# Is it true that $\Omega^k(E\oplus E)\simeq \bigoplus_{j=0}^k \Omega^{k-j}(E)\otimes_{C^\infty(M)} \Omega^j(F)$?

If $E\longrightarrow M$ is a vector bundle then we can associate a new vector bundle $$\Lambda^k E\longrightarrow M$$ whose fiber over $p\in M$ is $\Lambda^k E_p$, that is, the $p$-th exterior power of $E_p$.

Let us denote by $\Omega^k(E)$ the $C^\infty(M)$-module of sections of $\Lambda^k E$.

Take $F\longrightarrow M$ another vector bundle and let $E\oplus F\longrightarrow M$ the whitney sum of $E$ and $F$. Is it true that there is an isomorphism of $C^\infty(M)$-modules $$\Omega^k(E\oplus F)\simeq \bigoplus_{j=0}^k \Omega^{k-j}(E)\otimes_{C^\infty(M)} \Omega^j(F)?$$

Thanks.

$$\Omega^k(E \oplus F) = \Gamma(\Lambda^k(E \oplus F)) \cong \Lambda^k(\Gamma(E \oplus F)) \cong \Lambda^k(\Gamma(E) \oplus \Gamma(F)) \cong \\ \bigoplus_{j=0}^k \Lambda^j(\Gamma(E)) \otimes_{C^{\infty}(M)} \Lambda^{k - j}(\Gamma(F)) \cong \bigoplus_{j=0}^k \Gamma(\Lambda^j(E)) \otimes_{C^{\infty}(M)} \Gamma(\Lambda^{k-j}(E)) = \\ \bigoplus_{j=0}^k \Omega^j(E) \otimes_{C^{\infty}(M)} \Omega^{k-j}(F).$$