3
$\begingroup$

Is there a definition (and theory) of $\pi_n(X),n>1$ of algebraic varieties/schemes analogous to that of etale $\pi_1(X)$? Particularly,

Is there a non-trivial long exact sequence $...\longrightarrow \pi_2(B)\longrightarrow \pi_1^{et}(F) \longrightarrow \pi_1^{et} (E) \longrightarrow \pi_1^{et}(B) \longrightarrow \pi_0(F)\longrightarrow .. $ where $F\longrightarrow E \longrightarrow B$ is a sequence of morphisms of algebraic varieties/schemes? Is there a simple example of such a sequence ?

Motivation: I have a general context (outside of algebraic geometry, in model theory) where $\pi_1$'s and covering spaces may possibly make some sense; and I want to see whether they could help define $\pi_2$. I am most interested to see a simple example of such a sequence.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
1
$\begingroup$

You might be interested in Michael Artin, Barry Mazur: Etale Homotopy. Lecture Notes in Mathematics 100, Springer-Verlag 1969.

See also http://epub.uni-regensburg.de/14043/1/MP101.pdf and 42/PMIHES_1973_42_5_0/PMIHES_1973_42_5_0.pdf">http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1973_42/PMIHES_1973_42_5_0/PMIHES_1973_42_5_0.pdf

Improve this answer
$\endgroup$
2
  • $\begingroup$ Thanks! However, is there a simple example of such a sequence which is not trivial (and as simple as possible)? (i'm updating the question with this..). $\endgroup$
    – mmm
    Commented Dec 15, 2011 at 20:05
  • $\begingroup$ What about the third equation on the first page of Friedlander's numdam article? $\endgroup$
    – user19475
    Commented Dec 15, 2011 at 20:10

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .