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.