  We extend the Cox-Hu-Keel construction of the Cox rings to any proper
birational morphisms of normal noetherian schemes. It allows the representation
of any proper birational morphism by a map of schemes with mild singularities
with torus actions. In a particular case, the notion generalizes the
combinatorial construction of Satriano and the recent construction of multiple
weighted blow-ups on Artin-stacks by Abramovich-Quek. The latter can be viewed
as an extension of stack theoretic blow-ups by Abramovich, Temkin, and
Wlodarczyk, a similar construction of McQuillan and the author's recent
cobordant blow-ups at weighted centers to a more general situation of arbitrary
locally monomial centers. We show some applications of this operation to the
resolution of singularities over a field of any characteristic.
