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Any rational map between affine spaces, projective spaces or toric varietiescan be described in terms of their affine, homogeneous, or Cox coordinates. Weshow an analogous statement in the setting of Mori Dream Spaces. More precisely(in the case of regular maps) we show there exists a finite extension of theCox ring of the source, such that the regular map lifts to a morphism from theCox ring of the target to the finite extension. Moreover the extension onlyinvolves roots of homogeneous elements. Such a description of the map can beapplied in practical computations.

Maps of Mori Dream Spaces in Cox coordinates. Part I: existence of descriptions