论文标题
在交换DG环上平滑平整图
Smooth flat maps over commutative DG-rings
论文作者
论文摘要
我们研究在派生的代数几何形状中出现的光滑图。给定一个地图$ a \ to b $在非阳性的noetherian dg-rings之间，该dg ring $ 0 $ $ 0 $，我们在toën-vezzosi的意义上且仅当它在kontsevich的意义上是材料上的平滑时，就表明它是平稳的。然后，我们表明$ b $是$ b \ otimes^{\ mathrm {l}} _ a b $在本地具有明确的半无分辨率作为Koszul Complect的完美DG-MODULE。作为一个应用，我们表明（衍生）Hochschild同源性和同一个同源物之间的强烈形式在这种情况下。
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 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.
