Cohomology of 
in the space of vector fields
Notations
Summary of results for 
Exact sequences
Computation of 
Summary of result
Computation
and 
Computation of 
Summary of result
Computation
Computation of 
Computation of 
and vanishing of 
Notations
Let
denote the singular supermanifold parametrized by bosonic
and fermionic
satisfying the pure spinor constraint:
(The space
introduced in
〚
Definition of 〛
is the direct product of two copies of
, and the space parametrized by
.)
Let
denote the algebra of polynomial functions on
, and
the space of polynomial vector fields. Consider the odd nilpotent vector field
:
The commutation
is a nilpotent operator on
.
We will now compute the cohomology of this operator.
Any vector field
can be written as
The condition
is needed because
is constrained to satisfy Eq. (
18).
Consider the subsheaf
consisting of vectors of the form
(in other words,
). Its space of sections is:
We observe that
is invariant under the action of
.
Therefore, we can think of both
and
as complexes with the differential
.
Summary of results for
In the rest of this Section we will explain the computation.
Exact sequences
Consider the short exact sequence of complexes:
The corresponding long exact sequence in cohomology of
is:
Computation of
Summary of result
We use the following segment of the long exact sequence:
The cohomology groups participating in this segment have the following description:
This implies:
We will now explain the computation.
Computation
The space
is generated as an
-module, by the following vector fields:
However
is
not a free
module, because there is a relation:
It is zero because both 
and 
can be extended to elements of 
commuting with
:
and
For any tensor
, consider vector fields of the form:
Such vector fields generate
.
But some of them are
-exact:
Therefore the vector fields of the form Eq. (
29) with
of the form:
are zero in
. This implies that
is generated by
the vector fields of the form:
A vector field of Eq. (
29) is zero in cohomology iff:
Vector fields of the form (
32) correspond to:
Notice that the section of
defined by Eq. (
32) can be extended
to a
-closed section of
:
This means that
is zero.
Eq. (
30) has the following refinement:
Computation of
Summary of result
Computation
We use the following segment of the long exact sequence:
is generated by:
is generated by:
Computation of
The linear map
is a bijection. More precisely:
Therefore 
cancels with .
Computation of and vanishing of
The space of cocycles
is generated by:
where
and
are from Eq. (
26).
Since both
and
extend to
-closed sections of
by Eqs. (
27)
and (
28), the coboundary operator
is zero.
But some cocycles are exact. Indeed, as sections of
:
