Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $X,Y$ be noetherian connected schemes over an algebraically closed field $k$ and let $\overline{x},\overline{y}$ be geometric points on them. There is a canonical homomorphism

$\pi_1(X \times Y,(\overline{x},\overline{y})) \to \pi_1(X,\overline{x}) \times \pi_1(Y,\overline{y})$

This is known to be an isomorphism when $X$ or $Y$ is proper and geometrically integer. This follows from a more general exact sequence in $\pi_1$ related to a proper flat morphism with geometrically integer fibers.

Question 1. Is there a more direct proof of this isomorphism? The corresponding isomorphism in topology is trivial, is it really that difficult in algebraic geometry?

Question 2. Is the assumption on $X$ or $Y$ really needed? If yes, can you give an easy counterexample for the general case?

share|improve this question
    
The corresponding isomorphism in topology is only trivial if you define $\pi_1$ in terms of maps from the circle. If you define $\pi_1$ directly in terms of covering spaces, is it still trivial? –  Qiaochu Yuan Oct 11 '12 at 7:42
    
@Qiaochu: Which definition in terms of covering spaces do you mean? –  Martin Brandenburg Oct 12 '12 at 10:09

1 Answer 1

up vote 4 down vote accepted

The proof of the isomorphism is not so hard in SGA X.1.7.

A counterexample when none of $X, Y$ is proper is given in SGA X.1.10 with $X=Y$ equal to the affine line over a field of positive characteristic.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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