Derived Equivalences of Upper Triangular Differential Graded Algebras

TL;DR

This paper extends the concept of derived equivalences to upper triangular matrix differential graded algebras, showing under certain conditions how these structures relate through derived categories and bimodules.

Contribution

It generalizes known results from matrix rings to differential graded algebras, establishing conditions for derived equivalences involving bimodules and endomorphism DGAs.

Findings

Established a derived equivalence between upper triangular matrix DGAs under specific conditions.

Constructed the DG-bimodule M' from M and a DG-module X.

Provided a framework for understanding derived categories of triangular DGAs.

Abstract

This paper generalises a result for upper triangular matrix rings to the situation of upper triangular matrix DGA's. An upper triangular matrix DGA has the form (R,S,M) where R and S are differential graded algebras and M is a DG-left-R-right-S-bimodule. We show that under certain conditions on the DG-module M and with the existance of a DG-R-module X, from which we can build the derived category D(R), that there exists a derived equivalence between the upper triangular matrix DGAs (R,S,M) and (S,M',R'), where the DG-bimodule M' is obtained from M and X and R' is the endomorphism DGA of a K-projective resolution of X.