I would like to ask some questions about algebraic geometry:

  • Can someone provide an example of a morphism of schemes with finite fibers such that under a base change it doesn't have finite fibers? I guess it is related with the infinite extension of fields but I cannot find a precise example.

  • When is $\operatorname{Spec}(\mathbb{F}_p[x,y]/(xy^2-m))$ an irreducible scheme, for $m\in \mathbb{Z}$?


If $K \subset L$ is an extension of fields, then Spec $L \to$ Spec $K$ certainly has finite fibres: there is only one fibre, and it is a single point. Now consider the base-change over $L$, i.e. Spec $L\otimes_K L \to $ Spec $L$. This again has only one fibre: is it finite?

If $L$ is finite over $K$, the answer is yes, since in that case $L\otimes_K L$ is an Artinian $L$-algebra.

But in general the answer is no; e.g. if $L = K(x)$, then $L\otimes_K L$ has infinite Spec.

This is why in EGA quasi-finite is defined as having finite fibres and being of finite type; with this definition it is stable under base-change.

If $p \not\mid m$ then $\mathbb F_p[x,y]/(xy^2 - m) = \mathbb F_p[y,y^{-1}]$ (with $x$ being identified with $my^{-2}$), and hence is an integral domain. Thus its Spec is irreducible; indeed, it is $\mathbb A^1 \setminus \{0\}$.

If $p \mid m,$ then $\mathbb F_p[x,y](xy^2 - m) = \mathbb F_p[x,y]/(xy^2)$, and so its Spec is the union of two lines, one of them doubled, and so is neither reduced nor irreducible.

  • 1
    $\begingroup$ Dear Matt, great answer! The scheme $Spec(\mathbb C(x)\otimes_\mathbb C \mathbb C(y))$ is the one-dimensional scheme obtained by removing from the plane all closed points and all generic points of vertical or horizontal lines, right? $\endgroup$ – Georges Elencwajg May 23 '12 at 12:20
  • 1
    $\begingroup$ Dear Georges, Thanks for the kind words, and yes, that's a good way to describe Spec $(\mathbb C(x)\otimes_{\mathbb C} \mathbb C(y))$. Best wishes, $\endgroup$ – Matt E May 23 '12 at 14:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.