We study smooth maps that arise in derived algebraic geometry. Given a map \(A \rightarrow B\) between non-positive commutative noetherian DG-rings which is of flat dimension 0, we show that it is smooth in the sense of Toën–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.

我们研究在衍生代数几何中出现的平滑图。给定平面为0的非正向交换Noetherian DG环之间的映射\（A \ rightarrow B \），我们证明，当且仅当它在同构中是光滑的时，它在Toën-Vezzosi的意义上是光滑的。 Kontsevich的感觉。然后，我们证明B是\（Btimes ^ {{{\ mathrm {L}}} _ AB B}上的理想DG模块，在本地具有作为Koszul复数的显式半自由分辨率。作为一个应用，我们证明了在这种情况下，（派生的）Hochschild同源性和同调性之间存在范登伯格二重性的强形式。