Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

What is the étale fundamental group of projective space over an algebraically closed field?

In char = 0 it is trivial (Lefschetz principle), as well as in dimension 1 (Riemann-Hurwitz).

share|improve this question
    
That's actually an interesting point even about P^1, given that A^1 is extremely not simply-connected. –  Ben Webster Jul 4 '10 at 10:56

1 Answer 1

up vote 6 down vote accepted

It is a birational invariant (for smooth proper connected schemes over a field, ultimately due to Zariski-Nagata purity of the branch locus), and its formation is compatible with products (for proper connected schemes over an algebraically closed field), so we can replace projective $n$-space with the $n$-fold product of copies of the projective line to conclude. Likewise, due to limit arguments and invariance of the etale site with respect to finite radiciel surjections (such as a finite purely inseparable extension of a ground field), it suffices to take the ground field to be separably closed rather than algebraically closed. This is all in SGA1.

share|improve this answer
    
Can we also argue inductively using the diagram Unknown control sequence $\mathbf{P}^n \leftarrow \mathrm{Bl}_x{\mathbf{P}^n} \to \mathbf{P}^{n-1}$? –  user6960 Jul 4 '10 at 12:07

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.