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?)



Detecting etale maps on reduced points
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?


