Suppose the map $d : \Omega^*(M) \rightarrow \Omega^*(M)$ (where $\Omega^*(M)$ is the space of differential forms on a smooth manifold $M$) is an antiderivation and $\omega \in \Omega^*(M)$ such that $\omega|_{U} = 0$ for some open subset $U \subseteq M $.
Antiderivation on a graded algebra $A = \bigoplus_{k=0}^{\infty} A^k$ is a $\mathbb{R}$ linear map $D : A \rightarrow A$ such that $D(\omega. \tau) = (D \omega). \tau + (-1)^k \omega . D \tau $ , where $\omega \in A^k $ and $\tau \in A^l$.
Show that $d(\omega)|_{U} = 0$.
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