Antonio Lorenzin

Some developments on existence and uniqueness of DG-enhancements

In this talk, we discuss some developments about the existence and the (strong) uniqueness of DG-enhancements for triangulated categories.
First, we deal with triangulated categories having a full strong exceptional sequence and explain when they admit a DG-enhancement.
Secondly, we consider the triangulated category of vector spaces over a field K, where the shift is the identity. It is well-known that this category does not have a unique DG-enhancement. However, the uniqueness holds for K-linear DG-enhancements. The proof uses the notion of intrinsical formality, a property of graded algebras.
Finally, we explore a new direction on strongly unique enhancements. In particular, this approach works for linearity over any ring; consequently, many examples can be generalized.