Cohomology rings, differential graded algebras and derived equivalences

TL;DR

This paper explores derived equivalences of differential graded algebras and their cohomology rings, extending Keller’s methods and addressing a specific open problem in the field.

Contribution

It constructs new derived equivalences for dg endomorphism algebras from triangles and standard equivalences, and relates these to cohomology ring equivalences.

Findings

Derived equivalences constructed via Keller's approach.

Cohomology rings of dg endomorphism algebras are also derived equivalent under certain conditions.

Confirmed a special case of Dugas's open problem.

Abstract

In this paper, we will consider derived equivalences for differential graded endomorphism algebras by Keller's approaches. First we construct derived equivalences of differential graded algebras which are endomorphism algebras of the objects from a triangle in the homotopy category of differential graded algebras. We also obtain derived equivalences of differential graded endomorphism algebras from a standard derived equivalence of finite dimensional algebras. Moreover, under some conditions, the cohomology rings of these differential graded endomorphism algebras are also derived equivalent. Then we give an affirmative answer to a problem of Dugas \cite{Dugas2015} in some special case.