Algebraic models and rational global spectra

In the first part of the thesis we define and study free global spectra: global spectra with non-trivial geometric fixed points only at the trivial group. We show that free global spectra often do not exist, and when they do, their homotopy groups satisfy strong divisibility conditions. The second part of the thesis is dedicated to the study of the algebraic model of rational global spectra for the family of finite groups as constructed in Schwede. We study the homological algebra of this category with a particular focus on the tensor triangulated geometry of its derived category. Along the way we make contact with the theory of representation stability and show that some algebraic invariants coming from global homotopy theory exhibit such a stability phenomenon. Finally, in the third part of the thesis we construct a symmetric monoidal algebraic model for the category of rational cofree G-spectra for all compact Lie groups G. The key ingredient in the proof is the Left Localization principle which gives mild hypotheses under which a Quillen adjunction between stable model categories descends to a Quillen equivalence between their left localizations. This last part is joint work with Jordan Williamson.