We develop a theory of residues for arithmetic surfaces, establish thereciprocity law around a point, and use the residue maps to explicitlyconstruct the dualizing sheaf of the surface. These are generalisations ofknown results for surfaces over a perfect field. In an appendix, explicit localramification theory is used to recover the fact that in the case of a localcomplete intersection the dualizing and canonical sheaves coincide.
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