TITLE: What is the "Lefschetz Principle" (examples) ?
QUESTION [6 upvotes]: Hi there,
 can anyone explain to me what the "Lefschetz Principle" is by some clear "classical"
examples (not relying explicitly on model theory, say).
 Thanks !

REPLY [6 votes]: The simplest example of an application of the Lefschetz principle is to prove the residue theorem (in characteristic zero): The sum of the residues of a differential on a smooth projective curve is zero. If you are given a curve and a differential over a field of characteristic zero, there is a finitely generated field over $\mathbb{Q}$ where the curve, the differential, the poles and the residues there, are all defined, so take this field and embed it in $\mathbb{C}$. The residue theorem over $\mathbb{C}$ is, of course, an immediate consequence of Cauchy's theorem. You can even deduce the positive characteristic case from this with a bit more effort.