For a vector bundle $\pi_A:A\longrightarrow M$ and a non-negative integer $k$ let us define $$\Omega^k(A):=\left\{\begin{array}{clc} C^\infty(M) &\textrm{if}& k=0\\ \displaystyle\mathsf{Hom}_{C^\infty(M)}(\Lambda^k \Gamma(A), C^\infty(M)) & \textrm{if} & k>0 \end{array}\right.,$$ where $\Gamma(A)$ is the $C^\infty(M)$-module of sections of $A$. There is a graded product $$\wedge:\Omega^i(A)\times \Omega^j(A)\longrightarrow \Omega^{i+j}(A)$$ defined by $$(\varepsilon\wedge \tau)(\alpha_1\wedge\ldots \wedge \alpha_{i+j}):=\sum_{\sigma\in\mathsf{Sh}(i, j)} \mathsf{sgn}(\sigma) \varepsilon(\alpha_{\sigma(1)}\wedge \ldots \wedge \alpha_{\sigma(i)})\tau(\alpha_{\sigma(i+1)}\wedge \ldots \wedge \alpha_{\sigma(i+j)}),$$ where $\mathsf{Sh}(i, j)$ is the set of all permutations $\sigma$ of the first $i+j$ integers such that $\sigma(1)<\ldots<\sigma(i)$ and $\sigma(i+1)<\ldots<\sigma(i+j)$. Then $(\Omega^\bullet(A), \wedge)$ is a graded algebra.

Now let us consider another vector bundle $\pi_B:B\longrightarrow N$ and a degree $-k$ map $\Phi:\Omega^i(B)\longrightarrow \Omega^{i-k}(A)$ such that that $$\Phi(\varepsilon_1\wedge \varepsilon_2)=\Phi(\varepsilon_1)\wedge \Phi(\varepsilon_2).$$ Is it true that such map is non-zero only for $k=0$ and $k=1$?

In other words, I'm asking wether there are only degree $0$ and degree $-1$ morphisms from $(\Omega^\bullet(A), \wedge)$ to $(\Omega^\bullet(B), \wedge)$?


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