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What is the standard reference for a definition , valid in all characteristics, of the residue in a point of a rational differential form on a curve?

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This is done for instance in Serre's Algebraic Groups and Class Fields, ch. II, §11. As Serre himself observes, the proof is somewhat artificial; a more elegant one is given in Tate's Residues of differentials on curves, Ann. Sci. E.N.S. 1 (1968), 149-159.

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