Iteration of Cox rings of klt singularities

In this article, we study the iteration of Cox rings of klt singularities (and Fano varieties) from a topological perspective. Given a klt singularity $(X,\Delta ;x)$, we define the iteration of Cox rings of $(X,\Delta ;x)$. The first result of this article is that the iteration of Cox rings ${\mathrm{Cox}}^{\left(k\right)}(X,\Delta ;x)$ of a klt singularity stabilizes for $k$ large enough. The second result is a boundedness one, we prove that for an $n$-dimensional klt singularity $(X,\Delta ;x)$, the iteration of Cox rings stabilizes for $k⩾c\left(n\right)$, where $c\left(n\right)$ only depends on $n$. Then, we use Cox rings to establish the existence of a simply connected factorial canonical (or scfc) cover of a klt singularity, with general fiber being an extension of a finite group by an algebraic torus. The scfc cover generalizes both the universal cover and the iteration of Cox rings. We prove that the scfc cover dominates any sequence of quasi-étale finite covers and reductive abelian quasi-torsors of the singularity. We characterize when the iteration of Cox rings is smooth and when the scfc cover is smooth. We also characterize when the spectrum of the iteration coincides with the scfc cover. Finally, we give a complete description of the regional fundamental group, the iteration of Cox rings, and the scfc cover of klt singularities of complexity one. Analogous versions of all our theorems are also proved for Fano-type morphisms. To extend the results to this setting, we show that the Jordan property holds for the regional fundamental group of Fano-type morphisms.