DG homological algebra, properties of ring homomorphisms, and the generalized Auslander-Reiten conjecture

This dissertation contains three aspects of my research that are listed as three joint papers with my advisor and a solo paper in the bibliography. Each paper will be discussed in a different chapter.;Chapter 1 contains the introduction to this dissertation. In this chapter we give the statements of the most important results discussed in Chapters 3–6.;Chapter 2 contains notation and background material for use in the subsequent chapters.;In Chapter 3, we prove lifting results for DG modules that are akin to Auslander, Ding, and Solberg's famous lifting results for modules.;Chapter 4 contains the complete answer to a question of Vasconcelos from 1974. We show that a local ring has only finitely many shift-isomorphism classes of semidualizing complexes. Our proof relies on certain aspects of deformation theory for DG modules over a finite dimensional DG algebra, which we develop.;In Chapter 5, we investigate Cohen factorizations of local ring homomorphisms from three perspectives. First, we prove a "weak functoriality'' result for Cohen factorizations: certain morphisms of local ring homomorphisms induce morphisms of Cohen factorizations. Second, we use Cohen factorizations to study the properties of local ring homomorphisms in certain commutative diagrams. Third, we use Cohen factorizations to investigate the structure of quasi-deformations of local rings, with an eye on the question of the behavior of CI-dimension in short exact sequences.;In Chapter 6 we show under some conditions that a Gorenstein ring R satisfies the Generalized Auslander-Reiten Conjecture if and only if so does R[x]. When R is a local ring we prove the same result for some localizations of R[x].