Ideal generated by differential forms

Question

I have troubles picturing what elements belong to a particular ensemble.

Let $\omega_1$,...,$\omega_r$ be differential 1-forms on a $C^\infty$ n-manifold that are independent at each point. Considering a complete base $\omega_1$,...,$\omega_r$,...,$\omega_{r+1}$,...,$\omega_{n}$, could someone give me some examples and counter-examples of elements in the ideal $\mathscr{I}$ generated by $\omega_1$,...,$\omega_r$ ?

For example, what to do with $\omega^1\wedge\omega^{r+2}$ ?

Thank you,

JD

Answer 1

The ideal generated comprises a linear combination of differential forms containing at least one element of $\omega^1,...,\omega^r$ as a factor in exterior products with $\omega^{1},...,\omega^n$. The generation is therefore not made using the additive operation and the base $\omega^1,...,\omega^r$. This is consistent with the rest of the solution of the exercise. Source: the author of the textbook.