# Ideal generated by differential forms

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

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... the generated ideal in which ring? – Rasmus Jul 4 '12 at 14:59
Well, this comes from an exercise which does not provides the ring. Maybe the "implied" ring is the space of differential forms with $\wedge$ as the multiplication operator ? Does this make sense ? – vkubicki Jul 4 '12 at 18:42
Since an ideal is a subgroup, $\theta$ has to be a sum of 1-forms $\omega^1$, $\omega^2$,..., $\omega^r$, without wedge products. $\omega^1\wedge\omega^{r+2}$ is not part of the ideal ??? – vkubicki Jul 4 '12 at 20:32
In the exercise they say that if $\theta$ is an element of the ideal, then $\theta$ is a linear combination of exterior products where those forms appear as factors. How come 2 operators be used to generate the ideal ? – vkubicki Jul 4 '12 at 20:46

## 1 Answer

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.

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