Koszul property
Some quadratic algebras have special property: Koszulity.
If we introduce nonhomogeneity in the Koszul quadratic algebra, it is enough to verify the Jacobi identity for three elementary letters. If it is satisfied, then the nonhomogeneous algebra will be automatically consistent, in the sense of the PBW property:
where is the corresponding homogeneous quadratic algebra.
In other words, the leading term is well-defined.
In our case, the Koszul property is a special (nontrivial) property of the pure spinor constraint:
It turns out that besides the consistency of , the Koszul property also implies some relation between the cohomology of the pure spinor BRST operator and the cohomology of . This relation allows to understand the relation between integrated and unintegrated vertex operators without using the -ghost.