Recall that a morphism of schemes $X\rightarrow Y$ is regular if it's flat with geometric fibres that are regular schemes.


Fix a field $k$ and consider a morphism $f:X\rightarrow Y$ of noetherian $k$-schemes, do we have an equivalence of:


1/ the cotangent complex $L\_{X/Y}$ is concentrated in degree zero and is a flat module.


2/ $f$ is a regular morphism.


(2 implies 1 follows from Popescu's theorem).



characterisation of regular morphisms
Yes, these are equivalent. See [Srikanth Iyengar's write-up on Andr\'e-Quillen homology](http://homepages.math.uic.edu/~bshipley/iyengar.pdf). Specifically, Theorem 9.5 (together with Proposition 5.9) proves exactly what you want, and a little more. The theorem is actually true for morphisms between Noetherian schemes, no base field required.


As you probably expect, this paper makes heavy use of simplicial module categories. It does, however, contain a lovely introduction to them. 


