Properties of triangular matrix and Gorenstein differential graded algebras

Abstract

The main goal of this thesis is to investigate properties of two types of Differential Graded Algebras (or DGAs), namely upper triangular matrix DGAs and Gorenstein DGAs. In doing so we extend a number of corresponding ring theory results to the more general setting of DGAs and DG modules. Chapters 2 and 3 contain background material. In chapter 2 we give a brief summary of some important aspects of homological algebra. Starting with the definition of an abelian category we give the construction of the derived category and the definition of derived functors. In chapter 3 we present the basics about Differential Graded Algebras and Differential Graded Modules in particular extending the definitions of the derived category and derived functors to the Differential Graded case before providing some results on Recollement of DGAs, Dualising DG-modules and Gorenstein DGAs. Chapters 4 and 5 contain the bulk of the work for the Thesis. In chapter 4 we look at upper triangular matrix DGAs and in particular we generalise a result for upper triangular matrix rings to the situation of upper triangular matrix differential graded algebras. An upper triangular matrix DGA has the form [R M / 0 S] where R and S are DGAs and M is a DG R-Sop-bimodule. We show that under certain conditions on the DG-module M, and given the existence 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 M / 0 S] and [ S M’ / 0 R’], where the DG-bimodule M0 is obtained from M and X, and R0 is the endomorphism differential graded algebra of a K-projective resolution of X. In chapter 5 we turn our attention to Gorenstein DGAs and generalise some results from Gorenstein rings to Gorenstein DGAs. We present a number of Gorenstein Theorems which state, for certain types of DGAs, that being Gorenstein is equivalent to the bounded and finite versions of the Auslander and Bass classes being maximal. We also provide a new definition of a Gorenstein morphism for DGAs by considering a DG bimodule as a generalised morphism of DGAs. We then show that some existing results for Gorenstein morphism extend to these "Generalised Gorenstein Morphisms". We finally conclude with some examples of generalised Gorenstein morphisms for some well known DGAs.