# characterisation of regular morphisms

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).

• It this known when $Y = \operatorname{Spec} k$ with $k$ algebraically closed? It might in principle be possible to construct some tower $X_i$ over $Y = \operatorname{Spec} k$ such that the image of $H_j(L_{X_i/Y})$ dies in $H_j(L_{X_{i+1}/Y})$ for example, yet $X = \lim X_i$ is not regular. – R. van Dobben de Bruyn Jun 15 '17 at 17:14