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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$\begingroup$ See this question : mathoverflow.net/questions/143806/computing-cotangent-complex. $\endgroup$– LibliCommented Aug 20, 2016 at 16:55
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2$\begingroup$ That is not exactly true. See Corollary 3 in [sciencedirect.com/science/article/pii/S0022404997001035] for a characterization of vanishing in all degrees $\geq 3$. From it, its not difficult to see an example of a non-complete intersection $A \to A/(x)$ such that cohomology vanishes in degrees $\geq 3$. $\endgroup$– A.GCommented Aug 21, 2016 at 21:20
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