As far as I understand, deformation theory of schemes may be calculated via the cotangent complex. I have read that in general the cotangent complex may be difficult to compute.
However, I have a much more naive question about it. Leafing through the basic references about the cotangent complex (by Quillen and Illusie), I only find rings. An affine schemes is the prime spectrum of a ring, so I imagine that the cotangent complex of an affine scheme would use this ring. But then affine schemes don't admit any deformations, so one doesn't need the cotangent complex to calculate deformations of affine schemes.
Is there any general method of calculating the cotangent complex of a (quasi-)projective scheme? Does one calculate the cotangent complex by starting from the affine charts and then trying to glue the cotangent complexes?