  The paper consist of two parts. In the first part we introduce flags of
lattices and associated injective systems of (non-normal) cones and projective
systems of (non-normal) affine toric varieties. We study the associated field
of multidimensional Laurent power series. In the second part we use the
resolution of singularities techniques to study the geometry near a complete
flag of subvarieties and the Parshin's residues. The first part plays the role
of a standard coordinate neighborhood for Parshin's points.
