Derived intersections and free dg-Lie algebroids

Abstract

We study free dg-Lie algebroids over arbitrary derived schemes, and compute their universal enveloping and jet algebras. We also introduce derived twisted connections, and relate them with lifts on twisted square zero extensions. This construction allows us to provide new conceptual approaches of existing results concerning the derived deformation theory of subschemes. Contents 1. Introduction 2. Recollections 2.1. DG-modules 2.2. Graded derivations and DG-Lie algebroids 2.3. Universal enveloping algebra of a dg-Lie algebroid 2.4. Coproduct and jets 2.5. Generalities on bimodules 3. Atiyah bimodules 3.1. Construction via anchored modules 3.2. Duality for Atiyah bimodules 3.3. Derived connections and the HKR class 4. Universal enveloping algebra of free Lie algebroids 4.1. Structure theorem 4.