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I'd like to ask some basic things in algebraic geometry.

Suppose I have a map $\phi:V\to W$, between affine varieties over $k=\mathbb{C}$. for any point $y \in W$. The scheme theoretical fiber is defined to be $Spec(k[V]\otimes_{k[W]} k )$. The question is when this is reduced?
I'm seeking an answer like: for generic point in the image of $\phi$, this is true. I also like to ask a proper reference for this (hope not in EGA, but sections in some textbook. such as Hartshorne).

Thank you very much!

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1 Answer 1

up vote 8 down vote accepted

If you are over a field of char. $0$, then yes, it is true that the fibre over a generic point of the image is reduced. To see this, first replace $W$ with the closure of the image of $\phi$, so as to assume that $\phi$ is dominant. Then the morphism $\phi$ corresponds to an injection of finite-type domains over $k$, say $A \hookrightarrow B$. (The injectivity follows from the dominance of $\phi$.) Now let's look at the generic fibre of $\phi$: this is Spec of the tensor product $K(A) \otimes_A B$ (here $K(A)$ denotes the fraction field of $B$), which is a localization of $B$, and so a domain (and hence reduced). Now if we are in char. zero, the reduced $K(A)$-algebra $K(A)\otimes_A B$ is necessarily geometrically reduced (i.e. stays reduced after extending to an algebraic closure of $K(A)$), and the property of fibres being geometrically reduced is open on the base, i.e. on Spec $A$; thus the fibres over an open subset of the image of $\phi$ will be reduced. (See e.g. Thm. 2.2 of these notes. You can find this in many places (though mabye not Hartshorne); these notes are just what turned up near the top of a quick google search.)

In char. $p$, this need not be true. E.g. consider the Frobenius map $\mathbb A^1 \to \mathbb A^1$, given by $x \mapsto x^p$. Then the fibre over every (closed) point is non-reduced. (The fibre over the generic point is reduced, but not geometrically reduced.) Nevertheless, if the generic fibre (in the scheme-theoretic sense) is geometrically reduced, then the fibre over an open set of closed points will be reduced too.

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Thank you very much! –  Hoxide Jul 11 '12 at 5:17

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