# What is the “Lefschetz Principle” (examples) ?

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 !

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How about you tell us in which context you saw that, because just like that it's not even clear what kind of people should answer that ! Also, have you at least looked on google/wikipedia ??? You really should read the howtoask for future questions ! – Amin May 16 '12 at 13:32
I corrected the question, and hope it is better now. As I misread the term, thinking it was "Lipschitz", google did not give much :-) – THC May 16 '12 at 13:48

Considering your choice of tags, it's plausible that you actually mean the Lefschetz principle. In its vaguest form, this principle says that algebraic geometry behaves "the same way" over $\mathbf C$, over an arbitrary algebraically closed field of characteristic zero, and over an arbitrary algebraically closed field of sufficiently large characteristic. This is typically used (e.g. in conjunction with GAGA) to deduce algebraic statements from things proved analytically over the complex numbers. See What does the Lefschetz principle (in algebraic geometry) mean exactly? for more information.

On the other hand, you might also be referring to H. Davenport, "On a principle of Lipschitz", 1951.

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You are right - I misread the term in a paper - this is corrected now. Thanks ! – THC May 16 '12 at 13:47

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.

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what is "bit more effort" ? – Alexander Chervov May 16 '12 at 18:54
The principle of permanence of algebraic identities. – Felipe Voloch May 16 '12 at 19:38
@FelipeVoloch Please, can you explain what the principle of permanence of algebraic identities is, or give a good source for it (I find it mentioned, but not explained!)? – Jose Brox Jul 12 at 11:52
@JoseBrox Serre "Groupes algebriques et corps de classes" very end of chapter II gives the argument. – Felipe Voloch Jul 12 at 22:04
I see that it is also called "the principle of prolongation of algebraic identities". For possible future reference, it is simply explained in ldc.usb.ve/~berry/preprints/irr.pdf – Jose Brox Jul 13 at 0:40

Harish-Chandra uses this buzzword also as a guiding principle in the sense: Every thing what is true for reductive Lie groups, should be expected to be true for $p$-adic reductive algebraic groups as well. I actually thought of asking a similar question a while back. I figured out later the same thing what Dan Petersen is mentioning, the terminology seems to originate from algebraic geometry.

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Hi, do you know in which place did Harish-Chandra write this guiding principle literally? – user148212 Feb 2 '15 at 1:25