1. ## graded Ideal and differential forms

Hello,

i have a proof of a statement, but i don't understand it very well. I have concrete questions about it and it would be very nice, when someone can help me.

Let E be a subbundle of TM, we define the (graded) ideal $\displaystyle J=\bigoplus_{k=1}^n J^k \; in \; \Omega(M)$ as follows $\displaystyle w \in J^k \;<=> w(X_1,...,X_k)=0$ for any sections $\displaystyle X_i$ of E

Claim: J is locally generated by q linearly independent 1-forms:

Pf: Choose a local frame$\displaystyle X_1,...,X_n$ of TM, s.t.$\displaystyle X_1,...,X_{n-q}$ form a frame of E.
There is the dual frame of differential 1-forms $\displaystyle w_1,...,w_n$ of TM* and the linearly independent 1-forms $\displaystyle w_{n-q+1},..,w_n$ clearly generate the ideal J.

Why do they generate the Ideal J????
What does it mean "generate"? the $\displaystyle w_i$ are 1-forms, i.e. sections: $\displaystyle M->TM^*, w_i (p)\in T^{*}_p M$.
and J is the direct sum of k-forms, for k=1,...,n?

Can you please explain it for me?

Regards

2. means any element in J can be expressed in terms of these q forms.