Fibration sequences in étale homotopy theory arising from geometric fibres - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T01:29:07Z http://mathoverflow.net/feeds/question/19977 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19977/fibration-sequences-in-etale-homotopy-theory-arising-from-geometric-fibres Fibration sequences in étale homotopy theory arising from geometric fibres K.J. Moi 2010-03-31T17:08:53Z 2010-03-31T17:08:53Z <p>Let $R = \mathbb{Z} [ \frac{1}{p}]$ for some prime number $p$ and <code>$GL_{n,R}$</code> be the general linear group scheme over $R$. The bar construction gives a simplicial scheme <code>$BGL_{n,R}$</code> over the constant simplicial scheme $Spec(R)$. If $q$ is a prime different from $p$ we can pull <code>$BGL_{n,R}$</code> back along a map $Spec( \bar{\mathbb{F}_q}) \to Spec(R)$ to get <code>$BGL_{n,\bar{\mathbb{F}_q}}$</code>. Here <code>$\bar{\mathbb{F}_q}$</code> is an algebraic closure of $\mathbb{F}_q$. The simplicial scheme <code>$BGL_{n,\bar{\mathbb{F}_q}}$</code> has the nice property that if we apply Friedlander's étale topological type functor, defined <a href="http://books.google.com/books/p/princeton?id=_DfLrwUEvS0C&amp;printsec=frontcover&amp;cd=1&amp;source=gbs_ViewAPI&amp;hl=en#v=onepage&amp;q=&amp;f=false" rel="nofollow">here</a>, and then p-complete, we get something that is equivalent to the $p$-completion tower <code>$\{ (\mathbb{Z}/p)_s BGL_n( \mathbb{C}) \}_s$</code>. (Here <code>$BGL_{n}( \mathbb{C})$</code> means the singular simplicial set of the classifying space of the Lie group).</p> <p>Several articles state that the sequence<code>$$(BGL_{n,\bar{\mathbb{F}_q}})_{ét} \to (BGL_{n,R})_{ét} \to Spec(R)_{ét}$$</code> becomes a fibration sequence after $p$-completing the $BGL$ terms, but I haven't been able to find any proof or argument supporting this anywhere. Does anyone know of a proof or argument for this? </p> <p>In the article <a href="http://www.nd.edu/~wgd/Dvi/Exotic.Cohomology.GLnZhalf.pdf" rel="nofollow">Exotic cohomology for $GL_n(\mathbb{Z} [ \frac{1}{2}])$</a> the reader is referred to <a href="http://books.google.com/books/p/princeton?id=_DfLrwUEvS0C&amp;printsec=frontcover&amp;cd=1&amp;source=gbs_ViewAPI&amp;hl=en#v=onepage&amp;q=&amp;f=false" rel="nofollow">Étale homotopy of simplicial schemes</a> but I have only been able to find a proof of the $p$-adic equivalence I mentioned above, not of the fibration sequence. In <a href="http://www.jstor.org/stable/2000179" rel="nofollow">Algebraic and étale k-theory</a> it is used several times. </p> <p>I hope this question isn't too narrow for Mathoverflow.</p> <p>The reason that I ask is that I would like to have similar fibration sequences for other group schemes and I hope they will be fibration sequences for the same reason that the one above is.</p>