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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?

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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
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.

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