Suppose I have a morphism of schemes for which I know the relative cotangent complex is trivial, and the map on reduced subschemes is an isomorphism. Is the map an isomorphism? More generally, given a morphism of schemes with zero relative cotangent complex, which is of finite presentation on the reduced points. Is the map of finite presentation, and thus etale?

(Maybe a better way to phrase this is - what's the reference for these statements? are they in SGA or in Illusie somewhere?)


Illusie, Complexe cotangent et deformations I, Prop. 3.1.1 (p. 203) is essentially the second thing you asked. Just a technical point: I don't think people would use the term "etale" unless the morphism is locally finitely presented or something like that (you seem to be wanting to assume that only at the level of reduced schemes or something?). Without thinking too hard, though, Prop. 3.1.2 (same page) says that L_{X/Y} of perfect amplitude in [0,0] implies f is formally smooth...surely also your condition implies it's formally etale, which is what you're asking in general (without finiteness assumption), no?

  • $\begingroup$ Hmm.. probably being dense, what I really need is the first statement, which I can deduce from the second but I don't immediately see how it follows from just formal smoothness. What's a reference to say a formally smooth map which is an isomorphism on reduced points an isomorphism? $\endgroup$ – David Ben-Zvi Dec 16 '09 at 18:05

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