Consider the projective space $\mathbf P^n$ over some field $k$, with assumptions on the field as necessary. There is a complex of $k[x_0,\ldots,x_n]$-modules $\Lambda^0\xrightarrow{\mathrm d}\cdots\xrightarrow{\mathrm d}\Lambda^n$, where $\Lambda^i = \bigwedge^i\Omega_{k[x_0,\ldots,x_n]/k}$ for $\Omega_{k[x_0,\ldots,x_n]/k}$ the dual space of the $k$-vector space $\langle\frac{\partial}{\partial x_0}, \ldots, \frac{\partial}{\partial x_n}\rangle$. The operation $\mathrm d: \Lambda^i\to \Lambda^{i+1}, f\mathrm dx_{j_1}\wedge\ldots\wedge\mathrm dx_{j_i} \mapsto \sum_l \frac{\partial f}{\partial x_l} \mathrm dx_{l}\wedge\mathrm dx_{j_1}\wedge\ldots\wedge\mathrm dx_{j_i}$ is a derivation. There is another derivation $\Delta:\Lambda^i\to \Lambda^{i-1}$ with $\Delta(\mathrm d\alpha)+\mathrm d(\Delta\alpha)=\alpha \deg\alpha$ for homogeneous $\alpha$ wrt. the grading $\deg f\mathrm dx_{j_1}\wedge\ldots\wedge\mathrm dx_{j_i} = \deg f + i$.
Questions:
What does $\Delta$ do? In degree $0$, it seems to me that the contraction $\omega\mapsto\sum_l x_l \frac{\partial}{\partial x_l}\lrcorner\omega$ has the desired property. What is it in higher degrees?
The text I am trying to read defines $M^i=\ker\Delta|_{\Lambda^i}$ and calls $\tilde M^i$ the sheaf of differential $i$-forms. Why doesn't one take all of $\Lambda^i$?
Especially for question 2., answers appealing to intuition rather than rigorousness are highly welcome!