Cox sheaves on graded schemes, algebraic actions and F1-schemes

Abstract

This Thesis extends the theory of Cox sheaves from the classical setting of algebraic (pre)varieties to that of (graded) schemes, the latter being spaces which locally look like sets of (homogeneously) prime ideals of (graded) algebras. The base of these algebras can be the ring of integers, the field of complex numbers or the multiplicative monoid {0,1}, which is also called the 'field' F1. Applying the equivalence of graded schemes of finite type and quasi-torus actions we obtain a theory of Cox sheaves for quasi-torus actions, in particular for (quasi-)toric (pre)varieties.

We study and characterize the properties of section algebras of Cox sheaves as well as the morphisms constructed from them, which are called characteristic spaces. Here, the basic results do not require Noetherianity or finiteness conditions. Of special interest is also the connection between characteristic spaces of schemes over F1 and toric characteristic spaces of toric prevarieties. Finally, we consider morphisms from characteristic spaces of actions to toric characteristic spaces and generalize embedding results from Wlodarcyk and Hausen.