arXiv:0911.0590 An explicit approach to residues on and dualizing sheaves of arithmetic surfaces from arXiv Front: math.AG by Matthew Morrow We develop a theory of residues for arithmetic surfaces, establish the reciprocity law around a point, and use the residue maps to explicitly construct the dualizing sheaf of the surface. These are generalisations of known results for surfaces over a perfect field. In an appendix, explicit local ramification theory is used to recover the fact that in the case of a local complete intersection the dualizing and canonical sheaves coincide.
arXiv:0911.2951 Zariski decompositions on arithmetic surfaces from arXiv Front: math.AG by Atsushi Moriwaki In this paper, we establish the Zariski decompositions of arithmetic R-divisors of continuous type on arithmetic surfaces and investigate several properties. We also develop the general theory of arithmetic R-divisors on arithmetic varieties.
arXiv:1101.1883 Grothendieck’s trace map for arithmetic surfaces via residues and higher adeles from arXiv Front: math.AG by Matthew Morrow We establish the reciprocity law along a vertical curve for residues of differential forms on arithmetic surfaces, and describe Grothendieck’s trace map of the surface as a sum of residues.
http://mathoverflow.net/questions/70942/minimal-model-of-a-surface-over-spec-mathbbz
nLab page on Arithmetic surfaces