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Is there an example of a non-lci morphism $X\rightarrow Y$ for which the entire cotangent complex (or just Andre-Quillen cohomology) can be explicitly computed? I believe it is a theorem of Avramov that in the non-lci case the cohomology is nonzero in every degree $\geq 1.$ There are explicit formulas (Lichtenbaum-Schlessinger) for the cohomologies in degree $0,1,2$; I don't know any example computations beyond degree $2$.

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