Abstract
We study smooth maps that arise in derived algebraic geometry. Given a map A -> B between non-positive commutative noetherian DG-rings which is of flat dimension 0, we show that it is smooth in the sense of Toen-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 circle times(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.