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I'm not sure I should admit this in public, but although I am aware of the precise formulations using first order logic and beyond (mentioned in the above answers), I tend not to use them. Rather I view the Lefschetz principle as more of a philosphical principle of what ought to be possible in general, and do the necessary verifications as and when I need them (but perhaps only implicitly). I suspect this attitude is pretty common among many algebraic geometers.

To give an example, for many years the only known proofs* of the Kodaira vanishing theorem were analytic. But since coherent cohomology behaves well under field extensions, Kodaira vanishing is valid over arbitrary (not necessarily algebraically closed) fields of characteristic $0$. On the other hand, for certain kinds of arguments, one needs a big enough field to carry out the argument. This typically happens when one is forced to remove a countable union of exceptional sets. Curiously, the Noether-Lefschetz theorem is one such case. Here the Lefschetz principle in the most naive sense won't work.

*(Added Footnote.) There is now an algebraic proof due to Deligne and Illusie, which involves reduction to positive characteristic. This is yet another kind of transfer.

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I'm not sure I should admit this in public, but although I am aware of the precise formulations using first order logic and beyond (mentioned in the above answers), I tend not to use them. Rather I view the Lefschetz principle as more of philosphical principle of what ought to be possible in general, and do the necessary verifications as and when I need them (but perhaps only implicitly). I suspect this attitude is pretty common among many algebraic geometers.

To give an example, for many years the only known proofs of the Kodaira vanishing theorem were analytic. But since coherent cohomology behaves well under field extensions, Kodaira vanishing is valid over arbitrary (not necessarily ) algebraically closed) fields of characteristic $0$. On the other hand, for certain kinds of arguments, one needs a big enough field to carry out the argument. This typically happens when one is forced to remove a countable union of exceptional sets. Curiously, the Noether-Lefschetz theorem is one such case. Here the Lefschetz principle in the most naive sense won't work.

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I'm not sure I should admit this in public, but although I am aware of the precise formulations using first order logic and beyond (mentioned in the above answers), I tend not to use them. Rather I view the Lefschetz principle as more of philosphical principle of what ought to be possible in general, and do the necessary verifications as and when I need them (but perhaps only implicitly). I suspect this attitude is pretty common among many algebraic geometers.

To give an example, for many years the only known proofs of the Kodaira vanishing theorem were analytic. But since coherent cohomology behaves well under field extensions, Kodaira vanishing is valid over arbitrary (not necessarily) algebraically closed fields of characteristic $0$. On the other hand, for certain kinds of arguments, one needs a big enough field to carry out the argument. This typically happens when one is forced to remove a countable union of exceptional sets. Curiously, the Noether-Lefschetz theorem is one such case. Here the Lefschetz principle in the most naive sense won't work.