We classify the localizing tensor ideals of the derived categories of mixed Tate motives over certain algebraically closed fields. More precisely, we prove that these categories are stratified in the sense of Barthel, Heard and Sanders. A key ingredient in the proof is the development of a new technique for transporting stratification between categories by means of Brown--Adams representability, which may be of independent interest.

An abstract homotopy theory is a situation in which one has a category with a class of ``weak equivalences'' that one would like to invert. One recent description of a homotopy theory is as an ``∞-category,'' which is like a category with extra structure that is made to contain homotopical data. Every ∞-category can be flattened into an ordinary category called its ``homotopy category,'' in a way that inverts the weak equivalences.

Among homotopy theories, there are certain ones that are called stable homotopy theories. Accordingly, there is a notion of stable ∞-categories which formalizes them. The homotopy category of a stable ∞-category has a canonical structure of a triangulated category.

A triangulated category is a category which is equipped with some structure that serves as a ``weak'' or homotopical version of an exact sequence. The theory of stable ∞-categories has some advantages over that of triangulated categories, since ∞-categories retain some homotopical information that is lost in the passage to the homotopy category. This thesis attempts to do two things: (1) to lay out a self-contained exposition of the theory of ∞-categories, and (2) to describe the relationships between stable ∞-categories and the triangulated structure on their homotopy categories.

This thesis explores the tensor triangular classification of pseudo-coherent complexes over a commutative noetherian ring with particular emphasis on the case of a discrete valuation ring. In the latter case, we derive key connections between thick tensor-ideals of pseudo-coherent complexes and the asymptotic behavior of torsion degree sequences. This leads to a description of the Balmer spectrum of the derived category of pseudo-coherent complexes as the spectral space associated via Stone duality with a certain lattice of asymptotic equivalence classes of monotonic sequences of natural numbers. We also generalize some of these results from discrete valuation rings to the ring of integers.

For each object in a tensor triangulated category, we construct a natural continuous map from the object's support---a closed subset of the category's triangular spectrum---to the Zariski spectrum of a certain commutative ring of endomorphisms. When applied to the unit object this recovers a construction of P. Balmer. These maps provide an iterative approach for understanding the spectrum of a tensor triangulated category by starting with the comparison map for the unit object and iteratively analyzing the fibers of this map via "higher" comparison maps. We illustrate this approach for the stable homotopy category of finite spectra. In fact, the same underlying construction produces a whole collection of new comparison maps, including maps associated to (and defined on) each closed subset of the triangular spectrum. These latter maps provide an alternative strategy for analyzing the spectrum by iteratively building a filtration of closed subsets by pulling back filtrations of affine schemes.

In this thesis we define the notion of a Galois extension of commutative rings, and present the analogue of the fundamental theorem of Galois theory in this setting. Following the work of Chase, Harrison, and Rosenberg, we show how the classical definition of a Galois extension of a field arises as a special case of this generalization. Furthermore, we generalize the notion of a Galois extension of commutative rings by replacing the Galois group with a Hopf algebra, leading to the notion of a Hopf Galois extension. We present the fundamental theorem in this context and show how the definition of a Galois extension of a commutative ring arises as a special case of this generalization.