TITLE: Take contraction wrt a vector field twice and define kernel mod image. Does that give anything interesting?
QUESTION [16 upvotes]: First, we make the following observation: let $X: M \rightarrow TM $ be a vector 
field on a smooth manifold. Taking the contraction with respect to $X$ twice gives zero, i.e. 
$$ i_X \circ i_{X} =0.$$ 
Is there any "name" for the corresponding "homology" group that one can define 
(Kernel mod image)? Has this "homology" group been studied by others (there are plenty of questions that one can ask........is it isomorphic to anything more familiar etc etc). 
Similarly, a dual observation is as follows: Let $\alpha$ be a one form; taking 
the wedge product with $\alpha$ twice gives us zero. One can again define kernel 
mod image. Does that give anything "interesting"? 
If people have investigated these questions, I would like to know a few references. 
My purpose for asking the "name" of the (co)homology group is so that I can make a google search using the name. I was unable to do that, since I do not know of any key words under this topic (or if at all it is a topic).

REPLY [3 votes]: These constructions certainly are very recognisable and meaningful in the algebraic context, that is if you think of algebraic differential forms on an affine algebraic variety. Namely, the corresponding complexes both implement versions of the Koszul complex for a sequence of elements $f_1,\ldots,f_n$ in $R$, where $R$ is the ring of functions on $M$, and $n=\dim M$. Indeed, the algebra of differential forms $\Omega^\bullet(M)$ is the exterior algebra $\Lambda_R^\bullet(R\,dx_1\oplus\ldots\oplus R\,dx_n)$, and for the given vector field $X=f_1\partial_1+\cdots+f_n\partial_n$, the differential $i_X$ on $\Omega^\bullet(M)$ is the $R$-algebra (super)derivation taking $dx_i$ to $f_i$, which essentially is a commonly used definition of the Koszul complex.
 For the $1$-form $\omega=f_1\,dx_1+\cdots+f_n\,dx_n$, the differential $\omega\wedge$ on $\Omega^\bullet(M)$, which as we remember, is $\Lambda_R^\bullet(R\,dx_1\oplus\ldots\oplus R\,dx_n)$, is dual to the differential corresponding to $i_X$ under the isomorphism of $R$-modules $$\mathop{\mathrm{Hom}}\nolimits_R(\Lambda_R^\bullet(R\,dx_1\oplus\ldots\oplus R\,dx_n),R)\simeq \Lambda_R^\bullet(R\,dx_1\oplus\ldots\oplus R\,dx_n).$$
In any case, the Koszul complex is acyclic if and only if $f_1,\ldots,f_n$ define a complete intersection (this I think also holds in the analytic case, I have no intuition about the smooth one). There are many references, e.g Principles of Algebraic Geometry by Griffiths & Harris, or Commutative Algebra by Bourbaki.