BRST cohomology of Type IIB in 
We want to construct some analogue of (
15), (
16) where the algebra
would be replaced with the superconformal algebra
.
The simplest naive guess: given any representation
of
, consider the same complex as in
(
15):
where now
is the space of
-th degree polynomials of
two pure spinors
and
and the operator
is defined just as a sum of two expressions like (
16):
This is almost correct, except for such an operator would not be nilpotent:
The way out is to impose some condition of
-invariance, either this:
or maybe this:
Imposing such an invariance condition makes
nilpotent. For example, in case of (
18):
This is zero because of the Jacobi identity and pure spinor constraints.
Eq. (19) is the correct definition of the BRST complex, and the one which actually comes from
the pure spinor formalism.
is the space of 
are ``co-invariants''. When 
is finite-dimensional, they are the same.
But we often need infinite-dimensional 
. In this case, we typically need (19).
