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:


There is a subtle difference between these two definitions. Here is the space of -invariants, while are ``co-invariants''. When is finite-dimensional, they are the same. But we often need infinite-dimensional such as . In this case, we typically need (19).
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.