Rationality Questions and the Derived Category

This document is roughly divided into four chapters. The first outlines basic preliminary material, definitions, and foundational theorems required throughout the text. The second chapter, which is joint work with Dr. Matthew Ballard, gives an example of a family of Fano arithmetic toric varieties in which the derived category is able to detect the existence of k-rational points. More succinctly, we show that if X is a generalized del Pezzo variety defined over a field k, then X contains a k-rational point (and is in fact k-rational, that is, birational to Pnk ) if and only if Db(X) admits a full étale exceptional collection. In the third chapter, which is joint work with Dr. Matthew Ballard, Dr. Alexander Duncan, and Dr. Patrick McFaddin, we describe, using techniques from Galois cohomology, a new invariant of reductive algebraic groups that captures precisely when this strategy will fail. Our main result characterizes this invariant in terms of coflasque resolutions of linear algebraic groups introduced by Colliot-Thélène. We determine whether or not this invariant is trivial for many fields. For number fields, we show it agrees with the Tate-Shafarevich group of the linear algebraic group, up to behavior at real places. In addition, we completely describe the cohomological invariants of a reductive algebraic group of degree 2 with values in a special torus, which generalizes a result of Blinstein and Merkurjev. In the final chapter, which is joint work with Dr. Matthew Ballard, Dr. Alexander Duncan, and Dr. Patrick McFaddin, we develop tools to understand the effect that twisting by a torsor has on the derived category. Applying these to the setting of arithmetic toric varieties reveals a surprising dichotomy in behavior split along the fault line of retract rationality. As a consequence of the general theory, we give a negative answer to a question of Bernardara and Bolognesi relating a categorical notion of dimension to rationality. Moreover, we show that a smooth projective toric variety over a field k possessing a full k-exceptional collection is automatically k-rational