# Differential using sheaves

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?

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