Koszul Duality for Symmetric Algebras and Derived Loop Spaces

Abstract

In this paper, we prove Koszul duality for symmetric algebras of connective complexes consisting of locally free sheaves of finite rank, which is not necessarily bounded below. As an application, we construct an equivalence between derived categories of derived loop spaces and the symmetric algebra of the tangent complex for arbitrary separated schemes of finite type over a field $k$ of characteristic zero. This can be regarded as a generalization of the classical duality between derived loop spaces and cotangent bundles by Kapranov in the smooth cases. We also give a geometrical interpretation of our equivalence in terms of derived formal stacks.