In this post I claimed that a formally smooth ring map has a cotangent complex which is quasi-isomorphic to a projective module sitting in degree 0. I thought this was in Illusie’s thesis. But when Wansu Kim asked me for a reference, and when I tried to find it today, I couldn’t find it.
Now I think it is simply wrong! I constructed what I think is a counter example and put it in the chapter on examples (search for cotangent complex). Let me know if I made a mistake… again.