Smooth flat maps over commutative DG-rings

Abstract

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 $B{\otimes }_A^{\mathrm{L}}B$ 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.

15 pages, final version, to appear in Math. Z

References in corpus (1)

• Koszul complexes over Cohen-Macaulay rings