  We study smooth maps that arise in derived algebraic geometry. Given a map $A
\to B$ between non-positive commutative noetherian DG-rings which is of flat
dimension $0$, we show that it is smooth in the sense of To\"{e}n-Vezzosi if
and only if it is homologically smooth in the sense of Kontsevich. We then show
that $B$, being a perfect DG-module over $B\otimes^{\mathrm{L}}_A B$ has,
locally, an explicit semi-free resolution as a Koszul complex. As an
application we show that a strong form of Van den Bergh duality between
(derived) Hochschild homology and cohomology holds in this setting.
