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Let $A$ be a vector bundle over $M$. For a non-negative integer $k$ let us write $$\Omega^k(A):=\Gamma(\Lambda^k A^*)$$, that is, the elements of $\Omega^k(A)$ are the sections of the vector bundle $\Lambda^kA^*\longrightarrow M$.

Suppose $B$ is a further vector bundle over $N$ and let $\Phi:A\longrightarrow B$ be a vector bundle morphism covering $\Phi_0:M\longrightarrow N$. Is there a relationship between $\Omega^k(B)$ and $\Omega^k(\Phi_0^*(B))$?

Of course we have and injection $\Omega^k(B)\longrightarrow \Omega^k(\Phi_0^*(B))$ induced by the map $\Phi_0^*(B)\longrightarrow B$. Is that all we can say?

Thanks.

Obs: I found that $$\Omega^k(\Phi_0^*(B))\simeq C^\infty(M)\otimes_{C^\infty(N)} \Omega^k(B),$$ where the $C^\infty(N)$-module structure on $C^\infty(M)$ is obtained using the morphism of algebras $\Phi_0^*:C^\infty(N)\longrightarrow C^\infty(M)$.

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