1}\mathcal{O}_N, \mathcal{O}_M)$ with $(F'X)_U([V,g]) = X_U(g \circ F|_{U} )$, where $V \subseteq N$ is an open s.t. $f(U) \subseteq V$.

So, we have a morphism of $\mathbb{R}_M-$modules $\,\,F': \mathcal{T}M \to \underline{Der}_{\mathbb{R}}(F^{-1}\mathcal{O}_N, \mathcal{O}_M)$. I would like to call it the derivative of $F$. On the other hand, I was expecting something like $F': \mathcal{T}M \to F^{-1} (\mathcal{T}N)$.

My question: is there an isomorphism of $F^{-1}\mathcal{O}_N-$modules $F^{-1} (\mathcal{T}N) \simeq \underline{Der}_{\mathbb{R}}(F^{-1}\mathcal{O}_N, \mathcal{O}_M)$? Am I missing something?

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  • $\begingroup$ Just out of curiosity, which book are you using to study this subject? $\endgroup$
    – Math
    May 3, 2020 at 17:15
  • $\begingroup$ @VictorHugo Ramanan - Global Calculus $\endgroup$
    – espacodual
    May 3, 2020 at 18:28

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