Smooth flat maps over commutative DG-rings

TL;DR

This paper characterizes smooth maps between non-positive commutative DG-rings in derived algebraic geometry, linking Toën-Vezzosi smoothness with Kontsevich's homological smoothness, and explores duality in Hochschild (co)homology.

Contribution

It establishes an equivalence between two notions of smoothness for DG-ring maps and provides explicit resolutions, advancing understanding in derived algebraic geometry.

Findings

Smoothness in the sense of Toën-Vezzosi is equivalent to homological smoothness of the DG-ring map.

B as a perfect DG-module admits a semi-free Koszul resolution locally.

A strong Van den Bergh duality between Hochschild homology and cohomology holds in this setting.

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 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.