If $\omega$ is compactly supported form then so is $d\omega$?

Question

If $\omega$ is a compactly supported differential form then so is $d\omega$. Is it true?

Answer 1

Yes. In local coordinates, if (using Einstein notation) $\omega=f_I\cdot dx^I$ (where $I$ are monotone sequences of appropriate length) we have: $$d\omega= \frac{\partial f_I}{\partial x_i} dx^i\wedge dx^I$$ In particular, support of $\omega$ is the union of supports of $f_I$, and that contains the union of supports of their partial derivatives, which contains the support of $d\omega$.