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I'm somewhat familiar with base change in scheme theory: sometimes a property of a morphism X \to Y survives a base change f:Z\to Y, meaning that X \times_{Y} Z \to Z also has this property.

Question: Is the base change in number theory and derived algebraic geometry the same thing as above? What would be the examples?

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What's base change in derived algebraic geometry? I would assume that the "derived" notion would be the obvious generalization of the ordinary notion. –  Kevin H. Lin Oct 12 '09 at 18:35
    
Please improve the title. –  Scott Morrison Oct 12 '09 at 18:48
    
Is it now better? –  Ilya Nikokoshev Oct 13 '09 at 3:53
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In number theory, base change can also refer to an operation on automorphic representations. If L/K is an extension of number fields, and pi is an automorphic representation of a reductive group G over K, then pi should "lift" to a new automorphic representation of G over L. This is the sense of the phrase used in, e.g., Langlands' book "Base Change for GL(2)". The existence of a certain kind of base change for GL(2) was used to prove the modularity of some mod 3 Galois representations, which in turn played a role in proving Fermat's last theorem.

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In number theory, base change refers to tensor product: the operation in the category of rings corresponding to fibred product in the category of (affine) schemes.

So, if A is a k-algebra, and K is a field extension of k (or less typically, another k-algebra), then the "base change of A to K" refers to A \otimes_k K.

(I would imagine in derived algebraic geometry it refers to a fibred product as usual, though I'm not sure.)

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To be more explicit, the base change you talk about here translates geometrically to base change along the map Spec K -> Spec k. –  Kevin H. Lin Oct 12 '09 at 18:57
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In number theory, base change of a scheme or a variety is with respect to the underlying ring or field, is viewing the same scheme/variety over an extended ring or field, but with the "same" set of equations.

For example given a curve over $\mathbb Q$, it is also a curve over any number field. Or given a scheme over spectrum of $\mathbb Z$ given by some equation, you can reduce it modulo a prime $p$ and obtain a scheme over $\mathbb F_p$.

When you are dealing with group schemes, moduli, motives, etc., such notions carry over through base change, modulo technical details.

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What do you mean by "such notions" in your last sentence? –  S. Carnahan Jul 24 '10 at 20:43
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Anweshi means that the base-changed objects are again groups schemes, moduli schemes, motives, etc. –  Emerton Jul 25 '10 at 2:34
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