Graded Deviations, Rigidity, and the Koszul Property

Differential graded algebras have played an important role in the study of infinite free resolutions over local commutative Noetherian rings. In particular, they have been a mechanism to import mathematical tools, such as the homotopy Lie algebra, from rational homotopy theory in order to study invariants of rings and the growth of resolutions. More recently, differential graded algebras and the homotopy Lie algebra have also been employed to study infinite graded free resolutions over graded rings. In this setting the algebras are bigraded by homological degree and the internal degree coming from the ring of interest. Chapters 2, 3, and 4 of this work establish important theory of these algebras in great generality. In chapter 5, we establish tools for computing the structure of these resolutions in particular internal degrees, yielding new results about the existence of certain long exact sequences. We then specialize to the study of ℕ-graded rings over fields. In chapter 6 we prove results about the relationship between the structure of the homotopy Lie algebra and the Koszul and complete intersection properties, answering a conjecture of Ferraro in the negative. In chapter 7 we prove a new type of rigidity result unique to the graded case, which makes partial progress towards answering another conjecture of Ferraro in the affirmative. Motivated by a question of Conca, we apply this result to study the asymptotic properties of the slope of an algebra