Free-field resolutions of unitary representations of the $N=2$ Virasoro algebra: I

We construct a free-field resolution of unitary representations of the $N=2$ superconformal algebra. The resolution is degenerated by the action of one fermionic screening operator (which generates the “gravitational descendants” in the field theory picture). The irreducible representation is singled out from the free-field space as the cohomology of this screening operator.