Twisted complexes on a ringed space as a dg-enhancement of perfect complexes

TL;DR

This paper introduces a new geometric dg-enhancement of the derived category of perfect complexes on a ringed space using twisted complexes, providing a more intuitive and locally defined approach.

Contribution

It constructs a dg-functor from twisted complexes to perfect complexes, establishing a novel geometric dg-enhancement of the derived category.

Findings

The dg-functor is a dg-enhancement of the derived category.

Twisted complexes provide a geometric perspective on perfect complexes.

Applications and further topics related to twisted complexes are discussed.

Abstract

In this paper we study twisted complexes on a ringed space and prove that it gives a new dg-enhancement of the derived category of perfect complexes on that space. A twisted complex is a collection of locally defined sheaves together with the homotopic gluing data. In this paper we construct a dg-functor from twisted complexes to perfect complexes, which turns out to be a dg-enhancement. This new enhancement has the advantage of being completely geometric and it comes directly from the definition of perfect complex . In addition we will talk about some applications and further topics around twisted complexes.