Derived Picard groups of homologically smooth Koszul DG algebras

TL;DR

This paper establishes an isomorphism between the derived Picard groups of certain homologically smooth Koszul DG algebras and their Ext-algebras, and computes these groups for various important examples.

Contribution

It provides a new isomorphism relating derived Picard groups of Koszul DG algebras to their Ext-algebras and computes these groups for several key classes of DG algebras.

Findings

Derived Picard group is isomorphic to the opposite of the Ext-algebra's Picard group.

Computed Picard groups for trivial, free, and some non-trivial DG algebras.

Established a method to analyze derived Picard groups in this context.

Abstract

In this paper, we show that the derived Picard group of a homologically smooth Koszul connected cochain DG algebra is isomorphic to the opposite group of the derived Picard group of its finite dimensional local Ext-algebra. As applications, we compute the derived Picard groups of some important DG algebras such as trivial DG polynomial algebras, trivial DG free algebras and several non-trivial DG down-up algebras and DG free algebras.