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Let $A$ be a vector bundle over a manifold $M$. We can assotiate a graded algebra $(\Omega(A), \wedge)$ where $$\wedge:\Omega^i(A)\times \Omega^j(A)\longrightarrow \Omega^{i+j}(A),$$ is given by $$(\varepsilon\wedge \tau)(\alpha_1, \ldots, \alpha_{i+j})=\sum_{\sigma\in\mathsf{Sh}(i, j)}\mathsf{sgn}(\sigma)\varepsilon(\alpha_{\sigma(1)}, \ldots, \alpha_{\sigma(i)})\tau(\alpha_{\sigma(i+1)}, \ldots, \alpha_{\sigma(i+j)}).$$

Why aren't there any derivations of degree inferior to $-1$ of this graded algebra $(\Omega(A), \wedge)$?

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If $d$ is of degree $ < -1$, then $d$ is zero restricted to $1$-form and by the formula you just wrote every form of higher degree will also be send to zero.

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