Algebraic models of modules over equivariant Eilenberg-MacLane spectra

The main subject is the study of modules over Eilenberg-MacLane rings in categories of equivariant spectra. The introduction presents a brief history of the subject and the main objects of study including spectra and Mackey functors. The second chapter introduces HF2 in C2-spectra. We describe the isotropy separation sequence and the Tate square. We then introduce model categories of diagrams and the cellularisation principle. We then provide the main proof for the Quillen equivalence between HF2-modules and algebraic quasicoherent diagrams using reduction to fixed points, Shipley’s theorem, and intrinsic formality. Finally, as a short aside, we employ this method to classify rational C2-spectra. The third chapter introduces main concepts of algebraic geometry, including prime spectra, schemes, and sheaves on curves such as ℙ1. We explain how sheaf cohomology is relates to the homotopy groups of HF2. We then use the classification of coherent sheaves on ℙ1 to classify finitely generated HF2-modules. The second section computes the coefficients of H(A⊗k) and provides an algebraic model related to quasicoherent sheaves on the cuspidal cubic. Finally, we compute a few torsion modules and discuss a classification result of simple torsion-free coherent sheaves on the singular curve. The brief fourth chapter begins with computing the coefficients of HF3 in C3-spectra, and continues with the proof of the algebraic model for the category of modules. We explain that this model is no longer formal and compute a few examples. The fifth and final chapter describes two methods of computing the coefficients of HF2 in C4-spectra, yielding two different algebraic models. We explain how the second model can be pursued further to obtain an algebraic model based on a Tate cube, and how this method can be generalised to any case C2n using n-dimensional Tate cubes.