Geometric Theory of Parshin's Residues. Toric Neighborhoods of Parshin's Points

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.