Relative cohomology

Let us consider the BRST cohomology:

(40)

where  .    The cohomology of this complex will be called:

(41)

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On the other hand, consider the relative cohomology group:

(42)

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We claim that (42) coincides with (41):

(43)

In fact (41) is the usual (slightly generalized) BRST cohomology of the Berkovits formalism.
And (42) is a new description.