Cotangent complex & formal smoothness

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.

3 thoughts on “Cotangent complex & formal smoothness

  1. Pingback: Formally smooth « Stacks Project Blog

  2. Is it possible that a non-algebraic stack can have a cotangent complex, e.g., the stack of all flat, proper, locally finitely presented morphisms? If we want to remove the stack issue, what if we look at the functor of equivalence classes of such families for which every geometric fiber has trivial automorphism group scheme?

    • A few days back I wondered if there is perhaps a notion of the cotangent complex which uses “formally smooth” as its basic model, instead of the more usual “polynomial algebra” thing. I’m afraid this would lead one to have to consider topological rings, formal schemes, etc.

      Anyway, this may be related to your question, because in the example you mention we have (formal) families over formal schemes which aren’t algebraizable that should be taken into account when computing the cotangent complex (if it exists).

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